MINIMAL SURFACES, SURFACES OF CONSTANT ... - Purdue Math

MINIMAL SURFACES, SURFACES OF CONSTANT MEAN CURVATURE AND ISOPERIMETRY IN SUB-RIEMANNIAN GROUPS DONATELLA DANIELLI, NICOLA GAROFALO, AND DUY-MINH NHIEU

Contents 1. Introduction 2. Carnot groups 3. Some important examples 4. Some useful properties of Carnot groups 5. Horizontal connection, mean curvature and minimal surfaces 6. Domains with special symmetries in groups of Heisenberg type 7. Perimeter measure and its first variation 8. Sub-Riemannian calculus on hypersurfaces 9. Horizontal Laplace-Beltrami operator 10. Horizontal Minkowski formulas on hypersurfaces 11. Monotonicity formulas on hypersurfaces 12. Isoperimetric inequalities 13. Existence of isoperimetric sets in Carnot groups 14. Isoperimetric sets with partial symmetry in Hn References

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1. Introduction The purpose of the present paper is to develop some new directions of investigation in geometric measure theory and calculus of variations. The object of our study are some metric spaces whose underlying ambient is a Riemannian manifold M n with a distance generated by an assigned subbundle HM n of the tangent bundle. Such ambients are nowadays known as Carnot-Carathéodory spaces, CC spaces henceforth, and they take their name after the foundational paper of Carathéodory [Ca] on Carnot termodynamics. The Riemannian metric in such spaces is loosely speaking confined to the background, and what is most important instead is the sub-Riemannian structure generated by HM n . The tangent space to a CC space is itself a CC space, although of a special type. These manifolds possess a structure of Lie group with Date: September 25, 2003. Key words and phrases. Minimal surfaces, mean curvature, isoperimetric inequality, minimizers, best constant. First aurthor was supported in part by NSF Grant No. DMS-01.... Second author was supported in part by NSF Grant No. DMS-0070492. 1

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DONATELLA DANIELLI, NICOLA GAROFALO, AND DUY-MINH NHIEU

a graded Lie algebra. Such groups are called Carnot groups, and play a fundamental role in analysis, geometry, and in various branches of the applied sciences. To introduce our discussion we recall that during the past century the development of the theory of minimal surfaces has been one of the main driving forces in mathematics. Such development was prompted by the study of the problems of Plateau and Bernstein. Minimal surfaces also play a central role in the positive mass theorem from relativy due to Schoen and Yau. We intend to develop a theory of minimal surfaces in CC spaces. In fact, in this paper we mainly focus on the case of hypersurfaces in Carnot groups. a Carnot group of step r is a connected, simply-connected Lie group G whose Lie algebra admits a nilpotent stratification g = V1 ⊕ V2 ⊕ ... ⊕ Vr of step r. This means that [V1 , Vj ] = Vj+1 , j = 1, ..., r − 1, [V1 , Vr ] = {0}. Every layer Vj of g is assigned the corresponding formal degree j. Accordingly, the stratification of the Lie algebra carries a natural family of non-isotropic dilations defined by ∆λ ξ = λξ1 + λ2 ξ2 + ... + λr ξr , if ξ = ξ1 + ξ2 + ... + ξr ∈ g. Such dilations are transferred to G by means of the exponential map as follows δλ = exp ◦ ∆λ ◦ exp−1 . We indicate by Lg (g 0 ) = gg 0 , Rg (g 0 ) = g 0 g the operators of left- and right-translation on G. The bi-invariant Haar measure on G obtained by pushing forward Lebesgue measure on g via the r P exponential map will be denoted by dg. One has d(g ◦ δλ ) = λQ dg, where Q = j dim Vj is j=1

the homogeneous dimension of G. If m = dim V1 , let X1 , ..., Xm be a fixed orthonormal basis of the horizontal layer V1 . Continue to denote by X1 , ..., Xm the corresponding system of left-invariant vector fields on G. Such vector fields generate the so-called horizontal subbundle HG of the tangent bundle T G. The CC distance on G is defined by minimizing on the length of those piecewise C 1 paths γ : [a, b] → G such that γ 0 (t) ∈ Hγ(t) G, for every t ∈ [a, b] for which γ 0 (t) exists. The balls in such metric will be denoted by B(g, R) = {g 0 ∈ G | d(g, g 0 ) < R}. Since the Xj ’s are left-translation invariant, so is the CC distance, i.e. d(Lg (g 0 ), Lg (g 00 )) = d(g 0 , g 00 ), g, g 0 , g 00 ∈ G. One then has

(1.1)

|B(g, R)| = ωQ RQ ,

g ∈ G , R > 0 , ωQ = |B(e, 1)| .

(Hereafter, |E| indicates the Haar measure of E ⊂ G). We note that the vector fields X1 , ..., Xm are homogeneous of degree one with respect to {δλ }λ>0 , i.e. one has for any smooth function u (1.2)

Xj (u ◦ δλ ) = λ Xj u ◦ δλ .

The horizontal gradient of u with respect to the basis {X1 , ..., Xm } is defined by Xu = Pm Pm 2 1/2 . j=1 Xj uXj , and we let |Xu| = ( j=1 (Xj u) ) Our starting point is the following isoperimetric inequality which is a special case of Theorem 1.17 in [GN1]. Theorem 1.1. There exists a constant Creliso = Creliso (G) > 0 such that for every X-Caccioppoli set E ⊂ G one has for any g ∈ G and R > 0 min {|E ∩ B(g, R)|, |E c ∩ B(g, R)|}

Q−1 Q

≤ Creliso PX (E; B(g, R)) .

MINIMAL SURFACES, SURFACES OF CONSTANT MEAN CURVATURE, ETC.

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Here, the notation PX (E; B(g, R)) indicates the X-perimeter of E relative to the open set B(g, R), and the defining property of an X-Caccioppoli set is to have finite X-perimeter relative to any ball. In De Giorgi’s formulation of the Plateau problem one seeks to minimize the relative perimeter with respect to a given bounded open set Ω ⊂ Rn , subjected to the constraint of coincidence with an assigned Caccioppoli set L outside Ω. Another basic result established in [GN1] was the following existence theorem for the solution of the sub-Riemannian Plateau problem, see Theorems 1.23 and 1.25 in [GN1]. Theorem 1.2. Let G be a Carnot group, given a bounded open set Ω ⊂ G and a X-Caccioppoli set L ⊂ G, one can find an X-minimal surface in Ω, that is, an X-Caccioppoli set E ⊂ G such that PX (E; G) ≤ PX (F ; G) for every F that coincides with L outside Ω. Theorems 1.1 and 1.2 represent two basic building blocks for the development of geometric measure theory in sub-Riemannian spaces, and, as it turns out, they open the door to a host of very challenging questions. The purpose of the present paper is to simply lift the veil on a beautiful program which appears compelling and, yet, at a very primitive stage : understanding the nature of minimal surfaces, or more in general of solutions of partial differential equations of mean curvature type. Naturally, such a program requires as a first step introducing an appropriate notion of mean curvature and developing a corresponding sub-Riemannian calculus. This is in fact one of our main contributions. finish introduction ....

2. Carnot groups In this section we describe the basic geometric ambients which will enter in this paper. A sub-Riemannian space is a triple (M, HM, d) constituted by a connected Riemannian manifold M , with Riemannian distance dR , a subbundle HM ⊂ T M , and the Carnot-Carathéodory (CC) distance d generated by HM . Such distance is defined by minimizing only on those absolutely continuous paths γ whose tangent vector γ 0 (t) belongs to Hγ(t) M , see [NSW] and also [Be]. Riemannian manifolds are a special example of sub-Riemannian spaces. They correspond to the case HM = T M . The tangent space of a sub-Riemannian space is itself a sub-Riemannian space (or a quotient of such spaces), but of a special type. It is a graded Lie group whose Lie algebra is nilpotent. These groups, which owe their name to the foundational paper of Charathéodory [Ca] on Carnot thermodynamics, occupy a central position in the study of hypoelliptic partial differential equations, non-commutative harmonic analysis, sub-Riemannian geometry, and CR geometric function theory. They are called sub-Riemannian, or Carnot groups.

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A Carnot group of step r is a connected, simply connected Lie group G whose Lie algebra g admits a stratification g = V1 ⊕ ... ⊕ Vr which is r-nilpotent, i.e., [V1 , Vj ] = Vj+1 , j = 1, ..., r − 1, [Vj , Vr ] = {0}, j = 1, ..., r. A trivial example of (an abelian) Carnot group is G = Rn , whose Lie algebra admits the trivial stratification g = V1 = Rn . The simplest non-abelian example of a sub-Riemannian group of step r = 2 is the (2n + 1)-dimensional Heisenberg group Hn , which will be described below. Given a sub-Riemannian group G, by the above assumptions on the Lie algebra one immediately sees that any basis of the horizontal layer V1 generates the whole g. We will respectively denote by (2.1)

Lg (g 0 ) = g g 0 ,

Rg (g 0 ) = g 0 g ,

the operators of left- and right-translation by an element g ∈ G. If {e1 , ..., em } is a basis of the vector space V1 , we define left-invariant vector fields on G by the formula (2.2)

Xi (g) = (Lg )∗ (ei ) ,

g∈G,

i = 1, ..., m ,

where (Lg )∗ indicates the differential of Lg . The system X = {X1 , ..., Xm } identifies a basis for the so-called horizontal subbundle HG of the tangent bundle T G. The second order partial differential operator (2.3)

L =

m X

Xi2 ,

i=1

is called the sub-Laplacian associated with the basis X. Except for the abelian case when the P 2 2 step r = 1 and L is just the standard Laplacian ∆ = m i=1 ∂ /∂xi , such operator fails to be elliptic at every point of G. The exponential mapping exp : g → G defines an analytic diffeomorphism onto G. We recall the important Baker-Campbell-Hausdorff formula, see, e.g., sec.2.15 in [V], µ ¶ ª 1 1© (2.4) exp(ξ) exp(η) = exp ξ + η + [ξ, η] + [ξ, [ξ, η]] − [η, [ξ, η]] + ... , 2 12 where the dots indicate commutators of order four and higher. Each element of the layer Vj is assigned the formal degree j. Accordingly, one defines dilations on g by the rule ∆λ ξ = λ ξ1 + ... + λr ξr , provided that ξ = ξ1 +...+ξr ∈ g. Using the exponential mapping exp : g → G, these anisotropic dilations are then tansferred to the group G as follows δλ (g) = exp ◦ ∆λ ◦ exp−1 g . Henceforth, we assume that a scalar product < ·, · > is given on g for which the Vj0 s are mutually orthogonal. Let πj : g → Vj denote the projection onto the j-th layer of g. Since the exponential map exp : g → G is a global analytic diffeomorphism, we can define analytic maps ξj : G → Vj , j = 1, ..., r, by letting ξj = πj ◦ exp−1 . As a rule, we will use letters g, g 0 , g 00 , go for points in G, whereas we will reserve the letters ξ, ξ 0 , ξ 00 , ξo , η, for elements of the Lie algebra g. We let mj = dim Vj , j = 1, ..., r, and denote by N = m1 + ... + mr the topological dimension of G. The notation {ej,1 , ..., ej,mj }, j = 1, ..., r, will indicate a fixed orthonormal basis of the


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j − th layer Vj . For g ∈ G, the projection of the exponential coordinates of g onto the layer Vj , j = 1, ..., r, are defined as follows (2.5)

xj,s (g) = < ξj (g), ej,s >,

s = 1, ..., mj .

The vector ξj (g) ∈ Vj , j = 1, ..., r, will be routinely identified with the point (xj,1 (g), ..., xj,mj (g)) ∈ Rmj . It will be easier to have a separate notation for the horizontal layer V1 . For simplicity, we set m = m1 , and let (2.6)

{e1 , ... , em } = {e1,1 , ... , e1,m1 } .

We indicate with (2.7)

xi (g) = < ξ1 (g), ei >,

i = 1, ..., m ,

the projections of the exponential coordinates of g onto V1 . Whenever convenient, we will identify g ∈ G with its exponential coordinates (2.8)

def

x(g) = (x1 (g), ..., xm (g), x2,1 (g), ..., x2,m2 (g), ..., xr,1 (g), ..., xr,mr (g)) ∈ RN ,

and we will ordinarily drop in the latter the dependence on g, i.e., we will write g = (x1 , ..., xr,mr ). Using the Baker-Campbell-Hausdorff formula (2.4) we can express (2.2) using the coordinates (2.8), obtaining the following lemma. Lemma 2.1. For each i = 1, ..., m, and g = (x1 , ..., xr,mr ), we have mj r X X ∂ ∂ + bsj,i (x1 , ..., xj−1,m(j−1) ) Xi = ∂xi ∂xj,s j=2 s=1 r mj

=

XX ∂ ∂ + bsj,i (ξ1 , ..., ξj−1 ) , ∂xi ∂xj,s j=2 s=1

where each

bsj,i

is a homogeneous polynomial of weighted degree j − 1.

By weighted degree we mean that, as previously mentioned, the layer Vj , j = 1, ..., r, in the stratification of g is assigned the formal degree j. Correspondingly, each homogeneous monomial ξ1α1 ξ2α2 ...ξrαr , with multi-indices αj = (αj,1 , ..., αj,mj ), j = 1, ..., r, is said to have weighted degree k if mj r X X j( αj,s ) = k . j=1

s=1

Throughout the paper we will indicate by dg the bi-invariant Haar measure on G obtained by lifting via the exponential map exp the Lebesgue measure on g. One easily checks that Q

(d ◦ δλ )(g) = λ dg,

where Q =

r X

j dim(Vj ).

j=1

The number Q, called the homogeneous dimension of G, plays an important role in the analysis of Carnot groups. In the non-abelian case r > 1, one clearly has Q > N .

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We denote by d(g, g 0 ) the CC distance on G associated with the system X. It is well-known that d(g, g 0 ) is equivalent to the gauge pseudo-metric ρ(g, g 0 ) on G, i.e., there exists a constant C = C(G) > 0 such that C ρ(g, g 0 ) ≤ d(g, g 0 ) ≤ C −1 ρ(g, g 0 ),

(2.9)

g, g 0 ∈ G,

see [NSW], [VSC]. The pseudo-distance ρ(g, g 0 ) is defined as follows. Let |·| denote the Euclidean distance to the origin on g. For ξ = ξ1 + · · · + ξr ∈ g, ξj ∈ Vj , one lets  1/2r! r X (2.10) |ξ|g =  |ξj |2r!/j  , |g|G = | exp−1 g|g , g ∈ G, j=1

and defines ρ(g, g 0 ) = |g −1 g 0 |G .

(2.11)

Both d and ρ are invariant under left-translations (2.12)

d(Lg (g 0 ), Lg (g 00 )) = d(g 0 , g 00 ) ,

ρ(Lg (g 0 ), Lg (g 00 )) = ρ(g 0 , g 00 ) .

and dilations (2.13)

d(δλ (g 0 ), δλ (g 00 )) = λ d(g 0 , g 00 ) ,

ρ(δλ (g 0 ), δλ (g 00 )) = λ ρ(g 0 , g 00 ) .

Denoting respectively with (2.14)

B(g, R) = {g 0 ∈ G | d(g 0 , g) < R},

Bρ (g, R) = {g 0 ∈ G | ρ(g 0 , g) < R},

the CC ball and the gauge pseudo-ball centered at g with radius R, one easily recognizes that there exist ω = ω(G) > 0, and α = α(G) > 0 such that (2.15)

|B(g, R)| = ω RQ ,

|Bρ (g, R)| = α RQ ,

g ∈ G, R > 0.

3. Some important examples We now describe three important examples of Carnot groups. The former is the step 2 Heisenberg group Hn . Such group plays an ubiquitous role in analysis and geometry. The second example constitutes a non-trivial and important generalization of the Heisenberg group, the class of groups of Heisenberg type. The latter is the step 3 cyclic, or Engel group. This is an interesting example to keep in mind since it represents the next level of difficulty with respect to the Heisenberg group. The Heisenberg group Hn . The underlying manifold of this Lie group is simply R2n+1 , with the non-commutative group law (3.1)

1 g g 0 = (x, y, t) (x0 , y 0 , t0 ) = (x + x0 , y + y 0 , t + t0 + (< x0 , y > − < x, y 0 >)) , 2


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where we have let x, x0 , y, y 0 ∈ Rn , t, t0 ∈ R. Let (Lg )∗ be the differential of the left-translation (3.1). A simple computation shows that µ ¶ ∂ ∂ yi ∂ def (3.2) (Lg )∗ + , i = 1, ..., n , = Xi = ∂xi ∂xi 2 ∂t µ ¶ xi ∂ ∂ ∂ def − (Lg )∗ = Xn+i = , i = 1, ..., n , ∂yi ∂yi 2 ∂t µ ¶ ∂ ∂ def (Lg )∗ = X2n+1 = ∂t ∂t We note that the only non-trivial commutator is [Xi , Xn+j ] = − X2n+1 δij ,

i, j = 1, ..., n ,

therefore the vector fields {X1 , ..., X2n } constitute a basis of the Lie algebra hn = R2n+1 = V1 ⊕ V2 , where V1 = R2n × {0}t , V2 = {0}(x,y) × R. We notice that the sub-Laplacian associated with the basis {X1 , ..., X2n } is ¾ n ½ 2n 2 X 1 ∂ X ∂ ∂ 2 2 2 ∂ − xj , (3.3) L = Xj = ∆x,y + (|x| + |y| ) 2 + yj 4 ∂t ∂t ∂xj ∂yj j=1

j=1

which coincides with the real part of the complex Kohn Laplacian, see [St2]. The non-isotropic group dilations are (3.4)

δλ (g) = (λx, λy, λ2 t) ,

with homogeneous dimension Q = 2n + 2. A convenient renormalization of the gauge (2.10) is given by ¡ ¢1/4 (3.5) N (g) = (|x|2 + |y|2 )2 + 16 t2 . The importance of such function is connected with the discovery due to Folland [F1] that the fundamental solution of (3.3) is given by (3.6)

Γ(g) = Γ(g, e) =

CQ , N (g)Q−2

where CQ < 0 is an explicit constant. Groups of Heisenberg type. In a group G of step 2, with Lie algebra g = V1 ⊕ V2 , consider the linear mapping J : V2 → End(V1 ) defined by (3.7)

< J(η)ξ 0 , ξ 00 > = < [ξ 0 , ξ 00 ], η >,

η ∈ V2 ,

ξ 0 , ξ 00 ∈ V1 .

The algebraic properties of the mapping J have important repercussions on the geometric and analytic properties of Carnot groups of step 2. An immediate consequence of the definition of J is that (3.8)

< J(η)ξ, ξ > = 0,

for every η ∈ V2 , ξ ∈ V1 .

Definition 3.1. A Carnot group G of step 2 is called of Heisenberg type if for every η ∈ V2 , such that |η| = 1, the map J(η) : V1 → V1 is orthogonal.

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Groups of Heisenberg type are modelled on the Heisenberg group Hn , but their geometry is more intricated since the center can have arbitrary dimension. The introduction of such groups is due to A. Kaplan [K1], [K2], [K3]. Because of their symmetries, groups of Heisenberg type play a distinguished role in analysis and geometry. We stress that there exists a plentiful supply of them. For instance, the nilpotent component N in the Iwasawa decomposition G = KAN , where G is a simple group of rank one, is a group of Heisenberg type [CDKR]. Such groups N are called Iwasawa groups. When the center V2 of the group is one-dimensional, then (up to isomorphisms) a group of Heisenberg type is nothing but the Heisenberg group Hn . In such case there is only one Kaplan mapping J, given by the symplectic matrix in R2n . Definition 3.1 implies (3.9)

|J(η)ξ| = |η| |ξ| ,

η ∈ V2 ,

(3.10)

< J(η 0 )ξ, J(η 00 )ξ > = < η 0 , η 00 > |ξ|2 ,

(3.11)

< J(η)ξ 0 , J(η)ξ 00 > = |η|2 < ξ 0 , ξ 00 > ,

ξ ∈ V1 ,

η 0 , η 00 ∈ V2 , ξ ∈ V1 ,

η ∈ V2 ,

ξ 0 , ξ 00 ∈ V1 .

We next recall a result due to Kaplan [K1], which generalizes Folland’s formula (3.6). In a group of Heisenberg type G we consider the renormalized gauge (compare with (3.5)) ¡ ¢1/4 (3.12) N (g) = |x(g)|4 + 16|y(g)|2 . Let L be a sub-Laplacian associated with an orthonormal basis X of the first layer of the Lie algebra of G, and denote by Γ(g, g 0 ) the corresponding positive fundamental solution. There exists C(G) > 0 such that (3.13)

Γ(g, g 0 ) =

C(G) ρ(g, g 0 )Q−2

g, g 0 ∈ G, g 6= g 0 ,

where ρ(g, g 0 ) = N (g −1 g 0 ). The four-dimensional Engel group. We consider the group G = K3 , see ex. 1.1.3 in [CGr], whose Lie algebra is given by the stratification, G = V1 ⊕ V2 ⊕ V3 , where V1 = span{X1 , X2 }, V2 = span{X3 }, and V3 = span{X4 }, so that m1 = 2 and m2 = m3 = 1. We assign the commutators (3.14)

[X1 , X2 ] = X3

[X1 , X3 ] = X4 ,

all other commutators being assumed trivial. We observe right-away that the homogeneous dimension of G is Q = m1 + 2 m2 + 3 m3 = 7 .


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The group law in G is given by the Baker-Campbell-Hausdorff formula (2.4). In exponential P P coordinates, if g = exp(X), g 0 = exp(X 0 ), where X = 4i=1 xi Xi , X 0 = 4i=1 yi Xi , we have g ◦ g0 = X + X 0 +

ª 1 1 © [X, X 0 ] + [X, [X, X 0 ]] − [X 0 , [X, X 0 ]] . 2 12

A computation based on (3.14) gives (see also ex. 1.2.5 in [CGr]) µ ¶ 0 g ◦ g = x1 + y1 , x2 + y2 , x3 + y3 + P3 , x4 + y4 + P4 , where

1 (x1 y2 − x2 y1 ) , 2 µ ¶ 1 1 = (x1 y3 − x3 y1 ) + x21 y2 − x1 y1 (x2 + y2 ) + x2 y12 . 2 12 P3 =

P4

Using (2.2), (2.4) we find that a left invariant basis of the Lie algebra g is given by the vector fields µ ¶ x2 ∂ x3 x1 x2 ∂ ∂ (3.15) − − + , X1 = ∂x1 2 ∂x3 2 12 ∂x4 ∂ x1 ∂ x2 ∂ X2 = + + 1 , ∂x2 2 ∂x3 12 ∂x4 ∂ x1 ∂ X3 = + , ∂x3 2 ∂x4 ∂ X4 = . ∂x4

4. Some useful properties of Carnot groups In this section we list some symmetry properties of Carnot groups which will be useful in the sequel. Our first result is Lemma 5.2 in [GV2]. Proposition 4.1. Let G be a Carnot group, then (4.1)

Xi xj = δij ,

Lxj = 0 , i, j = 1, ..., m .

As a consequence, we find |X(|x|2 )|2 = 4 |x|2 .

(4.2) One also has (4.3)

Xi ys =

1 < [ξ1 , Xi ], Ys > , 2

Lys = 0 , j = 1, ..., m , s = 1, ..., k .

The following result is Lemma 3.3 in [DGN2].

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Lemma 4.2. In a Carnot group G of step 2 for any s = 1, ..., k, one has < X(|x|2 ), Xys > ≡ 0.

(4.4)

Let l = 1, ..., k be fixed, and denote by y 0 (g) the (k − 1)−dimensional vector obtained from y(g), by removing the component yl (g). One has < X(|x|2 ), X(|y 0 |2 ) > = 0.

(4.5)

In the next lemma we recall some key properties of groups of Heisenberg type, see Lemma 3.3 in [DGN2]. Lemma 4.3. Let G be a group of Heisenberg type. For any fixed l = 1, ..., k, one has < X(yl ), X(|y 0 |2 ) > = 0.

(4.6)

|X(yl )|2 =

(4.7)

1 |x|2 . 4

|X(|y 0 |2 )|2 = |x|2 |y 0 |2 .

(4.8)

The next result will be used several times in the sequel. Lemma 4.4. Let G be a group of Heisenberg type with gauge given by (3.12), then the horizontal gradient of N is given by ¤ 1 £ 2 |ξ1 | ξ1 + 4 J(ξ2 ) ξ1 . XN = 3 N Proof. Insert..... ¤

5. Horizontal connection, mean curvature and minimal surfaces The main objective of this section is the introduction of two new concepts in sub-Riemannian geometry, namely that of tangential horizontal gradient on a hypersurface, and that of subRiemannian mean curvature. Although such notions could be developed in the greater generality of a Carnot-Carathéodory space, we will presently confine the attention to the important setting of a Carnot group G, with Lie algebra g = V1 ⊕ ... ⊕ Vr , with an orthonormal basis {e1 , ..., em } of the horizontal layer V1 , and corresponding system X = {X1 , ..., Xm } of generators, where Xi (g) = (Lg )∗ (ei ), g ∈ G. We consider a C 2 bounded open set Ω ⊂ G and we assume for convenience that there exists a globally defined φ ∈ C 2 (G) such that Ω = {g ∈ G | φ(g) < 0} ,


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and for which |∇φ| ≥ α > 0 in a neighborhood O of S = ∂Ω. Here, we have denoted by ∇φ the Riemannian gradient of φ. The Riemannian Gauss map ν : O → SN −1 is defined by ν(g) = ∇φ(g)/|∇φ(g)|, g ∈ O. We recall the definition of the characteristic set of S. Definition 5.1. A point go ∈ S = ∂Ω is called characteristic with respect to the system X, if one has < X1 (go ), ∇φ(go ) >= ... =< Xm (go ), ∇φ(go ) >= 0, or equivalently (5.1)

Xi (go ) ∈ Tgo S ,

i = 1, ..., m .

The characteristic set of S, Σ = ΣS,X , is the collection of all characteristic points of S with respect to X. We next state a basic result due to Derridj [De1], [De2]. Theorem 5.2. Let Ω be a bounded C ∞ open set in a sub-Riemannian space M of dimension N , then denoting with H s the s-dimensional Hausdorff measure constructed with the Riemannian distance one has HN −1 (Σ) = 0 . The following sharp result concerning the characteristic set has been recently proved by Magnani [Ma1], [Ma2]. Theorem 5.3. Let G be a sub-Riemannian group and denote by Hs the s-dimensional Hausdorff measure constructed with the Carnot-Carathéodory distance. For any C 1 surface of codimension k one has HQ−k (Σ) = 0 . In, particular the characteristic set of a C 1 hypersurface has zero HQ−1 -measure. The next definition plays a basic role in the sequel. Definition 5.4. We define the horizontal normal Y X : O → HG, relative to the basis X, by the formula (5.2)

YX =

m X

< ν, Xj > Xj .

j=1

The horizontal Gauss map ν X is defined by (5.3)

ν X (g) =

Y X (g) , |Y X (g)|

whenever (5.4)

|Y X (g)| =

|Xφ(g)| 6= 0 . |∇φ(g)|

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We note explicitly that Y X is the projection of the normal ν on the horizontal subbundle HG ⊂ T G generated by X. Throughout the following discussion we will denote by OX the natural domain of ν X , i.e., the open set OX = {g ∈ O | |Y X (g)| 6= 0} = {g ∈ O | |Xφ(g)| = 6 0} . An obvious consequence of the definition which, however, will be important in the sequel is |ν X |2 ≡ 1 ,

(5.5)

in OX .

One also has (5.6)

< ν X , Y X > = |Y X | ,

YX − < YX , νX > νX = 0 ,

and (5.7)

νX =

Xφ . |Xφ|

The development of sub-Riemannian calculus on a hypersurface hinges on the next definition. Definition 5.5. Consider a function u ∈ Γ1 (O). The tangential horizontal gradient on S with respect to X is defined as follows def

δX u = Xu − < Xu, ν X > ν X at every point g ∈ S \ Σ. We note right-away the following consequences of (5.5) and of Definition 5.5 (5.8)

< δX u, ν X > ≡ 0

in OX ,

and (5.9)

|δX u|2 = |Xu|2 − < Xu, ν X >2 .

Proposition 5.6. The tangential horizontal gradient depends only on the values of the function on S, i.e., if v ∈ Γ1 (O) is such that u ≡ v on S, then δX u ≡ δX v . Proof. It is enough to assume u, v ∈ C 1 (O), and then pass to the limit. By the assumption u ≡ v on S we know that ∇(u − v) = kν for some function k : S → R. This gives for every i = 1, ..., m Xi (u − v) = k < ν, Xi > = k Y X,i , or equivalently X(u − v) = k Y X . We thus find δX (u − v) = X(u − v) − < X(u − v), νX > νX = k [Y X − < Y X , ν X > ν X ] = 0 , where in the last equality we have used (5.6). ¤


13

We are now ready to introduce another basic notion, that of sub-Riemannian, or X-mean curvature. We recall that the Riemannian mean curvature H of the hypersurface S is given by the formula µ ¶ ∇φ (N − 1) H = div , |∇φ| where N denotes the topological dimension of G and we have indicated by div the Riemannian divergence. We notice that with the agreement that S = ∂{g ∈ G | φ(g) < 0} such formula respects the orientation of the (N − 1)-dimensional manifold S. In fact, when G = RN and φ(x) = |x| − r, we obtain the familiar fact H ≡ r−1 . Definition 5.7. We define the X-mean curvature of S at points of S \ Σ as m X (Q − 1) HX = δX,i νX,i . i=1

If go ∈ Σ we let HX (go ) =

lim

g→go ,g∈S\Σ

HX (g) ,

provided that such limit exists, finite or infinite. We do not define the X-mean curvature at those points go ∈ Σ at which the limit does not exist. Finally, we call HX = HX ν X the X-mean curvature vector . Definition 5.8. A C 2 hypersurface S is called X-minimal if its X-mean curvature with respect to any defining function φ of S vanishes everywhere. ˆ = G × R and the fact that δ ˆ = Xi , i = 1, ..., m, Remark 5.9. Discuss the product group G X,i ˆ ˆ in on S = {(g, s) ∈ G | s = 0}. Prove that for such S one has HXˆ ≡ 0, so that S is X-minimal ˆ G. L ˆ = L on such S................ X

The following result shows that when S is a ruled hypersurface over the first layer of the Lie algebra, then its X-mean curvature coincides with the classical Riemannian mean curvature of its projection. This has several important consequences, which will be presented in the subsequent sections. Theorem 5.10. Suppose that the hypersurface S is vertical, i.e., it can be represented in the form (5.10)

S = {g ∈ G | h(x1 (g), ..., xm (g)) = 0} ,

where h ∈ C 2 (Rm ), and there exist an open set ω ⊂ Rm and α > 0 such that |∇h| ≥ α in ω. Under these assumptions, the characteristic set of S is empty, and the X-mean curvature of S is given by m−1 (5.11) HX (g) = H(x(g)) , Q−1 where H(x(g)) represents the Riemannian mean curvature of the projection πV1 (S) of S onto the horizontal layer V1 . In particular, S is X-minimal if and only if πV1 (S) is a classical minimal surface in V1 ' Rm .

14


Proof. Insert proof...... ¤ Remark 5.11. We will prove in Theorem 7.4 that a X-minimal hypersurface is a critical point of the X-perimeter. To state the next proposition we need to introduce some relevant material. For a function u : G → R we introduce the symmetrized horizontal Hessian of u at g ∈ G as the m × m matrix with entries ½ ¾ def 1 (5.12) u,ij = Xi Xj u + Xj Xi u , i, j = 1, ..., m . 2 Setting HessX u = (u,ij ), the mapping g → HessX u(g) defines a 2-covariant tensor on the subbundle HG. We notice that the sub-Laplacian associated with the basis X is given by Lu = tr HessX u . We also consider the following nonlinear operator m def X (5.13) L∞ u = u,ij Xi u Xj u , i,j=1

which by analogy with its by now classical Euclidean ancestor we call the ∞-sub-Laplacian. Proposition 5.12. At every point of S \ Σ one has in terms of the defining function φ of S Lφ L∞ φ (Q − 1) HX = − . |Xφ| |Xφ|3 Proof. Henceforth, we adopt the summation convention over repeated indices. Definitions 5.7 and 5.5 give (5.14)

(Q − 1) HX = δX,i νX,i = Xi (νX,i ) − Xj (νX,i ) νX,j νX,i µ ¶ |νX |2 = Xi (νX,i ) − Xj νX,j 2 = Xi (νX,i ) ,

where in the last equation we have used (5.5). At this point we invoke (5.7), which gives µ ¶ Xi φ Lφ L∞ φ Xi (νX,i ) = Xi = − . |Xφ| |Xφ| |Xφ|3 This completes the proof. ¤ Remark 5.13. We stress that equation 1 H = |Xφ|3 then using Proposition 5.12 it is

if we consider the m × m matrix H = (Hij ) defined by the © ª |Xφ|2 HessX φ − HessX φ(Xφ) ⊗ Xφ , easy to recognize that (Q − 1) HX = trace H .


15

The matrix H incorporates important geometric information on the horizontal Gauss map ν X of S. The reader is referred for this aspect to the paper [GP]. It is interesting to consider a nonlinear operator which interpolates in an appropriate sense between the X-mean curvature operator in Definition 5.7, and the operator L∞ . Consider the one-parameter family of quasilinear equations (5.15)

Lp u =

m X

Xj (|Xu|p−2 Xj u) = 0 ,

1 + < b, y(g) > = γ} , where a ∈ Rm , b ∈ Rk , and γ ∈ R. We have the following interesting result. Proposition 6.3. The X-mean curvature of any hyperplane Π in a group of Heisenberg type vanishes identically. Proof. Let φ(g) =< a, x(g) > + < b, y(g) > − γ be the defining function of Π. Since we want to use Proposition 5.12, we need to compute the three relevant quantities |Xφ|2 , Lφ, and L∞ φ. As a first step we compute Xφ. Identifying a and b respectively with the elements P Pk a= m j=1 aj Xj ∈ V1 and b = s=1 bs Ys ∈ V2 , then Proposition 4.1 gives k 1 1 X bs < [ξ1 , Xj ], Ys > = aj + Xj φ = aj + < [ξ1 , Xj ], b > . 2 2 s=1


17

In a group of step 2 we can use the Kaplan mapping J to infer from the last formula 1 (6.4) Xφ = a + J(b) ξ1 . 2 The computation of Lφ is easy, since Proposition 4.1 gives (for any Carnot group) (6.5)

Lφ ≡ 0 .

We are left with computing L∞ φ. From (6.4) we find 1 1 2 (6.6) |Xφ|2 = |a|2 + |J(b)ξ1 |2 + < J(b)ξ1 , a > = |a|2 + |b| |x|2 + < J(b)ξ1 , a > 4 4 where in the second equality we have used (3.9). From (6.6) we obtain 1 2 (6.7) X(|Xφ|2 ) = |b| X(|x|2 ) + X(< J(b)ξ1 , a >) . 4 We next use the observation < J(b)ξ1 , a >= − < ξ1 , J(b)a > to obtain X(< J(b)ξ1 , a >) = − X(< ξ1 , J(b)a >) = − J(b) a . Inserting this information in (6.7) we conclude 1 2 (6.8) X(|Xφ|2 ) = |b| ξ1 − J(b) a . 2 From this formula and from (6.4) we obtain (6.9) 1 < X(|Xφ|2 ), Xφ > 2 ½ ¾ 1 1 2 1 2 1 = |b| < a, ξ1 > + |b| < J(b)ξ1 , ξ1 > − < J(b)a, a > − < J(b)a, J(b)ξ1 > 2 2 4 2 ½ ¾ 1 1 1 2 |b| < a, ξ1 > − < J(b)a, J(b)ξ1 > , = 2 2 2

L∞ φ =

where we have used (3.8). Finally, (6.9), and (3.11) allow to infer that (6.10)

L∞ φ = 0 .

Inserting (6.5) and (6.10) in Proposition 5.12 we conclude that HX ≡ 0. ¤ Remark 6.4. Proposition 6.3 is rather striking since it gives a first indication that in subRiemannian geometry a formulation of the Bernstein problem based on a verbatim analog of its Euclidean predecessor fails. In this respect we observe explicitly that the “linear” functions g → ys (g), s = 1, ..., k, grow in fact quadratically at infinity, and yet the characteristic planes Πs = {ys (g) = 0} are complete X-minimal surfaces according to Definition 5.8. Another instance of this unsettling phenomenon is provided by the quadratic surface in the first Heisenberg group H1 n xy o , S = g = (x, y, t) ∈ H1 | t = 2 which is also a complete X-minimal surface. We note that its characteristic set is non-empty, and in fact it is given by the line Σ = {(x, y, t) ∈ H1 | x = t = 0}. We also emphasize that, in addition to being a X-minimal surface according to Definition 5.8, the surface S also minimizes the X-perimeter. This example, along with many others, has been found by Scott Pauls [Pa].

18


We next want to consider another interesting geometric situation, namely that of a hypersurface S in the Heisenberg group Hn whose defining function has spherical symmetry with respect to the horizontal coordinates. Definition 6.5. Let G be a Carnot group of step 2 with Lie algebra g = V1 ⊕ V2 . A function φ : G → R is said to have partial symmetry with respect to the identity e ∈ G if φ(g) = Φ(|x(g)|, y(g)) , for some function Φ : [0, ∞) × Rk . Suppose that φ ∈ C 2 (Hn ) has partial symmetry, then we can write (6.11)

φ(g) = Φ(|z|, t) ,

g = (z, t) ∈ Hn .

From (3.3) we note that for such a function one has with r = |z| Q−3 r2 Φr + Φtt . r 4 Furthermore, if ψ(g) = Ψ(|z|, t) is another function having partial symmetry, then a calculation gives

(6.12)

Lφ = Φrr +

(6.13)

< Xφ, Xψ > = Φr Ψr +

r2 Φt Ψt . 4

In particular, (6.13) implies |Xφ|2 = Φ2r +

(6.14)

r2 2 Φ . 4 t

Lemma 6.6. Let φ a function of the type (6.11), then its ∞-horizontal Laplacian is given by the formula (6.15)

r2 r r4 L∞ φ = Φrr Φ2r + Φrt Φr Φt + Φr Φ2t + Φtt Φt 2 4 16 + ! *Ã µ ¶ µ ¶ r2 Φr Φrr Φr 4 Φrt = , r2 r4 r Φt Φ t Φ Φ + Φ 4 rt 16 tt 4 r

Proof. We begin with noting that from (5.13) we can write L∞ φ in the following alternative form 1 < X(|Xφ|2 ), Xφ > . L∞ φ = 2 We now apply (6.13) with the choice ψ = |Xφ|2 obtaining µ ¶ 1 r2 (6.16) L∞ φ = Φr Ψr + Φt Ψt . 2 4 Using (6.14) we find r2 2 Φ )r 4 t r 2 r2 Φr + Φt + Φrt Φt , 2 2

Ψr = (Φ2r + = 2 Φrr


19

r2 2 Φ )t 4 t r2 = 2 Φrt Φr + Φtt Φt . 2 Substituting the last two equations in (6.16) we reach the conclusion. Ψt = (Φ2r +

¤ We are now ready to write the formula for the X-mean curvature of a hypersurface having partial symmetry in Hn . Proposition 6.7. Let Ω be a C 2 domain in Hn whose boundary S is the level set of a function 2 φ of the type (6.11). At every point g = (z, t) ∈ S such that Φ2r + r4 Φ2t 6= 0 we have (6.17) (Q − 1) HX = µ Φ2r +

1 r2 2 4 Φt

¶3/2

¶ ½µ ¶µ Q−3 r2 r2 2 2 Φr + Φt Φrr + Φr + Φtt 4 r 4 µ ¶¾ r2 r r4 2 2 − Φrr Φr + Φrt Φr Φt + Φr Φt + Φtt Φt . 2 4 16

Proof. It is a direct consequence of Proposition 5.12, of the identities (6.12), (6.14), and of Lemma 6.6. ¤ The following consequence of Proposition 6.7 is of special interest. Proposition 6.8. Under the hypothesis of Proposition 6.7 suppose, in particular, that ¡ |z|2 ¢ φ(z, t) = u − t, 4 for some C 2 function u : [0, ∞) → R. For every point point g = (z, t) ∈ S such that z 6= 0 the X-mean curvature at g is given by (Q − 1) HX =

2 s u00 (s) + (Q − 3) u0 (s) (1 + u0 (s)2 ) , √ 2 s (1 + u0 (s)2 )3/2

Proof. We presently have Φ(r, t) = u

¡ r2 ¢ − t, 4

so that r 0 u , Φt = − 1 , 2 r2 00 1 0 Φrr = u + u Φrt = Φtt = 0 . 4 2 The first two equations in (6.18) give

(6.18)

(6.19)

Φr =

Φ2r +

r2 2 r2 Φt = (1 + (u0 )2 ) , 4 4

s=

|z|2 . 4

20


whereas we obtain from the latter Q−3 r2 r2 00 Q−2 0 (6.20) Lφ = Φrr + Φr + Φtt = u + u . r 4 4 2 Using (6.18) we find the following formula (6.21) ½ ¾ r r4 r2 0 r2 00 0 r2 Φrt Φr Φt + Φr Φ2t + Φtt Φt = u u u + (1 + (u0 )2 ) . L∞ φ = Φrr Φ2r + 2 4 16 8 2 At this point we substitute (6.19), (6.20), (6.21) in (6.17) obtaining, after some magic cancellations due to the symmetries of the geometry involved, the following beautiful formula n 2 o r r2 00 + Q−3 u0 (1 + (u0 )2 ) u 4 4 2 (6.22) (Q − 1) HX = n o3/2 02 r2 (1 + u ) 4 Setting s = r2 /4 in (6.22) we reach the conclusion. ¤ If we consider the characteristic cones M |z|2 } , M ∈R, 4 then it is interesting to compute their X-mean curvature. We stress that the hypersurface SM = ∂CM has an isolated characteristic point at the identity e = (0, 0). Applying Proposition 6.8 we obtain the following result. CM = {g = (z, t) ∈ Hn | t >

Proposition 6.9. The C ∞ hypersurface SM = ∂CM has X-mean curvature at g = (z, t) given by the formula (Q − 3)M 1 (Q − 1) HX = p , z= 6 0. (1 + M 2 ) |z| In particular, we conclude that for the characteristic hyperplane S0 = {t = 0} we have, see Proposition 6.3, HX ≡ 0

(f latness) ,

whereas as soon as M 6= 0 one has lim HX = ± ∞

z→0

(blow − up) ,

with the limit being +∞ for a convex cone (M > 0), and −∞ for a concave cone (M < 0). Proof. It suffices to apply Proposition 6.8 with u(s) = M s. ¤ Remark 6.10. We stress that if we look at the paraboloids SM with the glasses of the subRiemannian geometry of Hn , then with respect to the nonisotropic group dilations (z, t) → (λz, λ2 t) these sets look like “cones”. If one accepts the consequences of this observation it should not be surprising that, despite the C ∞ character of the surface, the X-mean curvature at the “vertex” becomes infinite. This negative phenomenon is of course absent in Riemannian


21

geometry, where smooth surfaces always have bounded curvatures. This example is also interesting since it sheds light on another aspect of sub-Riemannian spaces, namely the influence of the characteristic set on the curvature of the space. The hyperplane S0 = {t = 0} has a “good characteristic point” at e, whereas for the paraboloid SM , M > 0, the identity is a “bad” characteristic point. Here, such distinction between good and bad is based on an appropriate understanding of the flatness of the surface near the point in question. We notice in this respect that whereas the hyperplane S0 admits an interior and exterior tangent gauge ball at e (the latter has a contact of order four with the tangent plane), the “cones” SM fail to satisfy such property. Proposition 6.1 gives another example of “good” characteristic point.

7. Perimeter measure and its first variation In this section we recall the definition of X-perimeter introduced in [CDG1] and prove that X-minimal hypersurfaces are critical points of the X-perimeter, Theorem 7.4. Let HG denote the subbundle of T G generated by the first layer V1 of the Lie algebra g. Given an open set Ω ⊂ G, we let F(Ω) = {ζ ∈ Co1 (Ω, HG) | |ζ|∞ = sup |ζ| ≤ 1} . Ω

For a function u ∈

L1loc (Ω),

the X-variation of u with respect to Ω is defined by Z m X V arX (u; Ω) = sup u Xi ζi dg . ζ∈F (Ω)

G

i=1

We say that u ∈ L1 (Ω) has bounded X-variation in Ω if V arX (u; Ω) < ∞. The space BVX (Ω) of functions with bounded X-variation in Ω, endowed with the norm ||u||BVX (Ω) = ||u||L1 (Ω) + V arX (u; Ω) , is a Banach space. Definition 7.1. Let E ⊂ G be a measurable set, Ω be an open set. The X-perimeter of E with respect to Ω is defined by PX (E; Ω) = V arX (χE ; Ω) , where χE denotes the indicator function of E. We say that E is a X-Caccioppoli set if χE ∈ BVX (Ω) for every Ω ⊂⊂ G. Following classical arguments [EG], one obtains from the Riesz representation theorem. Theorem 7.2. Given an open set Ω ⊂ G, let E ⊂ G be a X-Caccioppoli set in Ω. There exist a Radon measure ||∂X E|| in Ω, and a ||∂X E||-measurable function ν E X : Ω → HG, such that |ν E X (g)| = 1

for

||∂X E|| − a.e.

g∈Ω,

22


and for which one has for every ζ ∈ Co1 (Ω; HG) Z X Z Z m E Xj ζj dg = < ζ, ν X > d||∂X E|| = E j=1

Ω

Ω

< ζ, d[∂X E] > .

Let E ⊂ G be a C 1 domain, with Riemannian outer unit normal ν. If ζ ∈ Co1 (Ω; HG), we have Z X Z m m X Xi ζi dg = ζi < Xi , ν > dHN −1 . E i=1

∂E∩Ω i=1

From this observation, and from Theorem 7.2, we conclude the following result. Proposition 7.3. Let E ⊂ G be a C 1 domain. For every open set Ω ⊂ G, and any ζ ∈ Co1 (Ω; HG), one has Z Z E < ζ, ν X > d||∂X E|| = < ζ, Y X > dHN −1 , ∂E∩Ω

Ω

where Y X is defined in (5.2). Moreover, (7.1)

d||∂X E|| = |Y X | d (HN −1 b∂E) ,

and one has (7.2)

Z ||∂X E||(Ω) = PX (E; Ω) =

∂E∩Ω

|Y X | dHN −1 .

Theorem 7.4. Let φ ∈ C 1 (G) with |∇φ| ≥ α > 0, and set Ωt = {g ∈ G | φ(g) < t}. We then have Z d HX PX (Ωt ; G) = (Q − 1) dHN −1 . dt ∂Ωt |∇φ| In particular, if S = ∂Ω0 is X-minimal, then Ω = Ω0 is a critical point of the X-perimeter since d PX (Ωt ; G)|t=0 = 0 . dt Proof. Using Federer’s coarea formula we can write Z Z t Z Z t |Xφ| (7.3) |Xφ| dg = dHN −1 dτ = PX (Ωτ ; G) dτ , −∞ Ωt −∞ ∂Ωτ |∇φ| where the second equality is a consequence of (5.4) and of (7.2). The identity (7.3) gives Z d (7.4) PX (Ωt ; G) = |Xφ| dg . dt Ωt We now compute Z Z (7.5) |Xφ| dg = Xi φ νX,i dg Ωt Ωt Z Z = φ < Xi , ν > νX,i dHN −1 − φ Xi νX,i dg Ωt Z∂Ωt Z = φ dσX − φ Xi νX,i dg ∂Ωt Ωt Z = t PX (Ωt ; G) − φ Xi νX,i dg . Ωt


From (7.4), (7.5) we find (7.6)

d PX (Ωt ; G) = {t PX (Ωt ; G)} − t dt

Z ∂Ωt

23

Xi νX,i dHN −1 . |∇φ|

Equation (7.6) easily implies the desired conclusion. ¤

8. Sub-Riemannian calculus on hypersurfaces In this section we establish some basic integration by parts formulas which play a fundamental role in the study of hypersurfaces of mean curvature type. Such formulas are formally reminiscent of the classical ones. However, an important difference is that the ordinary surface measure is replaced by the X-perimeter measure. Furthermore, they contain an additional term which is due to the non-trivial commutation relations. Such term prevents the corresponding horizontal Laplace-Beltrami operator from being formally self-adjoint in L2 (S, dσX ) in general. Since the framework we work in does not lend itself to a preferred choice of coordinates, we have developed an approach which is coordinate-free. In fact, our proofs sligthly simplify several of the classical formulas for hypersurfaces in Rn which are derived by writing S as a graph. We begin by introducing some notation. Given a hypersurface S ⊂ G, with defining function φ, we will denote by (8.1)

dσX = |Y X | dHN −1 bS ,

the X-perimeter measure supported on S. We remark that if E = {g ∈ G | φ(g) < 0} is the open set such that S = ∂E, then according to (7.1) we have dσX = d||∂X E|| , and this justifies the name of X-perimeter measure. Given an open set Ω ⊂ G the notation Γk (Ω) indicates the class introduced by Folland and Stein [F2] of functions which admit continuous derivatives up to order k with respect to the vector fields X1 , ..., Xm . We denote by Γko (Ω) the subspace of functions with compact support in Ω. Our main result in this section is the following intrinsic divergence theorem. Theorem 8.1 (Integration by parts). Consider a C 2 hypersurface S ⊂ G in a sub-Riemannian group G. Let S = {g ∈ G | φ(g) = 0} ⊂ O, where O is an open subset of G. If u ∈ Γ1o (O) we have Z Z (8.2) {δX,i u + cS,i u} dσX = (Q − 1) u HX νX,i dσX . S

S

where the dσX -measurable functions cS,i on S are defined by Pm j=1 [Xi , Xj ]φ νX,j (8.3) cS,i = . |Xφ|

24


Moreover, the vector field cS = i.e., one has

Pm

i=1 cS,i Xi

(8.4)

is perpendicular to the horizontal Gauss map ν X ,

< cS , ν X > = 0 .

Proof. For every t ∈ R we define Ωt = {g ∈ G | φ(g) < t}. Federer’s coarea formula gives Z Z t Z |Xφ| (8.5) δX,i u |Xφ| dg = δX,i u dHN −1 dτ . |∇φ| Ωt −∞ ∂Ωτ Recalling (8.1) and (5.4), we obtain from (8.5) Z Z d δX,i u dσX = δX,i u |Xφ| dg . (8.6) dt Ωt ∂Ωt This important observation allows us to reduce the computation of the surface integral to that of an integral over the solid region Ωt . The hope is that a small miracle happens, and the R integral Ωt δX,i u|Xφ|dg can be expressed only in terms of solid integrals on Ωt . If that is the case, then by writing each of these integrals by means of the coarea formula as for (8.5), we can differentiate them with respect to the foliating parameter t. We turn to the computations. Recalling Definition 5.5, we have Z Z Z (8.7) δX,i u |Xφ| dg = Xi u |Xφ| dg − Xj u νX,j νX,i u |Xφ| dg , Ωt

Ωt

Ωt

where we have adopted the summation convention over repeated indices. Integrating by parts in the first integral in the right-hand side of (8.7) we find Z Z Z (8.8) Xi u |Xφ| dg = u < Xi , ν > |Xφ| dHN −1 − u Xi (|Xφ|) dg Ωt ∂Ωt Ωt Z Z Xi Xj φ νX,j |Xφ| dg , = u Xi φ dσX − u |Xφ| ∂Ωt Ωt where we have used the fact that in a sub-Riemannian group G one always has div Xi = 0. We next integrate by parts in the second integral in the right-hand side of (8.7), obtaining Z Z (8.9) Xj u νX,j νX,i |Xφ| dg = u < Xj , ν > νX,j νX,i |Xφ| dHN −1 Ωt ∂Ωt Z − u Xj (νX,j νX,i |Xφ|) dg Ωt Z Z = u < Xφ, ν X > νX,i dσX − u Xj (νX,j ) νX,i |Xφ| dg ∂Ωt Ωt Z + u νX,j Xj (νX,i |Xφ|) dg Ωt Z Z = u Xi φ dσX − u Xj (νX,j ) νX,i |Xφ| dg ∂Ωt Ωt Z Xj Xi φ νX,j |Xφ| dg , + u |Xφ| Ωt where we have used < Xφ, ν X > νX,i = Xi φ, which follows recalling that ν X = Xφ/|Xφ| at every non-characteristic point. Inserting (8.8), (8.9) into (8.7), and keeping in mind the


definition (8.3) of cS,i , we finally obtain Z Z Z (8.10) δX,i u |Xφ| dg = u Xj (νX,j ) νX,i |Xφ| dg − Ωt

Ωt

Ωt

25

u cS,i |Xφ| dg .

Formula (8.10) is what we were hoping for. Proceeding now as in (8.5), and applying (8.6), we conclude for every t ∈ R Z Z Z (8.11) δX,i u dσX = u Xj (νX,j ) νX,i dσX − u cS,i dσX . ∂Ωt

∂Ωt

∂Ωt

Recalling (5.14), and noting that S = ∂Ω0 , we conclude that (8.2) holds. Finally, we have < cS , ν X > = cS,i νX,i =

m X 1 [Xi , Xj ]φ Xi φ Xj φ = |Xφ|3

 m X

1 |Xφ|3 

=

i,j=1

Xi Xj φ Xi φ Xj φ −

i,j=1

m X i,j=1

  Xj Xi φ Xi φ Xj φ 

1 {L∞ φ − L∞ φ} = 0 . |Xφ|3

=

This establishes (8.4) and completes the proof. ¤

Remark 8.2. We emphasize that in the abelian case G = Rm , we have Xi = ∂/∂xi , i = 1, ..., m, and thereby cS,i ≡ 0. In this case formula (8.2) recaptures the classical integration by parts formula on a hypersurface, see for instance [MM], [GT].

Remark 8.3. We note explicitly that when G is the Heisenberg group Hn the matrix ([Xi , Xj ]φ)m i,j=1 appearing in the definition (8.3) is given by φt S, where ¶ µ 0 −I S = I 0 is the symplectic matrix in R2n (we have denoted with I the identity matrix in Rn ). If ζ denotes a horizontal vector, then we have Pm < S(Xφ), ζ > i,j=1 [Xi , Xj ]φ Xj φ ζi = φt . (8.12) < cS , ζ > = 2 |Xφ| |Xφ|2 We have the following notable consequence of Theorem 8.1. Theorem 8.4. In a sub-Riemannian group G suppose that the hypersurface is given by (8.13) If u ∈ Co1 (O) we have

S = {g ∈ G | φ(g) = h(x1 , ..., xm )} . Z

(8.14) S

Z δX,i u dσX = (Q − 1)

S

u HX νX,i dσX .

26


Proof. It suffices to observe that if the defining function of S depends only on the variables in the first layer V1 of g, then [Xi , Xj ] φ ≡ 0 ,

i, j = 1, ..., m .

This implies cS ≡ 0, see (8.3). The conclusion thus follows from (8.2). ¤

9. Horizontal Laplace-Beltrami operator We next introduce a partial differential operator on S which plays an important role in the development of the theory. For the abelian group G = Rm such operator is the standard Laplace-Beltrami operator on a hypersuface. Definition 9.1. The horizontal Laplace-Beltrami operator on S is defined as follows m m X def X (9.1) LX u = δX,i δX,i u + cS,i δX,i u , i=1

i=1

where the functions cS,i are given by (8.3). Remark 9.2. It is important to keep in mind that when S is a vertical hypersurface given by (8.13), then the operator LX is given by LX =

m X

δX,i δX,i .

i=1

In such case it is easy to show that LX is formally self-adjoint in L2 (S, dσX ). One basic raison d’être for the operator LX is in the following sub-Riemannian Stokes’ theorem which follows from Theorem 8.1. Corollary 9.3. Let u ∈ Γ2o (O), then we have Z (9.2) LX u dσX = 0 . S

When S is a compact hypersurface without boundary, then (9.2) holds for any u ∈ Γ2 (O) Proof. It suffices to take δX,i u instead of u in Theorem 8.1. Formula (8.2) gives, with LX u defined by (9.1), Z Z LX u dσX = (Q − 1) HX < δX u, ν X > dσX = 0 , S

S

since < δX u, ν X >= 0 σX − a.e. on S. ¤


27

Corollary 9.4. Let u ∈ Γ1 (O), then for every ζ ∈ Γ2o (O) we have Z Z (9.3) < δX u, δX ζ > dσX = − u LX ζ dσX . S

S

Proof. We take u δX,i ζ, instead of u, in Theorem 8.1. ¤ ˆ = G × R one Remark 9.5. In connection with Remark 5.9 we see that for the product group G has LXˆ = L on such S................ We note immediately the following formulas which are verified by direct computation from the definition. Lemma 9.6. Let u, v ∈ Γ2 (O), F ∈ C 2 (R), then (9.4)

LX (uv) = u LX v + v LX u + 2 < δX u, δX v > ,

(9.5)

LX (F ◦ u) = (F 00 ◦ u) |δX u|2 + (F 0 ◦ u) LX u .

The next result provides a very useful mean for computing LX u on S using the vector fields X1 , ..., Xm in the ambient group G. Proposition 9.7. Let u ∈ Γ2 (O), then we have on S \ Σ LX u = Lu + < cS , Xu > − < HessX u ν X , ν X > − (Q − 1) < Xu, ν X > HX . Proof. We begin with Definition 5.5 which gives δX,i u = Xi u − < Xu, ν X > νX,i . Applying δX,i again and using the summation convention over repeated indices we find (9.6)

δX,i δX,i u = δX,i (Xi u) − δX,i (< Xu, ν X >) νX,i − < Xu, ν X > δX,i νX,i .

We now compute the terms in the right-hand side of (9.6). (9.7)

δX,i (Xi u) = Xi Xi u − < X(Xi u), ν X > νX,i = Lu − Xj Xi u νX,i νX,j = Lu − < HessX u ν X , ν X > .

Next, equation (5.5) gives (9.8)

δX,i (< Xu, ν X >) νX,i = Xi (Xj u νX,j ) νX,i − < X(< Xu, ν X >), ν X > νX,i νX,i = 0.

Finally, we find from Definition 5.7 (9.9)

< Xu, ν X > δX,i νX,i = (Q − 1) < Xu, ν X > HX .

We now substitute (9.7), (9.8), (9.9) in (9.6). To reach the conclusion we only need to observe that thanks to (8.3) one has < cS , δX u > = < cS , Xu > − < Xu, ν X >< cS , ν X > = < cS , Xu > .

28


¤ The next result is an interesting (and important) consequence of Proposition 9.7. Proposition 9.8. If the function u is constant on S, then LX u = 0 . Proof. First of all, let us notice that, thanks to Proposition 5.6, we can without restriction assume that u ≡ 1 in O. Under such hypothesis the conclusion now follows trivially from Proposition 9.7. ¤ We next introduce the notions of p-Dirichlet integral and of p-harmonic function on an hypersurface. Such notions are central to the development of geometric pde’s. Definition 9.9. Given 1 < p < ∞ we define the (X, p)-Dirichlet integral of a function u ∈ Γ1 (O) on the hypersurface S as Z 1 EX,S (u) = |δX u|p dσX . p S We say that u is (X, p)-subharmonic (-superharmonic) in S if for every ζ ∈ Γ1o (O), ζ ≥ 0, one has Z S

|δX u|p−2 < δX u, δX ζ > dσX ≤ 0 (≥ 0) .

We say that u is (X, p)-harmonic in S if u is simultaneously (X, p)-subharmonic and superharmonic. When p = 2 we simply say that u is X-harmonic in S. According to Corollary 9.4 we can, and will, adopt the following alternative notion of Xsubharmonicity. Definition 9.10. A function u ∈ L1 (S, dσX ) is called X-subharmonic in S if Z (9.10) 0 ≤ u LX ζ dσX , for every ζ ∈ Co2 (O) , ζ ≥ 0 . S

Remark 9.11. We stress that we have defined the operator δX in its natural domain OX . However, thanks to Theorem 5.3 we know that HQ−1 (S ∩ (O0 )c ) = 0. On the other hand, if S = ∂Ω, where Ω ⊂ G is a C 1 domain, then Magnani also proved (see Theorem 4.10 in [Ma2]) that HQ−1 is mutually absolutely continuous with respect to the perimeter measure dσX , therefore the characteristic set Σ of S is a set of zero σX -measure. These considerations allow to conclude that (X, p)-Dirichlet integral is well defined.


29

10. Horizontal Minkowski formulas on hypersurfaces In what follows we consider a hypersurface S in a Carnot group G. We suppose that e ∈ S and consider the unique fundamental solution Γ(g) = Γ(g, e) of L with singularity at e and vanishing at ∞. Introduce the function ρ = ρe ∈ C ∞ (G \ {e}) defined by def

(10.1)

1 − Q−2

ρ(g) = Γ(g)

.

We will need the following useful result. Lemma 10.1. One has in G \ {e} Lρ =

Q−1 |Xρ|2 . ρ

Proof. Let h(t) = t−1/(Q−2) , then ρ = h(Γ) and we obtain from the chain rule Lρ = h00 (Γ) |XΓ|2 + h0 (Γ) L Γ = h00 (Γ) |XΓ|2 ,

in G \ {e} ,

where we have used the fact that LΓ = 0 in G \ {e}. Using the definition of h(t), an elementary calculation gives Q−1 − 1 −2 Lρ = Γ Q−2 |XΓ|2 . 2 (Q − 2) On the other hand, it is easy to check that Q−1 Q−1 − 1 −2 |Xρ|2 = Γ Q−2 |XΓ|2 . 2 ρ (Q − 2) The latter two equations establish the lemma. ¤ By choosing u = F ◦ ρ in Proposition 9.7 we obtain the following. Proposition 10.2. Let G be a Carnot group with ρ given by (10.1). Suppose that e 6∈ S. For F ∈ C 2 (R) the horizontal Laplace-Beltrami operator of F ◦ ρ on S ∩ Σc is given by LX F (ρ) = F 00 (ρ) |δX ρ|2 + F 0 (ρ) < cS , Xρ > − F 0 (ρ) < HessX ρ ν X , ν X > Q−1 0 F (ρ)|Xρ|2 − (Q − 1) F 0 (ρ) < Xρ, ν X > HX . + ρ Proof. Applying (9.5) in Lemma 9.6 we find (10.2)

LX F (ρ) = F 00 (ρ) |δX ρ|2 + F 0 (ρ) LX ρ .

Invoking Proposition 9.7 we obtain (10.3) LX ρ = Lρ + < cS , Xρ > − < HessX ρ ν X , ν X > − (Q − 1) < Xρ, ν X > HX . Q−1 = |Xρ|2 + < cS , Xρ > − < HessX ρ ν X , ν X > − (Q − 1) < Xρ, ν X > HX , ρ where in the second equality we have used Lemma 10.1. We now substitute (10.3) in (10.2) to reach the conclusion. ¤

30


Proposition 10.2 has some deep consequences when the ambient group possesses some special symmetries. What we are alluding to is the following fundamental identity. Theorem 10.3. Let G be a group of Heisenberg type, then in view of (3.13) we have ρ = N (up to an immaterial multiplicative factor C = C(G) > 0), where N is the non-isotropic gauge (3.12). One has in G \ {e} (10.4)

|XN |2 ξ1 ⊗ ξ1 3 IV1 + 2 − XN ⊗ XN N N3 N Ã k ! X 2 + < J(Ys )ξ1 , Xi >< J(Ys )ξ1 , Xj > N3

HessX N =

s=1

where IdV1 denotes the identity on V1 ∼

,

i,j=1,...,m

Rm .

Proof. We introduce the function φ(g) = |x(g)|4 + 16 |y(g)|2 = N (g)4 .

(10.5) Clearly, we have (10.6)

Xj φ = 4 N 3 Xj N ,

Xi Xj φ = 4 N 3 Xi Xj N + 12 N 2 Xi N Xj N ,

therefore 1 3 XN ⊗ XN . HessX φ − 3 4N N We are left with computing HessX φ. From (6.21) in [DGN3] we have

(10.7)

HessX N =

HessX (|x|4 ) = 4 |x|2 IV1 + 8 ξ1 ⊗ ξ1 .

(10.8)

On the other hand, formula (6.16) in [DGN3] gives Ã k !m X 1 (10.9) HessX (|y|2 ) = < J(Ys )ξ1 , Xi >< J(Ys )ξ1 , Xj > 2 s=1

.

i,j=1

Combining (10.8), (10.9) we obtain Ã 2

(10.10) HessX φ = 4 |x| IV1 + 8 ξ1 ⊗ ξ1 + 8

k X

!m < J(Ys )ξ1 , Xi >< J(Ys )ξ1 , Xj >

s=1

. i,j=1

If we use (10.10) into (10.7) we find (10.11) |x|2 2 ξ1 ⊗ ξ1 HessX N = IV1 + 2 + 3 3 N N N3

Ã k X s=1

!m < J(Ys )ξ1 , Xi >< J(Ys )ξ1 , Xj > i,j=1

3 − XN ⊗ XN . N Recalling (6.1), we obtain (10.4). ¤


31

Theorem 10.4. Let G be a group of Heisenberg type. If S ⊂ G is a C 2 hypersurface with e 6∈ S, we have (10.12)

< HessX N ν X , ν X > =

|δX N |2 2 + ΨS , N N

where (10.13)

ΨS

1 = N2

def

" 2

< ξ1 , ν X >

+

k X

# 2

< J(Ys )ξ1 , ν X >

− < XN, ν X >2 .

s=1

Moreover, we have (10.14)

ΨS ≥ 0

on

S ,

and ΨS ≡ 0 if S is a gauge sphere {g ∈ G | N (g) = R}. Proof. Equation (10.12) follows immediately from Theorem 10.3, and from (5.9). The inequality (10.14) is a consequence of the following considerations. From (10.5) and (10.6) we find Xφ = 4 |ξ1 |2 ξ1 + 16 X(|y|2 ) ,

(10.15) therefore (10.16)

< XN, ν X > =

|ξ1 |2 4 < ξ1 , ν X > + 3 < X(|y|2 ), ν X > . 3 N N

From (6.24) in [DGN3] we obtain (10.17)

Xi (|y|2 ) =

k X

ys < J(Ys )ξ1 , Xi > ,

s=1

hence (10.18)

< X(|y|2 ), ν X > = < J(ξ2 )ξ1 , ν X > ,

where we have used the equation ξ2 = (10.19)

Pk

s=1 ys J(Ys ).

Substitution of (10.18) in (10.16) gives

¤2 1 £ 2 |ξ1 | < ξ1 , ν X > + 4 < J(ξ2 )ξ1 , ν X > 6 N # " k X |ξ1 |4 + 16|ξ2 |2 < J(Ys )ξ1 , ν X >2 ≤ < ξ1 , ν X >2 + N6 s=1 " # k X 1 2 2 = < ξ1 , ν X > + < J(Ys )ξ1 , ν X > , N2

< XN, ν X >2 =

s=1

where we have used Cauchy-Schwarz inequality. From the latter estimate (10.14) follows. Finally, suppose that S is a gauge sphere {g ∈ G | N (g) = R}. From [DGN2] we know that the characteristic set of S is given by Σ = {g ∈ S | x(g) = 0}. At every non-characteristic point we have XN νX = . |XN |

32


We thus obtain at such points ΨS

(10.20)

1 = 2 N |XN |2

( < XN, ξ1 >2 + ½µ

k X

) < XN, J(Ys )ξ1 >2

− |XN |2

s=1

¶2 1 |ξ1 4 + < J(ξ2 )ξ1 , ξ1 > = N 2 |XN |2 N3 N3 ¶2 ¾ k µ X |ξ1 |2 4 + < ξ1 , J(Ys )ξ1 > + 3 < J(ξ2 )ξ1 , ξ1 > − |XN |2 N3 N s=1 Ã !2 ¾ ½ k k X |ξ1 |8 1 4 X + ys0 δss0 |ξ1 |2 − |XN |2 = N 2 |XN |2 N6 N3 0 s=1 s =1 ½ ¾ 8 4 1 |ξ1 | 16|ξ1 | |ξ2 |2 + − |XN |2 = N 2 |XN |2 N6 N6 |ξ1 |4 = − |XN |2 = 0 . N 4 |XN |2 |4

In deriving (10.20) we have used (3.8), (3.10) and (6.1). This establishes the sufficiency of the condition S = {g ∈ G | N (g) = R} for ΨS ≡ 0. ¤

Theorem 10.5. In a group of Heisenberg type G, let F ∈ C 2 (R). Given a compact hypersurface S ⊂ G, with e 6∈ S, the horizontal Laplace-Beltrami operator of the composition F ◦ N is given by (10.21)

½ ¾ F 0 (N ) F 00 (N ) − |δX N |2 + F 0 (N ) < cS , XN > N F 0 (N ) |XN |2 − (Q − 1) F 0 (N ) < XN, ν X > HX + (Q − 1) N ( ) k F 0 (N ) < ξ1 , ν X >2 1 X − 2 < J(Ys )ξ1 , ν X >2 − < XN, ν X >2 . + N N2 N2

LX F (N ) =

s=1

Proof. It suffices to combine Theorem 10.4 with Proposition 10.2 (in which we take ρ = N ) to reach the conclusion. ¤

Some remarks are in order at this point.

Remark 10.6. It is interesting to compare Theorem 10.5 with its Euclidean ancestor. Suppose in fact that S represents a hypersurface in Rn , with 0 6∈ S, and denote by r = r(x) = |x| the


33

ordinary Euclidean distance from the origin. One has the following formula for the LaplaceBeltrami operator ∆S of F (r) ½ ¾ F 0 (r) 00 (10.22) ∆S F (r) = F (r) − |δr|2 r n−1 0 n−1 0 + F (r) − F (r) < x, ν > H , r r where we have denoted by H the Riemannian mean curvature of S. Formula (10.21) in Theorem 10.5 incorporates this classical formula as a special case. To see this, consider the abelian Carnot group G = Rn , with Lie algebra g = V1 = Rn , so that all commutators are zero. In this case one has trivially N = r, hence XN = ξ1 /N . We thus find < ξ1 , ν X >2 − < XN, ν X >2 ≡ 0 , N2

k X

< J(Ys )ξ1 , ν X >2 ≡ 0 .

s=1

Keeping in mind that we also have cS ≡ 0, we see that (10.21) reduces to (10.22). Remark 10.7. As the next proposition shows, the sub-Riemannian term in (10.21) is nonnegative, and this distinguishes in a substantial way (10.21) from its Riemannian ancestor (10.22). A beautiful identity due to Minkowski states that for a compact C 2 hypersurface without boundary S ⊂ Rn one has Z (10.23) Hn−1 (S) = < x, ν > H dHn−1 , S

where H denotes the mean curvature of S and < x, ν > its supporting function. This formula is an immediate consequence of the choice F (r) = r2 /2 in equation (10.22) of Remark 10.6. With this choice we have in fact F 00 (r) − F 0 (r)/r = 0, and therefore integrating (10.22) over S we find µ 2¶ Z Z Z 1 r dHn−1 − dHn−1 = 0 , < x, ν > H dHn−1 = ∆S n−1 S 2 S S in view of Stokes’ theorem. We have the following interesting consequence of Theorem 10.5. Corollary 10.8. Let G be a group of Heisenberg type, then one has · µ 2¶ µ 2¶ ¸ µ 2¶ 1 N N N 2 LX + 2 ΨS − < cS , X > = |XN | − < X , ν X > HX . (Q − 1) 2 2 2 Proof. It suffices to choose F (t) = t2 /2 in (10.21) of Theorem 10.5. ¤ The following result represents a sub-Riemannian version of Minkowski’s formula (10.23). We stress that it does contain (10.23) as a special case. Theorem 10.9. Let S ⊂ G be a compact C 2 hypersurface in a group of Heisenberg type G, suppose that e 6∈ S, then µ 2¶ µ 2¶ ¸ Z Z Z · N 1 N 2 |XN | dσX = HX dσX + 2 Ψ S − < cX , X > dσX , 2 (Q − 1) S 2 S S where dσX is the perimeter measure introduced in Definition 9.9.

34


Proof. It follows by integrating the identity in Corollary 10.8 with respect to the measure dσX . The term containing LX (N 2 /2) is zero thanks to Corollary 9.3. ¤ We close this section with the following intriguing Conjecture: In a group of Heisenberg type G let S ⊂ G be a C ∞ compact hypersurface. Suppose that (10.24)

HX ≡ |XN |

on

S .

Is it true that there exists R > 0 such that S = {g ∈ G | N (g) = R}? Notice that if S is a gauge sphere of radius R, then thanks to Proposition 6.1 condition (10.24) is satisfied. We also notice then when G = Rm the conjecture is the celebrated soap bubble theorem of A.D. Alexandrov [A].

11. Monotonicity formulas on hypersurfaces One of the most fundamental properties of classical minimal surfaces S ⊂ Rm is the maximum volume growth expressed by the global inequality (11.1)

ωm−1 Rm−1 ≤ Hm−1 (S ∩ Be (x, R)) ,

x∈S ,

R>0,

where ωm−1 indicates the area of the unit ball in Rm−1 . This is a consequence of the following important result, see e.g. [Si], [MM]. Theorem 11.1. Let S ⊂ Rm be a C 2 hypersurface, with H being its mean curvature, then for every fixed x ∈ S the function Z Z r Hm−1 (S ∩ Be (x, r)) m−1 (11.2) r → + |H| dHm−1 dt , m−1 rm−1 0 t S∩Be (x,t) is non-decreasing. In this section we investigate related monotonicity results for hypersurfaces in Carnot groups. In what follows we consider a hypersurface S ⊂ O ⊂ G, where as before O denotes an open set. Our first result concerns vertical non-characteristic surfaces. Theorem 11.2. Let S ⊂ G be a surface of the type (5.10), and suppose that e ∈ S. If S is X-minimal then the function σX (S ∩ B(e, r)) rQ−1 is non-decreasing. In particular, there exists a constant ω = ω(G) > 0 such that for every r > 0

(11.3)

(11.4)

r →

σX (S ∩ B(e, r)) ≥ ω rQ−1 .


35

Proof. We present the proof only for groups of step 2, the details for the general case are completely analogous, and are left to the reader. Denoting by m = dim(V1 ), k = dim(V2 ), we have Q = m + 2k. We fix r1 < r2 . To prove the first part of the theorem we will show that (11.5)

σX (S ∩ B(e, r2 )) r2Q−1

−

σX (S ∩ B(e, r1 )) r1Q−1

≥ 0.

In view of (8.1) we have Z σX (S ∩ B(e, r)) =

S∩B(e,r)

|Xφ| dHN −1 . |∇φ|

Recalling that φ(g) = h(x(g)), we obtain Xφ = ∇x h, and therefore |Xφ| = |∇φ| = |∇x h|. This gives Z (11.6) σX (S ∩ B(e, r)) = dHN −1 S∩B(e,r) Z = dHN −1 {h(x)=0}∩{|x|4 +16|y|2 −4 |ξ1 |2

)

Proof. Definition 5.5 gives (11.23)

δX,i ζi = Xi ζi − Xj ζi νX,j νX,i = Q − Xi ζj νX,i νX,j ,

where in the second equality we have used Theorem 11.5. We now compute the second term in the right-hand side of (11.23). One has (11.24) Xi ζj νX,i νX,j =

Xi Xj N Xi N Xj N νX,i νX,j + N νX,i νX,j |XN |2 |XN |2

+ N Xj N Xi (|XN |−2 ) νX,i νX,j ½ ¾ < XN, ν X >2 < HessX N ν X , ν X > < XN, ν X >< X(|XN |2 ), ν X > = + N − |XN |2 |XN |2 |XN |4 ½ ¾ < HessX N ν X , ν X > < XN, ν X >< X(|XN |2 ), ν X > |δX N |2 + N − , = 1 − |XN |2 |XN |2 |XN |4


39

where in the last equality we have used (5.9). We now use (10.12) in Theorem 10.4. Substituting the latter in (11.24) we find (11.25) Xi ζj νX,i νX,j

½ ¾ |δX N |2 |δX N |2 2ΨS < XN, ν X >< X(|XN |2 ), ν X > = 1 − + N + − |XN |2 N |XN |2 N |XN |2 |XN |4 2ΨS N < XN, ν X >< X(|XN |2 ), ν X > = 1 + − , 2 |XN | |XN |4

where the function ΨS is defined by (10.13). Keeping in mind (6.1) we obtain (11.26) |x|2 < XN, ν X > < X(|x|2 ), ν X > − 2 N2 N3 < XN, ν X > < ξ1 , ν X > − 2 = 2 |XN |2 . 2 N N Substituting (11.26) in (11.25) we find < X(|XN |2 ), ν X > = < X(|x|2 N −2 ), ν X > =

(11.27) Xi ζj νX,i νX,j = 1 +

2ΨS < XN, ν X >< ξ1 , ν X > < XN, ν X >2 − 2 + 2 . |XN |2 N |XN |4 |XN |2

Finally, using (10.13) we conclude (11.28)

Xi ζj νX,i νX,j

2 = 1 + 2 N |XN |2 +

k X

½ < ξ1 , ν X >2 2

< J(Ys )ξ1 , ν X >

s=1

N < XN, ν X >< ξ1 , ν X > − |XN |2

¾ .

To reach the conclusion we now appeal to Lemma 4.4, which combined with (6.1) gives N < XN, ν X >< ξ1 , ν X > < J(ξ2 )ξ1 , ν X >< ξ1 , ν X > = < ξ1 , ν X >2 + 4 . 2 |XN | |ξ1 |2 Substitution of this equation in (11.28) gives the desired conclusion. ¤ We are now ready to state the main result. Theorem 11.7. Let G be a group of Heisenberg type, and S ⊂ G be a C 2 hypersurface. For g ∈ S and r > 0 we denote by Sg the left-translated surface Lg−1 (S) passing through the identity. Consider the quantity Z £ ¤ 1 (11.29) ωS (g, r) = Q−1 ΘSg − < cSg , ζ > dσX , r Sg ∩B(e,r) where the function ΘSg is the quantity introduced in Theorem 11.6, but relative to the hypersurface Sg , cS is defined by (8.3), and the horizontal vector field ζ is as in (11.18). We assume that ωS ≤ 0, and that for every r > 0 Z r dt (11.30) |ωS (g, t)| < ∞. t 0 Under these hypothesis, if S is X-minimal the function r →

σX (S ∩ B(g, r)) rQ−1

40


is non-decreasing. Proof. We choose ρ = N in Lemma 11.4, and observe that the horizontal vector field ζ in (11.18) satisfies the estimate (11.31)

|ζ| ≤

N , |XN |

wherever |XN | = 6 0. This estimate allows to conclude (11.32)

|ζ| |δX ρ| = |ζ| |δX N | ≤

N |δX N | ≤ N . |XN |

Using (11.32) and Theorem 11.6 in (11.14) we obtain Z Z Z (11.33) (Q − 1) λ(r − ρ) dσX + [< cS , ζ > − ΘS ] λ(r − ρ) dσX − r λ0 (r − ρ) dσX S S S Z |HX | ≤ (Q − 1) r λ(r − ρ) dσX . |XN | S We can rewrite the latter inequality as follows Z Z Q−1 (11.34) 0 ≤ λ0 (r − ρ) dσX − λ(r − ρ) dσX r S S Z Z 1 |HX | dσX . + [ΘS − < cS , ζ >] λ(r − ρ) dσX + (Q − 1) λ(r − ρ) r S |XN | S If we now define

Z H(r) = S

λ(r − ρ) dσX ,

then (11.34) implies µ ¶ Z H(r) d 1 0 ≤ + Q [ΘS − < cS , ζ >] λ(r − ρ) dσX dr rQ−1 r S Z Q−1 |HX | + Q−1 λ(r − ρ) dσX . r |XN | S

(11.35)

We integrate (11.35) on the interval (r1 , r2 ), where r1 , r2 < dist(e, ∂O), obtaining (11.36)

Z Z r2 Z |HX | 1 Q−1 λ(r − ρ) dσX dr + [ΘS − < cS , ζ >] λ(r − ρ) dσX dr 0 ≤ Q−1 Q |XN | S r1 r S r1 r Z Z 1 1 + Q−1 λ(r2 − ρ) dσX − Q−1 λ(r1 − ρ) dσX . r2 r1 S S Z

r2

For a fixed ² > 0 we now choose λ as before in Lemma 11.4, but satisfying λ ≡ 1 on the interval (², ∞). We obtain from (11.36) ¾ ½ Z r2 Z 1 1 |HX | (11.37) 0 ≤ + [ΘS − < cS , ζ >] dσX dr (Q − 1) Q−1 |XN | r r1 r S∩B(e,r) Z Z 1 1 + Q−1 dσX − Q−1 dσX . r2 r1 S∩B(e,r2 ) S∩B(e,r1 −²)


41

Letting ² → 0 we conclude ¾ ½ Z r2 Z 1 |HX | 1 (11.38) 0 ≤ (Q − 1) + [ΘS − < cS , ζ >] dσX dr Q−1 |XN | r r1 r S∩B(e,r) +

σX (S ∩ B(e, r2 )) r2Q−1

−

σX (S ∩ B(e, r1 )) r1Q−1

.

We now assume that S is X-minimal, i.e., HX ≡ 0. Using the assumption (11.30), we conclude from the latter inequality that the function r −→

σX (S ∩ B(e, r)) rQ−1

is non-decreasing. ¤ We now present a non-trivial example in which the hypothesis of Theorem 11.7 can be satisfied. Proposition 11.8. In a group of Heisenberg type G consider the characteristic hyperplanes Si = {g = (x(g), (y(g)) ∈ G | yi (g) = 0}, where i = 1, ..., k. For g = e we have for every r > 0, ωS (g, r) ≡ 0 . As a consequence of Theorem 11.7, we conclude that the function r −→

σX (S ∩ B(e, r)) rQ−1

is non-decreasing. Proof. We begin by observing that, thanks to Proposition 6.3, the hyperplanes Si are Xminimal surfaces. We next focus on S1 , since the remaining cases are treated in exactly the same manner. Let φ(x, y) = y1 be the defining function for S1 , then according to (4.3) the i-th component of Xφ is given by 1 1 < Xφ, Xi > = Xi y1 = < [ξ1 , Xi ], Y1 > = < J(Y1 )ξ1 , Xi > . 2 2 Hence, we obtain from (3.9) m

(11.39)

2

|Xφ|

1X 1 |ξ1 |2 = < J(Y1 )ξ1 , Xi >2 = |J(Y1 )ξ1 |2 = . 4 4 4 i=1

The horizontal normal to S1 is given by m

(11.40)

νX =

Xφ 1 X = < J(Y1 )ξ1 , Xi > Xi . |Xφ| |ξ1 | i=1

Using (11.40), we compute m

< J(Ys )ξ1 , ν X > =

1 X 1 < J(Ys )ξ1 , Xi >< J(Y1 )ξ1 , Xi > = < J(Ys )ξ1 , J(Y1 )ξ1 > = δs1 |ξ1 | . |ξ1 | |ξ1 | i=1

Therefore, (11.41)

k X s=1

2

< J(Ys )ξ1 , ν X > =

k X s=1

2 δs1 |ξ1 |2 = |ξ1 |2 .

42


We now observe that by using (11.40) m 1 X < ξ1 , J(Y1 )ξ1 > < ξ1 , ν X > = < ξ1 , Xi >< J(Y1 )ξ1 , Xi > = = 0. |ξ1 | |ξ1 | i=1

Hence (11.42)

4

< J(ξ2 )ξ1 , ν X >< ξ1 , ν X > = 0. |ξ1 |2

Combining (11.41) and (11.42) we conclude ( k ) X 1 < J(ξ )ξ , ν >< ξ , ν > 2 1 1 X X (11.43) ΘS1 = ≡ 2, < J(Ys )ξ1 , ν X >2 − 4 |ξ1 |2 |ξ1 |2 s=1

a.e. on S1 with respect to σX . We now turn our attention to < cS1 , ζ > where the components of ζ are given by (11.18). To this end, we recall the following useful commutator formula for Carnot groups of step r = 2 (see (5.10) in [DGN3]) (11.44)

[Xi , Xj ] =

k X

bsij

s=1

where

bsij

∂ , ∂ys

represent the group constants defined by the formula [Xi , Xj ] =

k X

bsij Ys .

s=1

The latter equation gives (11.45)

< J(Yr )Xi , Xj > = < [Xi , Xj ], Yr > =

k X

bsij δsr = brij .

s=1

In particular, (11.46)

b1ij

=< J(Y1 )Xi , Xj >. We can now compute < cS1 , ζ > =

m X

cS1 ,i ζi =

i=1

1 X [Xi , Xj ]φ ν X,j ζi |Xφ| i,j

(11.47) k 2N X X s Xi N (by (11.39),(11.40),(11.44) and (11.18)) = bij δs1 < J(Y1 )ξ1 , Xj > 2 |ξ1 | |XN |2 i,j s=1   m m 2N X X Xi N (by (11.45)) = < J(Y1 )Xi , Xj >< J(Y1 )ξ1 , Xj > 2 |ξ1 | |XN |2 i=1

=

j=1

m Xi N 2N X < J(Y1 )Xi , J(Y1 )ξ1 > 2 |ξ1 | |XN |2 i=1

=

m X 2N < ξ1 , Xi >< XN, Xi > |ξ1 |2 |XN |2 i=1

2N < XN, ξ1 > |ξ1 |2 |XN |2 2N 3 |ξ1 |4 (by (??)) = ≡ 2. |ξ1 |4 N 3 =

Hence, using (11.46) and (11.43) we conclude ωS1 ≡ 0.

¤


43

Finally, we present an example (using numerical means that) to show that in general, one cannot expect the function σX (S ∩ B(g, r)) r → rQ−1 to be non-decreasing on [0, ∞). This phenomena is an interesting one and sets the stage for further analysis of this quantity. Consider in H1 the characteristic plane Π = {(x, y, t) | t = 0} and a point go = (xo , yo , 0) ∈ Π. We compute σX (Π ∩ B(go , r))/r3 by using translation and this expression becomes σX (Lgo−1 (Π) ∩ B(e, r))

(11.48)

r3

,

where Lgo−1 (Π) = {go−1 g | g ∈ Π} = {(x − xo , y − yo , 2(xo y − yo x)) | x, y ∈ R} with defining function given by φ(x, y, t) = t − 2(xo y − yo x) . Ignoring the factor r3 for the moment, elementary computations show σX (Lgo−1 (Π) ∩ B(e, r)) Z = L

Z

−1 go

|Xφ| dσ

(Π)∩B(e,r) 1

= {t = 2(xo y−yo x) and

Z p = 2 1 + 4|zo |2

(x2 +y 2 )2 +t2

for the characteristic plane is given by ΘSg − < CSg , ζ > = 2

< z, zo >2 < z, zo > (xo y − yo x)2 < z, zo > +2 2 +4 2 2 |z + zo | |z| |z + zo | |z|4 |z + zo |2

where in the above, we have let zo = (xo , yo ) and z = (x, y). Note that ωS (g, r) does not satisfy the assumptions of Theorem 11.7, namely, it is not integrable near r = 0 nor is ωS (g, r) ≤ 0 throughout the entire range r > 0. Otherwise, our graph (figure 11.1) above would contradict Theorem 11.7. From the graph one can observe that the values of ωS (g, r) → 0 rapidly as r → ∞ and ωS (g, r) → −∞ as r → 0+ faster than −1/r → −∞.


ωS(g,r)

45

g = (10,0,0)

50 20

40

60

0 –50

–100

–150

–200

Figure 11.2. Graph of ωS (g, r) with g = (10, 0, 0)

80

100

r

46


12. Isoperimetric inequalities The appropriateness of the notion of X-perimeter in Carnot-Carathéodory geometry is witnessed by the isoperimetric inequalities. Similarly to their Euclidean counterpart, these inequalities play a fundamental role in the development of geometric measure theory. In this section we present some refinements of the results in [CDG1], [GN1]. Our first result is a special case of Theorem 1.4 in [CDG1]. Theorem 12.1. Let G be a Carnot group with homogeneous dimension Q. There exists a constant Ciso (G) > 0, such that for every go ∈ G, 0 < R < Ro , one has for every C 1 domain E ⊂ E ⊂ B(go , R) |E|(Q−1)/Q ≤ Ciso PX (E; B(go , R)) . Theorem 12.1 generalizes an earlier result of Pansu [P], who proved a related inequality for the first Heisenberg group H1 , but with the X-perimeter in the right-hand side replaced by the quantity H3 (∂E). Here, H3 indicates the 3-dimensional Hausdorff measure in H1 constructed with the Carnot-Carathéodory distance d(g, g 0 ) (recall that the homogeneous dimension of H1 is Q = 4, so 3 = Q − 1). Although Pansu’s result is dimensionally sharp, the fact that it involves the difficult to pin-down Hausdorff measure leads naturally to wonder about the connection between the latter and the X−perimeter. The answer to this question involves a substantial amount of work, and it was given to a large extent in [DGN1]. More recent and improved results on this subject can be found in [DGN2], [FSS2], [FSS3], [DGN5]. We mention that other isoperimetric and Gagliardo-Nirenberg type inequalities have been independently, and at various times, obtained by several authors [Va1], [CS], [BM], [FGW], [FLW], [MaSC]. We next extend Theorem 12.1 from C 1 domains to X-Caccioppoli sets. That such extension be possible is due to the following basic approximation result for functions in the space BVX , which is contained in Theorem 1.14 in [GN1]. Theorem 12.2. Let Ω ⊂ G be open, where G is a Carnot group. For every u ∈ BVX (Ω) there exists a sequence {uk }k∈N in C ∞ (Ω) such that (12.1) (12.2)

uk → u

in

L1 (Ω)

as

k→∞,

lim V arX (uk ; Ω) = V arX (u; Ω) .

k→∞

Theorem 12.3. Let G be a Carnot group with homogeneous dimension Q. With the same constant Ciso (G) > 0 of Theorem 12.1, for any go ∈ G, R > 0, one has for every X-Caccioppoli set E ⊂ E ⊂ B(go , R) |E|(Q−1)/Q ≤ Ciso PX (E; B(go , R)) .


47

Proof. In [CDG1] it was proved that Theorem 12.1 implies the following Sobolev inequality of Gagliardo-Nirenberg type: for every u ∈ Co1 (B(go , R)) (Z )(Q−1)/Q Z R Q/(Q−1) (12.3) |u| dg ≤ Ciso |Xu| dg . |B(go , R)|1/Q B(go ,R) B(go ,R) If now u ∈ BVX (B(go , R)), with supp u ⊂ B(go , R), then by Theorem 12.2 there exists a sequence {uk }k∈N ∈ Co∞ (B(go , R)) such that uk → u in L1 (B(go , R)), and V arX (uk ; B(go , R)) → V arX (u; B(go , R)), as k → ∞. Passing to a subsequence, we can assume that uk (g) → u(g), for dg-a.e. g ∈ B(go , R). Applying (12.3) to uk and passing to the limit we infer from the theorem of Fatou )(Q−1)/Q (Z R |u|Q/(Q−1) dg ≤ Ciso V arX (u; B(go , R)) , |B(go , R)|1/Q B(go ,R) for every u ∈ BVX (B(go , R)), with supp u ⊂ B(go , R). If now E ⊂ E ⊂ B(go , R) is a XCaccioppoli set, then taking u = χE in the latter inequality we reach the conclusion. ¤ The following is a basic consequence of Theorem 12.3. Theorem 12.4. Let G be a Carnot group with homogeneous dimension Q. With Ciso (G) as in Theorem 12.1 one has for any bounded X-Caccioppoli set |E|(Q−1)/Q ≤ Ciso PX (E; G) . Our next goal is to remove in Theorem 12.4 the restriction that the X-Caccioppoli set be bounded. This step will also be important in our proof of the existence of isoperimetric sets. We begin with a preliminary result. Lemma 12.5. Let Ω ⊂ G be an open set and E ⊂ G be a C 1 bounded domain. One has Z PX (E; Ω) = |Y X | dHN −1 , Ω∩∂E

where Y X is the projection onto HG of the outward unit normal to E defined as in Definition 5.4. For the proof of this lemma we refer the reader to [CDG1]. Theorem 12.6. Let G be a Carnot group with homogeneous dimension Q. With the same constant Ciso = Ciso (G) > 0 as in Theorem 12.4, for every X-Caccioppoli set E ⊂ G one has |E|(Q−1)/Q ≤ Ciso PX (E; G) . Proof. In view of Theorem 12.4 we only need to consider tha case of an unbounded XCaccioppoli set E. If PX (E; G) = +∞ there is nothing to prove, so we assume that PX (E; G) < +∞ and |E| < +∞. We consider the C ∞ X-balls BX (e, R) generated by the smooth pseudodistance ρ = ρe = Γ(·, e)1/(2−Q) , see [DGN5]. For any R > 0 we have (12.4)

PX (E ∩ BX (e, R); G) ≤ PX (E; BX (e, R)) + PX (BX (e, R); E) .

48


Thanks to the smoothness of BX (e, R) we have from Lemma 12.5 Z Z PX (BX (e, R); E) = |Y X | dHN −1 = ∂BX (e,R)∩E

∂BX (e,R)∩E

|Xρ| dHN −1 . |∇ρ|

Recalling that Γ(·, e) is homogeneous of degree 2 − Q, and therefore ρ is homogeneous of degree one, we infer that for some constant C(G) > 0, (12.5)

|Xρ(g)| ≤ C(G) .

This gives (12.6)

Z PX (BX (e, R); E) ≤ C(G)

By Federer co-area formula [Fe], we obtain Z Z ∞ Z |E| = χE dg = G

0

∂BX (e,R)∩E

∂BX (e,t)∩E

dHN −1 . |∇ρ|

dHN −1 dt < ∞ , |∇ρ|

therefore there exists a sequence Rk % ∞ such that Z dHN −1 (12.7) −→ 0 . k→∞ ∂BX (e,Rk )∩E |∇ρ| Using (12.7) in (12.6) we find (12.8)

lim PX (BX (e, Rk ); E) = 0 .

k→∞

From (12.4), (12.8), we conclude (12.9)

lim PX (E ∩ BX (e, Rk ); G) ≤ PX (E; G) .

k→∞

We next apply Theorem 12.4 to the bounded X-Caccioppoli set E ∩ BX (e, Rk ) obtaining |E ∩ BX (e, Rk )|(Q−1)/Q ≤ Ciso PX (E ∩ BX (e, Rk ); G) . Letting k → ∞ in the latter inequality, from (12.9), and from the relation lim |E ∩ BX (e, Rk )|(Q−1)/Q = |E|(Q−1)/Q ,

k→∞

we conclude that |E|(Q−1)/Q ≤ Ciso PX (E; G) . This completes the proof. ¤ Another type of isoperimetric inequalities are the so-called relative ones. They are essentially different in nature from the global ones and play a fundamental role in many geometric problems. Although we will not use them in the remainder of this paper, we nonetheless list them because of their relevance and for completeness. The first result is a special case of Theorem 1.18 in [GN1]. We recall that the notation P S indicates the class of Poincaré-Sobolev domains, which is essentially the largest one for which relative isoperimetric, or equivalently Gagliardo-Nirenberg inequalities hold. In fact, the following inclusions hold [GN1] N T A ⊂ (², δ) ⊂ P S ,


49

where we have respectively denoted by N T A and (², δ) the class of non-tangentially accessible, and that of (², δ)-domains with respect to the CC metric d(g, g 0 ) in G. Theorem 12.7. Let G be a Carnot group with homogeneous dimension Q. There exists a constant Creliso (G) > 0 such that for any P S domain, and every X-Caccioppoli set Ω ⊂ G, one has ½ ¾ Q−1 Q diam(Ω) c min |E ∩ Ω| , |E ∩ Ω| ≤ Creliso PX (E; Ω) . |Ω|1/Q In particular, since metric balls are P S domains, one obtains for g ∈ G, and R > 0, ½ ¾ Q−1 Q −1/Q c min |E ∩ B(g, R)| , |E ∩ B(g, R)| ≤ 2 ωG Creliso PX (E; B(g, R)) , where we have let ωG = |B(e, 1)|. It is worth observing at this moment that, in the abelian case when G = Rn , with its standard basis X = { ∂x∂ 1 , ..., ∂x∂n }, then Theorem 12.7 gives back the classical relative isoperimetric inequality due to De Giorgi, Federer, Maz’ya.

Since clearly PX (E; B(g, R)) ≤ PX (E; G), using the fact that B(g, R) % G, and letting R % ∞ in the second inequality of Theorem 12.7, we obtain the following useful consequence. Corollary 12.8. Let G be a Carnot group with homogeneous dimension Q. There exists a constant Creliso = Creliso (G) > 0, such that for every X-Caccioppoli set E ⊂ G, one has ½ ¾ Q−1 Q −1/Q c min |E| , |E | ≤ 2 ωG Creliso PX (E; G) . In [CG] it was proved that in a Carnot group of step 2 the gauge pseudo-balls Bρ (g, r), defined in (2.14), are N T A, and therefore P S domains, with respect to the Carnot-Carathéodory distance. We therefore obtain from the first part of Theorem 12.7. Theorem 12.9. Let G be a Carnot group of step 2 having homogeneous dimension Q. There exists a constant Creliso = Creliso (G) > 0, such that for every gauge pseudo-ball Bρ (g, r) ⊂ G, and any X-Caccioppoli set E ⊂ G, one has ½ ¾ Q−1 Q c ≤ Creliso PX (E; Bρ (g, r)) . min |E ∩ Bρ (g, r)| , |E ∩ Bρ (g, r)|

13. Existence of isoperimetric sets in Carnot groups We now turn to the basic question of the existence of the isoperimetric sets in Carnot groups. In view of the equivalence established in [GN1] between (i) of Theorem 12.6 and the embedding

50


of Gagliardo-Nirenberg type ½Z ¾(Q−1)/Q Q/(Q−1) (13.1) |u| dg ≤ Ciso V arX (u; G) , G

valid for any u ∈ BVX (G) with supp u bounded, we recognize at once that the existence of the isoperimetric minimizers also provides an answer to that of the existence of minimizers for (13.1). We mention that for the case p > 1 the existence of minimizers in the horizontal Sobolev embedding due to Folland and Stein [F2] was proved by D. Vassilev in his Ph.D. Dissertation [Vas]. In their deep paper [JL3] Jerison and Lee proved that when p = 2 all minimizers with finite energy in Hn are given by dilations and translations of the function ¶(Q−2)/4 µ 1 , φ(g) = (1 + |z|2 )2 + 16t2 and they also computed the Sobolev constant in the Sobolev embedding for Hn . In [GV2] the second named author and D. Vassilev proved that in any group G of Iwasawa type, all finite energy minimizers with spherical symmetry in the first layer must in fact be translated and dilated of the function µ ¶(Q−2)/4 m(Q − 2) (13.2) K(g) = , g ∈ G, (1 + |x(g)|2 )2 + 16|y(g)|2 where m = dim(V1 ), and (x(g), y(g)) represent the exponential coordinates of the point g ∈ G. This result generalizes, in part, that of Jerison and Lee to the higher codimensional case. The purpose of this section is to establish the existence of the isoperimetric sets. We mention that a different proof of this result has been recently independently given by Leonardi and Rigot [LR]. We begin by introducing the main object of present study. Definition 13.1. Given a Carnot group G with homogeneous dimension Q we define the isoperimetric constant of G as PX (E; G) αiso (G) = inf , E⊂G |E|(Q−1)/Q where the infimum is taken on all X-Caccioppoli sets E such that 0 < |E| < ∞.

Remark 13.2. We stress that, thanks to Theorem 12.6, the isoperimetric constant is strictly positive.

Proposition 13.3. In a Carnot group G one has for every measurable set E ⊂ G and every r>0 PX (E; G) = rQ−1 PX (δ1/r E; G) .

(13.3)

Proof. Let E ⊂ G be a measurable set. If ζ ∈ Co1 (G, HG), then the divergence theorem, and a rescaling, give Z X Z X Z X m m m (13.4) Xj∗ ζj (g) dg = − Xj ζj (g) dg = − rQ Xj ζj (δr g) dg , E j=1

E j=1

Er j=1


51

where we have let Er = δ1/r (E) = {g ∈ G | δr g ∈ E}. Since Xj (ζj ◦ δr ) = r (Xj ζj ◦ δr ) , we conclude (13.5)

Z

m X

E j=1

Z Xj∗ ζj dg = rQ−1

m X

E j=1

Xj∗ (ζj ◦ δr ) dg .

Formula (13.5) implies the conclusion. ¤ Proposition 13.3 asserts that the X-perimeter scales appropriately with respect to the nonisotropic group dilations. Since on the other hand one has |δ1/r E| = r−Q |E|, we easily obtain the following important scale invariance of the isoperimetric quotient. Proposition 13.4. For any X-Caccioppoli set in a Carnot group G one has (13.6)

PX (δ1/r E; G) PX (E; G) = , (Q−1)/Q |E| |δ1/r E|(Q−1)/Q

r>0.

Another equally important property which is however a trivial consequence of the left-invariance on the vector fields X1 , ..., Xm , and of the definition of X-perimeter, is the translation invariance of the isoperimetric quotient. Proposition 13.5. For any X-Caccioppoli set in a Carnot group G one has (13.7)

PX (Lgo (E); G) PX (E; G) = , (Q−1)/Q |Lgo (E)| |E|(Q−1)/Q

go ∈ G ,

where Lgo g = go g is the left-translation on the group. A fundamental property of the space BVX is the following special case of the compactness Theorem 1.28 proved in [GN1]. Theorem 13.6. Let Ω ⊂ G be a PS (Poincaré-Sobolev) domain. The embedding i : BVX (Ω) ,→ Lq (Ω) is compact for any 1 ≤ q < Q/(Q − 1). Combining Theorem 13.6 with Propositions 13.4 and 13.5 we can now establish the existence of isoperimetric sets. Theorem 13.7. Let G be a Carnot group, then there exists a bounded X-Caccioppoli set Fo (in fact, we show that Fo ⊂ B(e, 1)), such that PX (Fo ; G) = αiso (G) |Fo |(Q−1)/Q . The equality continue to be valid if we replace Fo by Lgo ◦ δλ (Fo ), for any λ > 0, go ∈ G.

52


Before proving Theorem 13.7, we analyze a technical point which is crucial in the proof of the above theorem. We define ½ ¾ PX (E; G) βiso (G) = inf | E ⊂ G, E is bounded . |E|(Q−1)/Q Proposition 13.8. αiso (G) = βiso (G) . Proof. Clearly we have αiso (G) ≤ βiso (G). We need only to establish the reverse inequality. But this is precisely what we have done in the proof of Theorem 12.6. Let in fact E be an arbitrary X-Caccioppoli set with |E|, PX (E; G) < ∞. Then for any R > 0 we have PX (E ∩ BX (e, R); G) ≥ βiso (G) . |E ∩ BX (e, R)|(Q−1)/Q

(13.8)

Let now {Rk }k∈N be the sequence in the proof of Theorem 12.6. As before we obtain (13.9)

lim

k→∞

PX (E ∩ BX (e, Rk ); G) PX (E; G) = . |E ∩ BX (e, Rk )|(Q−1)/Q |E|(Q−1)/Q

Thus, taking R = Rk in (13.8) and letting k → ∞, together with (13.9), we obtain βiso (G) ≤

PX (E ∩ BX (e, Rk ); G) PX (E; G) −→ . (Q−1)/Q k→∞ |E|(Q−1)/Q |E ∩ BX (e, Rk )|

By the arbitrariness of the X-Caccioppoli set E ⊂ G, taking the infimum in the latter inequality, we conclude αiso (G) ≥ βiso (G) . ¤ We are now ready to prove Theorem 13.7. Proof of Theorem 13.7. We define yet another constant: µ γiso (G) = inf {PX (E; G) | E ⊂ B(e, Ro ) and |E| = 1} where Ro =

2 |B(e, 1)|

¶1

Q

.

Clearly we have γiso (G) ≥ βiso (G). Once we show γiso (G) ≤ βiso (G), then in view of proposition 13.8, we can replace αiso (G) with γiso (G) in the statement of Theorem 13.7. To this end, let ² > 0 be given and consider {Ek }k∈N , a sequence of bounded X-Caccioppoli sets which is minimizing for βiso (G). For every k ∈ N let rk > 0 be such that δ1/rk (Ek ) ⊂ B(e, Ro ). Since Ek is bounded it is of course possible to select such a number rk . In view of Proposition 13.4 we have PX (δrk (Ek ); G) PX (Ek ; G) γiso (G) ≤ = ≤ βiso (G) + ² |δrk (Ek )|(Q−1)/Q |Ek |(Q−1)/Q if k is sufficiently large. Since ² is arbitrary, we have γiso (G) ≤ βiso (G). Hence, combining with our trivial observation earlier and proposition 13.8 we have γiso (G) = βiso (G) = αiso (G). Next, we establish the Theorem with γiso (G) in place of αiso (G). To accomplish this, consider a minimizing sequence {Fk } for γiso (G), that is (i) |Fk | = 1 (ii) Fk ⊂ B(e, Ro ) (iii) PX (Fk , G) < ∞


53

Consider the sequence of characteristic functions {χFk }k∈N . We claim that there exists C > 0 independent of k ∈ N, such that ||χFk ||BVX (B(e,Ro )) ≤ C .

(13.10)

To prove this claim notice first of all that ||χFk ||L1 (B(e,Ro ) = |Fk | = 1 . On the other hand, since {Fk } is a minimizing sequence for γiso (G), for large enough k we have PX (Fk ; B(e, Ro )) = PX (Fk ; G) ≤ γiso (G) + 1 . Discarding the first few terms of the sequence {Fk } if we must, we now have established (13.10). We can now invoke Theorem 13.6 to find a subsequence, which we continue to denote by {Fk }k∈N , and a function χ ∈ L1 (B(e, Ro )), such that χFk −→ χ k→∞

in L1 (B(e, Ro )) ,

χFk −→ χ k→∞

dg − a.e. in B(e, Ro ) .

The convergence a.e. implies that χ must be a characteristic function, i.e., χ = χFo for some measurable set Fo ⊂ B(e, Ro ). Since |Fk | → |Fo | We note explicitly that |Fo | = 1. Furthermore, by the lower semi-continuity of the perimeter we have PX (Fo , B(e, Ro )) ≤ lim inf PX (Fk ; B(e, Ro )) < ∞ , k→∞

and so Fo is a X-Caccioppoli set. The latter inequality also implies γiso (G) ≤ PX (Fo ; G) ≤ lim inf PX (Fk ; G) = γiso (G) . k→∞

We have thus proved that PX (Fo , G) = γiso (G) = γiso (G)|Fo |

Q−1 Q

.

This shows the existence of an isoperimetric set Fo ⊂ B(e, Ro ) with |Fo | = 1. By the translation and dilation invariance of the isoperimetric quotient we now reach the conclusion of the theorem. ¤ Once we know that isoperimetric sets do exist, and we also know that they are bounded X-Caccioppoli sets, then we can invoke the full strength of the theory developed in [DGN5]. For instance, we know that if Fo is an isoperimetric set, then every point go of its X-reduced ? F has positive density . boundary ∂X o

14. Isoperimetric sets with partial symmetry in Hn In Theorem 13.7 we have established the existence of the isoperimetric sets in any Carnot group. Of course, this leaves open the fundamental question of the characterization of such sets. In what follows we provide some interesting progress in this direction by finding the minimizer

54


of the X-perimeter among sets with cylindrical symmetry subject to a volume constraint in the Heisenberg group Hn , see Theorem 14.6 below. A remarkable property of these sets is that, similarly to their Riemannian predecessors, they have constant X-mean curvature. We conjecture that such sets, along with their left-translated and dilated, are all the isoperimetric sets in Hn . We plan to come back to these challenging problems in a future study. Our starting point presently is the representation of the X-perimeter in Lemma 12.5. Given a Carnot group G and a piece-wise C 1 domain given by a defining function φ, (14.1)

E = {g ∈ G | φ(g) < 0} ,

the latter gives (14.2) where Y X

Z

Z

|Xφ| dHN −1 , ∂E ∂E |∇φ| is introduced in Definition 5.4, and N indicates the topological dimension of G. PX (E; G) =

|Y X | dHN −1 =

Definition 14.1. A X-Caccioppoli set E ⊂ G is called an isoperimetric set if given V > 0, PX (E; G) is minimal among all X-Caccioppoli sets F ⊂ G with |F | = V . In what follows we restrict our attention to G = Hn and we establish some symmetry results of isoperimetric sets. Consider the map O : Hn → Hn defined by O(x, y, t) = (y, x, −t) . It is easy to see that the map O preserves Lebesgue measure (which is a bi-invariant Haar measure on Hn ). Lemma 14.2. Let E ⊂ Hn be measurable, then |E| = |O(E)|. Proof. Observe that |det(JacO )| = 1, and therefore Z Z |O(E)| = dg = |det(JacO )| dg = |E|. O(E)

E

¤ Perhaps, it is somewhat surprising that, under no additional symmetry assumptions, O also preserves the X-perimeter. Lemma 14.3. Let E ⊂ Hn be a piecewise C 1 domain with PX (E, Hn ) < ∞, then PX (O(E), Hn ) = PX (E, Hn ) . Proof. We give the proof for the first Heisenberg group H1 , leaving it to the reader to provide the easy modifications necessary to cover the higher-dimensional case. Suppose that E is as in (14.1) for some piecewise C 1 function φ : H1 → R with ∇φ > 0 on ∂E. Let W ⊂ R2 be a connected open set, and A : W → A(W ) ⊂ ∂E be a coordinate patch on a smooth part of ∂E, then Z Z q |Xφ| |Xφ ◦ A| dσ = EA GA − FA2 dα dβ, |∇φ| |∇φ ◦ A| A(W ) W


55

q where

EA GA − FA2 is the standard Riemannian volume density induced by A on A(W ). On

the other hand, it is easy to see that the defining function ψ of O(E) is given by ψ = φ ◦ O−1 def and A˜ = O ◦ A : W → ∂O(E) is a coordinate patch on a smooth portion of ∂O(E). We thus have

Z ˜ A(W )

|Xψ| dσ = |∇ψ|

Z W

˜q |X(φ ◦ O−1 ) ◦ A| EA˜ GA˜ − FA2˜ dα dβ, ˜ |∇(φ ◦ O−1 ) ◦ A|

q ˜ ). From the where EA˜ GA˜ − FA2˜ is the Riemannian volume density induced by A˜ on A(W definition of the map O it is easy to verify that (i) (ii) (iii) (iv)

O−1 = O. ˜ = |Xφ ◦ A|. X1 ψ ◦ O = X2 φ, X2 ψ ◦ O = X1 φ. Hence, |Xψ ◦ A| ˜ = |∇φ ◦ A| . ∇ψ ◦ O = (φy , φx , −φt ). Hence, |∇ψ ◦ A| EA = EA˜ , GA = GA˜ , FA = FA˜ .

Using (i)-(iv) we readily conclude that Z Z |Xφ| |Xψ| (14.3) dσ = dσ . ˜ A(W ) |∇φ| A(W ) |∇ψ| Since ∂E is a piecewise C 1 manifold, both ∂E and ∂O(E) can be covered by the same sequence of domains of integration W1 , W2 , ... (finite sequence if ∂E is compact, see, e.g., [STh]), and therefore we conclude from (14.3) that Z Z |Xφ| |Xψ| 1 PX (E; H ) = dσ = dσ = PX (O(E); H1 ). ∂E |∇φ| ∂O(E) |∇ψ| ¤ Remark 14.4. The proof of Lemma 14.3 shows in fact that for any open set Ω ⊂ Hn one has PX (O(E); O(Ω)) = PX (E; Ω). Using the above two lemmas, we can establish the following symmetry result for isoperimetric sets. In the sequel, we let Hn+ = {(x, y, t) ∈ Hn | t > 0} and Hn− = {(x, y, t) ∈ Hn | t < 0}. Theorem 14.5. Let E = {g ∈ Hn | φ(g) < 0} where φ is a piecewise C 1 function on Hn and E has partial symmetry, that is the defining function φ of E satisfies (14.4)

φ(z, t) = Φ(|z|2 , t)

for some Φ ∈ C 1 .

Suppose E is an isoperimetric set satisfying |E ∩ Hn+ | = |E ∩ Hn− | = |E|/2, then PX (E; Hn+ ) = PX (E; Hn− ) . Proof. We argue by contradiction, and assume that E satisfies |E ∩ Hn+ | = |E ∩ Hn− | = |E|/2, but (14.5)

PX (E; H1+ ) < PX (E; H1− ).

Let O be the mapping in Lemma 14.3 and F = (E ∩ H1+ ) ∪ O(E ∩ H1+ ) .

56


That is, F is obtained from E by taking the top half part of E together with the “reflection” of the top part of E across the horizontal plane {(z, t) ∈ Hn | t = 0}. We define ψ : Hn → R by ( φ(x, y, t) if t ≥ 0 ψ(x, y, t) = φ ◦ O(x, y, t) if t < 0 . Clearly, ψ is a piecewise C 1 function, and F = {g ∈ Hn | ψ(g) < 0}. Next, we observe that since E has partial symmetry, and the map O preserves |z|, we conclude O(E ∩{t = 0}) = E ∩{t = 0}. Hence, the construction of the set F do not introduce any “extra perimeter”. Now since ∂F is piecewise C 1 we have by Lemma 14.3, Remark 14.4, and by (14.5) PX (F ; H1 ) = PX (F ; H1+ ) + PX (F ; H1− ) = PX (E; H1+ ) + PX (O(E ∩ H1+ ); H1− ) = PX (E; H1+ ) + PX (O(E ∩ H1+ ); O(H1+ )) = 2PX (E; H1+ ) < PX (E; H1 ) . On the other hand, by the assumption on |E| and Lemma 14.2 we have |F | = |E ∩ H1+ | + |O(E ∩ H1+ )| = |E ∩ H1+ | + |E ∩ H1+ | = |E ∩ H1+ | + |E ∩ H1− | = |E|. This proves that E is not an isoperimetric set, thus giving a contradiction.

¤

In order to state the main result of this section, we introduce the following collection of sets E = {E ⊂ Hn | E

satisfies (i) − (ii)} ,

where (i) |E ∩ Hn+ | = |E ∩ Hn− | ; (ii) There exist R > 0, and C 1 functions u, v : [0, R] → R satisfying u(R) = v(R) = 0, such that ∂E ∩ Hn+ = {(z, t) | t = u(|z|2 /4)} and ∂E ∩ Hn− = {(z, t) | t = v(|z|2 /4)} . Theorem 14.6. Given V > 0, the variational problem min{PX (E; Hn ) | E ∈ E, |E| = V } has a unique solution Eo ∈ E. Furthermore, ∂Eo is given explicitly as the graph of the function ( Ã ! ) 2 2 1 p 2 R |z| πR −1 p t = ± |z| R − |z|2 − tan , |z| ≤ R . + 4 4 8 R2 − |z|2 n n The sign ± depends on whether we are considering ∂E ³ o ∩ H2+´, or ∂Eo ∩ H− . Finally, the set Eo is of class C 2 near its two characteristic points 0, ± πR , smooth away from them, and 8 S = ∂Eo has positive constant X-mean curvature given by

HX =

Q−2 1 . Q−1 R

Before giving the proof of Theorem 14.6, we recall some classical results from calculus of variations.


57

Definition 14.7 ( [Tr], p.56 ). A function f : R3 → R is said to be strongly convex on S ⊂ R3 if f = f (x, y, z) and its partial derivatives fy and fz are defined and continuous on this set and for every (x, y, z) and (x, y + v, z + w) ∈ S we have (14.6)

f (x, y + v, z + w) − f (x, y, z) ≥ fy (x, y, z)v + fz (x, y, z)w ,

with strict inequality except when v = 0 or w = 0. If the inequality (14.6) is not strict, we simply say that f is convex. Theorem 14.8 ( [Tr], p. 74 ). If D is a domain in R2 , such that for some constants λj , j = 1, 2, ..., p, f (x, y, z) and λj gj (x, y, z) are convex on [a, b] × D and at least one of these functions is strongly convex on this set, let h = f +

p X

λj gj .

j=1

The solution yo of the Euler-Lagrange differential equation d (14.7) hz (x, y(x), y 0 (x)) − hy (x, y(x), y 0 (x)) = 0 dx is the unique minimizer on

on (a, b)

D = {y ∈ C 1 [a, b] | y(a) = yo (a), y(b) = yo (b); (y(x), y 0 (x)) ∈ D} of the functional

Z

b

F (y) =

f (x, y, y 0 ) dx

a

under the constraints def

Gj (y) =

Z a

b

gj (x, y, y 0 ) dx = Gj (yo ),

j = 1, 2, ..., p .

We note that Definition 14.7 differs slightly from the ordinary notion of convexity in that we do not require the function to be convex (in the usual sense) in the first variable x. If we treat f as a function of (y, z), then the above notion of convexity is equivalent to the classical notion for f (x, ·, ·). We also point out that condition (14.6) can be replaced by the standard geometric definition (again, treating the first variable x as a constant) without referring to the gradient of f with respect to the variables y and z. Theorem 14.9. Given V > 0, there exist λ and R > 0 such that the function uo given by ¶ µ √ 2 πR2 1 p R 2 s −1 √ + (14.8) uo (s) = s(R2 − 4s) − tan . 2 4 8 R2 − 4s minimizes uniquely Z (14.9)

0

F (s, u, u ) = 2

2n

σ2n−1

R2 /4 p

1 + u0 (s)2 s(Q−3)/2 ds ,

0

subject to the constraint (14.10)

0

def

Z 2n

G(s, u, u ) = 2

σ2n−1

0

R2 /4

u(s) s(Q−4)/2 ds = V

58


over the set D = {u ∈ C 1 [0, R2 /4] | u(R2 /4) = 0} 2

Proof. Consider the functions f, g : [0, R4 ] × R+ × R given by p f (x, y, z) = 22n σ2n−1 x(Q−3)/2 1 + z 2 , g(x, y, z) = 22n σ2n−1 x(Q−4)/2 y . Now fy (x, y, z) = 0 gz (x, y, z) = 0

z , fz = 22n σ2n−1 x(Q−3)/2 √ 1 + z2 gy = 22n σ2n−1 x(Q−4)/2 .

√ Clearly, f is strongly convex according to definition 14.7 if the function h(z) = 1 + z 2 is 3 strictly convex in the usual sense. This conclusion follows from h00 (z) = (1 + z 2 ) 2 > 0. Similarly, we verify easily that λg is convex according to definition 14.7. Claim: given V > 0, one can determine R and λ (equivalently β) such that the function uo : [0, R2 /4] → R, the solution to the Euler-Lagrange equation (which depends on R and λ) given by (14.8) satisfies the contraint (14.10). Again, integration by parts and using (14.23) gives Q−1 Z R2 Z R2 4 4 s 2 2n (Q−4)/2 p 2 σ2n−1 u(s) s ds = Cn ds β2 − s 0 0 where 22n+1 σ2n−1 Q−2 Cn = , β = . Q−4 2λ This forces β < −R/2 since β 2 is under the square root sign (recall that β < 0). We observe that µ 2 ¶ Q−1 Q−1 Z R2 Z R2 2 4 4 s 2 R 1 p p lim ds ≤ ds = 21−Q RQ . lim − 2 2 R− 0 R 4 β −s β −s 0 β→− 2 β→− 2 This shows that if we choose β = −R/2, we have an integrable singularity. That is the function Q−1 Z R2 4 s 2 p h(R) = ds ≤ 21−Q RQ 2 R /4 − s 0 Observe that with this choice of β we have µ 2 ¶ Q−1 Q−1 Q−1 Z R2 Z R2 Z R2 2 4 4 4 3 s 2 s 2 R 1 p p p h(R) = ds ≥ ds ≥ ds = 21− 2 Q RQ . 2 2 2 2 2 R R 8 R /4 − s R /4 − s R /4 − s 0 8 8 The computations above shows h(R) → ∞ as R → ∞. Obviously, we also have h(0) = 0. It is easy to see that h is continuous on [0, ∞). Hence, given any V > 0, the intermediate value theorem gives an R = R(V ) ∈ (0, ∞) for which V . Cn This establishes our claim. To complete the proof, we note that once a choice of β is made, existence of the λ is established. Appealing to Theorem 14.8 we also obtain uniqueness. This h(R) =


59

shows that there is only one choice of the parameter β = −R/2, and with this choice of β, the function uo given by (14.8) is the unique minimizer of the variational problem. ¤ Proof of Theorem 14.6. Let E ∈ E be given and let u : [0, R] → R+ be the function describing ∂E ∩ Hn+ . The parameter R will be specified later. First, we derive the Euler-Lagrange equation associated to this problem. From assumption (i) in the definition of the collection E, from Theorem 14.5, and letting φ(z, t) = t − u(|z|2 /4) in (14.2), we obtain q Z r2 0 2 4 (1 + (u ) ) n q PX (E; H ) = 2 dHN −1 . 2 |z| 0.

64


Letting

r f (x, y, u, z1 , z2 ) =

y x (z1 − )2 + (z2 + )2 , 2 2

g(x, y, u, z1 , z2 ) = u

and finally h = f + λg , the above constrainted variation problem is then equivalent to the one without constraint. That is, minimize Z (14.33) h(x, y, u(x, y), ux (x, y), uy (x, y)) dxdy Ω

among the set D defined in (14.31) After an elementary computation we found the Euler-Lagrange equation of (14.33) to be (14.34)     y x ux (x, y) − 2 uy (x, y) + 2 ∂   + ∂ q  = λ. q y 2 y 2 ∂x ∂y x 2 x 2 (ux (x, y) − 2 ) + (uy (x, y) + 2 ) (ux (x, y) − 2 ) + (uy (x, y) + 2 ) Solving (14.34) directly is difficult. However, when u takes a special form, (i.e., cylindrically symmetric say) then (14.34) reduces to (14.20) and (14.8) is a solution (with s = (x2 + y 2 )/4). Our goal is to show that (14.8) is the unique minimizer of the variational problem in consideration. Theorem 14.12. Given V > 0, there exists λ > 0 and R > 0 such that the function φ(x, y) = uo ((x2 + y 2 )/4) where uo is from (14.8) is the unique minimizer of the variational problem (14.33). Proof. Clearly, the function g is convex. It is equally obvious that f is a strictly convex function of the arguments (u, z1 , z2 ). In view of the remarks following Theorem 14.8, and proceeding as in the proof of Theorem 14.8, we reach the conclusion. ¤ Remark 14.13. If one can demonstrate that certain symmetrization process that takes a function u : Ω ⊂ R2 → R+ into one that is cylindrically symmetric u? : B(0, R) → R+ , but at the same time, diminishes the perimeter of the graph of u, then Theorem 14.12 is superfluous. However, this is a difficult and an open problem and therefore, Theorem 14.12 can be considered as another symmetry result for isoperimetric sets. Remark 14.14. For the first Heisenberg group H1 the idea of using the Euler-Lagrange equation (14.15) above arose in some computations that Giorgio Talenti and the second named author carried in a set of unpublished notes in Oberwolfach in 1995. In his paper [Pa] Scott Pauls has independently computed the function (14.8) for the first Heisenberg group H1 , by directly solving the equation in Proposition 6.8. The isoperimetric sets in Hn in Theorem 14.6 were discovered by us in 1997 [DGN0]. The results in this section were presented by the second named author in the lecture: ”Remarks on the best constant in the isoperimetric inequality for the Heisenberg group


65

and surfaces of constant mean curvature”, Analysis seminar, University of Arkansas, April 12, 2001, (http://comp.uark.edu/ lanzani/schedule.html). They have been circulated in the preprint [DGN0] since 1999.

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[Vas] D. N. Vassilev, Yamabe equations in Carnot groups, Ph. D. Dissertation, Purdue University, 2000. Department of Mathematics, Purdue University, West Lafayette, IN 47907 E-mail address, Donatella Danielli: [email protected] Department of Mathematics, Purdue University, West Lafayette, IN 47907 E-mail address, Nicola Garofalo: [email protected] Department of Mathematics, Georgetown University, Washington DC 20057-1233 E-mail address, Duy-Minh Nhieu: [email protected]