arXiv:1812.05022v2 [math.DG] 6 Feb 2019

arXiv:1812.05022v1 [math.DG] 12 Dec 2018

SHARP GEOMETRIC INEQUALITIES FOR CLOSED HYPERSURFACES IN MANIFOLDS WITH NONNEGATIVE RICCI CURVATURE. VIRGINIA AGOSTINIANI, MATTIA FOGAGNOLO, AND LORENZO MAZZIERI

Abstract. In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension n ≥ 3. For every bounded open subset Ω ⊂ M with smooth boundary, we prove that ˆ n−1 H n−1 , n − 1 dσ ≥ AVR(g) S ∂Ω

where H is the mean curvature of ∂Ω and AVR(g) is the asymptotic volume ratio of (M, g). Moreover, the equality holds true if and only if (M \Ω, g) is isometric to a truncated cone over ∂Ω. An optimal version of Huisken’s Isoperimetric Inequality for 3-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue’s non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.

1. Introduction and main results The classical Willmore inequality [70] for a bounded domain Ω of R3 with smooth boundary says that ˆ H 2 dσ ≥ 16π, (1.1) ∂Ω

where H is the mean curvature of ∂Ω. Such an inequality has been extended in [17] to submanifolds of any co-dimension in Rn , for n ≥ 3. In particular, for a bounded domain Ω in Rn with smooth boundary there holds ˆ H n−1 n−1 |, (1.2) n − 1 dσ ≥ |S ∂Ω

with equality attained if and only if Ω is a ball. Implicit in this statement is the fact that the underlying metric by which H and dσ are computed is the Euclidean metric gRn . Note that the above rigidity statement can be rephrased by saying that the equality in (1.1) is fulfilled if and only if (∂Ω, g∂Ω ), where g∂Ω is the metric induced by gRn on the submanifold ∂Ω, is homothetic to (Sn−1 , gSn−1 ), where gSn−1 is the standard round metric. Recently, in [1], the Willmore-type inequality (1.2) and the corresponding rigidity statement have been deduced as a consequence of suitable monotonicity-rigidity properties of the function ˆ −(n−1) |Du|n−1 dσ, t ∈ (0, 1], (1.3) U (t) := t {u=t}

associated with the level set flow of the electrostatic potential u generated by the uniformly charged body Ω. In other words, u is the unique harmonic function in Rn \ Ω which vanishes at infinity and such that u = 1 on ∂Ω. More precisely, what is proven in [1] is that the function U is nondecreasing and that this monotonicity is strict unless Ω is a ball. Once this fact is established, the proof of (1.2) consists of a few lines. Indeed, exploiting first the global feature 1

2

V. AGOSTINIANI, M. FOGAGNOLO, AND L. MAZZIERI

of the monotonicity i.e. U (1) ≥ U (0+ ) and using then the asymptotic expansion at infinity of u and |Du| one gets ˆ |Du|n−1 dσ =: U (1) ≥ lim U (t) = (n − 2)n−1 |Sn−1 |. (1.4) t→0+

∂Ω

On the other hand, computing the derivative at t = 1 ˆ h U ′ (1) = (n − 2) |Du|n−2 H −

n−1 n−2

∂Ω

i |Du| dσ ,

and using U ′ (1) ≥ 0, we deduce that ˆ ˆ H n−1 Du n−1 n − 1 dσ ≥ n − 2 dσ, ∂Ω

(1.5)

∂Ω

where we have also applied the Hölder inequality. The coupling of the latter inequality with the former (1.4) yields the desired (1.2). In this paper, we show that the strategy described above can be adapted to a much more general setting, giving rise to a new Willmore-type inequality. More precisely, we obtain the following theorem, which is the main result of this paper. Theorem 1.1 (Willmore-type inequality). Let (M, g) be a complete noncompact Riemannian manifold with Ric ≥ 0 and Euclidean volume growth, of dimension n ≥ 3. If Ω ⊂ M is a bounded and open subset with smooth boundary, then ˆ H n−1 n−1 |, (1.6) n − 1 dσ ≥ AVR(g)|S ∂Ω

where AVR(g) ∈ (0, 1] is the asymptotic volume ratio of (M, g). Moreover, the equality holds if and only if (M \ Ω, g) is isometric to 1 n−1 |∂Ω| 2 r0 , +∞) × ∂Ω , dr ⊗ dr + (r/r0 ) g∂Ω , with r0 = . AVR(g)|Sn−1 | In particular, ∂Ω is a connected totally umbilic submanifold with constant mean curvature.

Let us remark that no connectedness assumption on Ω is required in the above statement. On the other hand, if equality holds in the Willmore-type inequality, then we get that ∂Ω is connected as a by-product of the rigidity statement combined with the fact that (M, g) has one end, see Proposition 2.10. We recall that since Ric ≥ 0 in the above statement, then, by the classical Bishop-Gromov Volume Comparison Theorem, the function (0, +∞) ∋ r 7−→ Θ(r) =

n|B(p, r)| r n |Sn−1 |

is nonincreasing. In particular, we have that the asymptotic volume ratio AVR(g) = lim Θ(r) r→+∞

is well defined. Moreover, we have that this limit does not depend on the point p ∈ M , and that limr→0+ Θ(r) = 1. Hence, we have that 0 ≤ AVR(g) ≤ 1. Moreover, AVR(g) = 1 if and only if (M, g) = (Rn , gRn ). Assuming Euclidean volume growth means assuming AVR(g) > 0. Observe in particular that for n = 3, if Ric ≡ 0 then (M, g) is isometric to (R3 , gR3 ) and consequently

SHARP GEOMETRIC INEQUALITIES FOR CLOSED HYPERSURFACES

3

AVR(g) = 1. On the other hand, for n ≥ 4 there exists an important class of complete noncompact Ricci flat Riemannian manifolds with 0 < AVR(g) < 1, that is the class of Ricci flat Asymptotically Locally Euclidean (ALE for short) manifolds. We refer the reader to Definition 4.13 for the precise notion. For the time being, we just recall that a n-dimensional Riemannian n manifold is ALE if it is asymptotic to (R \ {0})/Γ , gRn , where Γ is a finite subgroup of SO(n) acting freely on Rn \ {0}. This family of Riemannian manifolds is widely studied. In this regard, we first mention that in [6] it is proved that any Ricci flat manifold with Euclidean volume growth and strictly faster than quadratic curvature decay is actually ALE. Moreover, we point out that 4-dimensional Ricci flat ALE manifolds appear as important examples of gravitational instantons, that are noncompact hyperkhäler 4-manifolds with decaying curvature at infinity, introduced by Hawking in [34] in the framework of his Euclidean quantum gravity theory. An explicit example is given by the famous Eguchi-Hanson metric, introduced in [29], where n = 4, Ric ≡ 0 and Γ = Z2 . We remark that ALE gravitational instantons are completely classified in [41] and [42]. Concerning the general class of gravitational instantons, let us cite, after the important works of Minerbe [56, 57, 58], the recent PhD thesis [18], where gravitational instantons with strictly faster than quadratic curvature decay are classified. We refer the reader to this last mentioned work and to the references therein for a more complete picture on this subject. The following is Theorem 1.1 applied to ALE manifolds with nonnegative Ricci curvature. Corollary 1.2. Let (M, g) be an ALE Riemannian manifold without boundary with Ric ≥ 0, of dimension n ≥ 3. If Ω ⊂ M is a bounded and open subset with smooth boundary, then    |Sn−1 |  ˆ H n−1 , Ω ⊂ M bounded and smooth dσ inf = n − 1  card Γ 

(1.7)

∂Ω

Moreover, if equality holds, then M \ Ω is isometric to

r0 , +∞) × S

n−1

/Γ , dr ⊗ dr + r gSn−1 /Γ ,

2

with

r0 =

card Γ|∂Ω| |Sn−1 |

1 n−1

,

(1.8)

for some r0 > 0 and some finite subgroup Γ of SO(n). In particular, (∂Ω, g∂Ω ) is homothetic to n−1 S /Γ, gSn−1 /Γ .

We observe at once that on ALE manifolds the rigidity is much stronger, being characterized by cones whose cross sections are homothetic to Sn−1 /Γ. Notice also that if Γ is trivial one recovers the classical Willmore inequality (1.2). Moreover, we point out that (1.7) also says that, in every ALE manifold with nonnegative Ricci curvature, the lower bound we find for the Willmore-type functional is actually an infimum. This fact holds true for a larger class of manifolds. Indeed, as proved in Theorem 4.12, it is sufficient to assume the hypotheses of Theorem 1.1 together with a quadratic curvature decay condition. Understanding metric and topological consequences of curvature decay conditions is a very interesting and widely studied problem in geometric analysis. Dropping any attempt to be complete, we refer the interested reader to the aforementioned [6], to the seminal [15], to [62], where the case n = 3 is considered, to [73] and references therein. To make the picture more complete, let us also mention that Willmore-type inequalities are proven in [3] for asymptotically flat (AE) static metrics in the framework of General Relativity, and in [64] for integral 2-varifolds in Cartan-Hadamard manifolds.

4


Theorem 1.1 will be deduced as a consequence of the monotonocity-rigidity properties of the function U defined as in (1.3), where now u is the unique solution to the problem    ∆u = 0 in M \ Ω, (1.9) u = 1 on ∂Ω,   u(q) → 0 as d(O, q) → +∞,

with O being a fixed point in Ω and d a distance function on M . Observe that the hypotheses of Theorem 1.1, namely the nonnegativity of the Ricci tensor and the Euclidean volume growth, guarantee the existence of the solution to problem (1.9), as explained in Section 2. Once that the monotonicity-rigidity of U is known, the proof of Theorem 1.1 consists of exactly the same formal steps outlined in (1.4)–(1.5), the main difference being the careful computation of the limit value lim U (t) = AVR(g)(n − 2)n−1 |Sn−1 |. t→0+

We remark that whereas in the classical Euclidean context the limit was deduced from the pointwise asymptotic expansion of u and |Du| at infinity, here it will be deduced from some quite delicate integral asymptotic expansions, in the spirit of [23]. This is an important technical difference from the Euclidean case. Such integral asymptotics will be worked out in Section 4. For completeness, we state the monotonicity-rigidity result concerning U in the wider generality of Theorem 1.3 below. Indeed, the same monotonicity-rigidity properties are shared by the whole family of functions {Uβ }, with β ≥ (n − 2)/(n − 1), where Uβ : (0, 1] −→ R is defined as ˆ n−1 −β n−2 Uβ (t) = t |Du|β+1 dσ. (1.10) {u = t}

Note that Uβ coincides with the function U defined in (1.3) when β = n − 2. Moreover, such a Monotonicity-Rigidity Theorem holds for a wider class of manifolds than the ones with Euclidean volume growth. Namely, we prove it for any complete noncompact Riemannian manifold with Ric ≥ 0 admitting a solution to (1.9). This class of manifolds coincides with the thoroughly studied class of nonparabolic ones, as we are going to see in Section 2. Theorem 1.3 (Monotonicity-Rigidity Theorem for nonparabolic manifolds). Let (M, g) be a nonparabolic Riemannian manifold with Ric ≥ 0. Given a bounded and open subset Ω ⊂ M with smooth boundary, let u be the solution to problem (1.9) and let Uβ be the function defined in (1.10). Then, for every β ≥ (n − 2)/(n − 1), the function Uβ is differentiable, with derivative ˆ h i n−1 dUβ −β n−2 (t) = β t |Du|β H − n−1 |D log u| dσ, (1.11) n−2 dt {u=t}

where H is the mean curvature of the level set {u = t} computed with respect to the unit normal vector field ν = −Du/|Du|. The derivative of Uβ fulfills ˆ n−1 dUβ β u2−β n−2 |Du|β−2 Ric(Du, Du) (t) = 2 dt t {u 0 ≥´ β β dt ∂Ω |Du| dσ 1

where Uβ is defined in (1.10) and its derivative satisfies (1.12). (ii) If (M, g) is parabolic, then 1 1 dΨβ sup H∂Ω ≥ − ´ (0) ≥ 0, |Dψ|β dσ β ds ∂Ω ∂Ω

where Ψβ is defined in (1.16) and its derivative satisfies (1.17).

Kasue’s Theorem then follows as a corollary.

(1.18)

(1.19)


7

Corollary 1.7 (Kasue’s Theorem). Let (M, g) be a complete noncompact Riemannian manifold with Ric ≥ 0, of dimension n ≥ 3. Assume Ω ⊂ M is a bounded and open subset with smooth boundary. Assume also that H∂Ω ≤ 0 on ∂Ω. Then (M \ Ω, g) is isometric to a Riemannian product ([0, +∞) × ∂Ω, dr ⊗ dr + g∂Ω ). In particular, ∂Ω is a totally geodesic connected submanifold of (M, g). Theorem 1.1 and Corollary 1.7 can also be gathered in a single general statement, see Corollary 5.2. Remark 1.8. It is worth pointing out that if (M, g) is a Riemannian cylinder, then M \ Ω can have two connected components. In this situation, problem (1.15) has to be set in one of these two ends. The results above involving parabolic manifolds, and in particular Theorem 1.6, can still be proved with trivial modification in this case. Similarly, one can also deal with unbounded Ω, provided that ∂Ω is a compact hypersurface. Finally, we combine our sharp Willmore-type inequality (1.6) with curvature flows techniques along the lines of an argument presented by Huisken in [35]. We obtain a characterization of the infimum of the Willmore functional in terms of the isoperimetric ratio of 3-manifolds with nonnegative Ricci curvature, refining the analogous result stated in the aforementioned contribution. This is the content of the following theorem. Theorem 1.9 (AVR(g) = isoperimetric constant). Let (M, g) be a nonparabolic 3-manifold with Ric ≥ 0. Then ˆ H2 dσ |∂Ω|3 ∂Ω inf = inf = AVR(g), (1.20) 16π 36π|Ω|2 where the infima are taken over bounded and open subsets Ω ⊂ M with smooth boundary. In particular, the following isoperimetric inequality holds for any bounded and open Ω ⊂ M with smooth boundary |∂Ω|3 ≥ AVR(g). (1.21) 36π|Ω|2 Moreover, equality is attained in (1.21) if and only if M = R3 and Ω is a ball. Beside the characterization of the isoperimetric constant in terms of the Asymptotic Volume Ratio, the novelties with respect to [35] lie in the rigidity statement and in the fact that the infimum of the Willmore functional is taken over the whole class of bounded open subsets Ω with smooth boundary, and not just over outward minimizing subsets. All of these improvements substantially come from our optimal Willmore-type inequality (1.6). It is worth noticing that the above theorem can be rephrased in terms of a Sobolev inequality with optimal constant. This is the content of the following corollary. Corollary 1.10 (AVR(g) = Sobolev constant). Let (M, g) be a Riemannian manifold with Ric ≥ 0. Then 3 ˆ |Df | dσ M (1.22) inf ˆ 2 = AVR(g). 1,1 f ∈W0 (M ) 3/2 |f | 36π M

Once Theorem 1.9 is established, (1.22) is obtained by very standard tools. We refer the reader to [63, pages 89-90] for a complete proof of the well known equivalence between the isoperimetric and the Sobolev inequality. On this regard, it is worth observing that relations

8


between isoperimetry and mean curvature functionals date back to Almgren [5], while a first derivation of isoperimetric inequalities through a curvature flow has been obtained by Topping in the case of curves [67]. Isoperimetric inequalities in Rn and in Cartan-Hadamard manifolds through curvature flows have been established by Schulze in [65] and [64], while the application to manifolds with nonnegative Ricci curvature is suggested in the already mentioned [35]. Actually, as pointed out in the discussion following [20, Theorem 5.13], a positive isoperimetric/Sobolev constant for complete noncompact Riemannian manifolds with Ric ≥ 0 and Euclidean volume growth can be deduced via the techniques introduced by Croke in [27] and [28]. However, it is known that such constant is not optimal. Other strictly related issues about isoperimetry in noncompact manifolds with Ric ≥ 0 are treated in [59] and [19]. This paper is organized as follows. In Section 2 we review, for ease of the reader, the theory of harmonic functions on Riemannian manifolds with nonnegative Ricci curvature we are going to employ along this work. In Section 3 we introduce the conformal formulation of problem (1.9). In this setting, we prove (the conformal version of) Theorem 1.3. In Section 4 we work out the integral asymptotic estimates for the electrostatic potential on manifolds with nonnegative Ricci curvature. With these estimates at hand, we prove Theorem 1.1 and Corollary 1.2. In Section 5 we prove the Monotonicity-Rigidity Theorem for parabolic manifolds and deduce Theorem 1.6 and Corollary 1.7. In Section 6, we prove Theorem 1.9. Finally, we included an Appendix where we describe the relations between our monotonicity formulas and some of those obtained by Colding and Colding-Minicozzi in [21] and [26]. 2. Harmonic functions in exterior domains In this section we are mainly concerned with characterizing Riemannian manifolds for which problems (1.9) and (1.15) admit a solution. We are going to see that complete noncompact nonnegatively Ricci curved manifolds enjoying existence to (1.9) are the nonparabolic ones, namely, manifolds admitting a positive Green’s function, while those admitting a solution to (1.15) are the parabolic ones. Nothing of substantially new appears in this section, we are just re-arranging and applying classical results contained in [45], [46], [47], [49], [71], and [68]. See also the nice survey [33] , where in addition the relation with the Brownian motion on manifolds is explored, the lecture notes [44] and references therein for a more general account on the vast subject of harmonic functions on manifolds. Other important works in this field will be readily cited along the paper. Anyway, the results gathered in this preliminary section are spread in a huge literature, and frequently the results do not appear exactly in the form we need or a with a precise proof. For this reason, we include the most relevant ones. Important gradient bounds are discussed too. Along this section, we denote by D the Levi-Civita connection of the Riemannian manifold considered, and by ∆ the related Laplacian. For any two points p, q ∈ M , we let d(p, q) be their geodesic distance. Moreover, it is understood that we are always dealing with manifolds of dimension n ≥ 3. 2.1. Preliminaries. Let us first recall the following celebrated and fundamental gradient inequality, first provided by Yau in [71]. See also the nice presentation in [63]. Theorem 2.1 (Yau’s inequality). Let (M, g) be a complete noncompact Riemannian manifold with Ric ≥ 0. Let u be a positive harmonic function defined on a geodesic ball B(p, 2R) of center p and radius 2R. Then there exists a constant C = C(n) > 0 such that C |Du| ≤ . R x∈B(p,R) u sup

(2.1)


9

We now apply the above inequality to a harmonic function v defined in a geodesic annulus B(p, R1 ) \ B(p, R0 ). We obtain a decay estimate on the gradient of u that we will employ several times along this paper. Proposition 2.2. Let (M, g) be a complete noncompact Riemannian manifold with Ric ≥ 0, and let u be a positive harmonic function defined in a geodesic annulus B(p, R1 ) \ B(p, R0 ), with R1 > 3R0 . Then, there exists a geometric constant C = C(n) such that u , (2.2) |Du|(q) ≤ C d(p, q) R1 + R0 for any point q such that 2R0 ≤ d(p, q) < . In particular, if u is a harmonic function 2 defined in M \ B(p, R0 ), then u (2.3) |Du|(q) ≤ C d(p, q) for any point q with 2R0 ≤ d(p, q). R1 + R0 . Then the ball B q, d(p, q) − r0 is all 2 contained in the annulus B(p, R1 ) \ B(p, R0 ). In particular, by Yau’s inequality (2.1) we have Proof. Let q be such that 2R0 ≤ d(p, q)
0. Combining (2.6) with (2.11), we get that the minimal Green’s function goes to 0 at infinity, i.e. for any fixed p in M lim G(p, q) = 0. (2.9) d(p,q)→∞

12


An easy application of Laplace Comparison Theorem then gives the following well known fact. Lemma 2.12. Let (M, g) be a nonparabolic Riemannian manifold with Ric ≥ 0, and let G be its minimal Green’s function. Then, for any fixed pole p we have d2−n (p, q) ≤ G(p, q).

(2.10)

for any q 6= p in M . Proof. Let r be the function mapping a point q in M to d(p, q). By the Laplacian Comparison Theorem, we have n−1 ∆r ≤ r in the sense of distributions (see e.g. [20, Theorem 1.128]). Therefore, we have, in the sense of distributions, ∆r 2−n = (n − 2) (n − 1)r −n − r 1−n ∆r ≥ 0,

and then the the function r 2−n − G(p, ·) is sub-harmonic. By the maximum principle, for any ε > 0 and R > ε max

(r 2−n − G(p, ·)) =

B(p,R)\B(p,ε)

max

∂B(p,R)∪∂B(p,ε)

(r 2−n − G(p, ·)).

We conclude by passing to the limit as ε → 0 and R → ∞, taking into account the asymptotic behavior at the pole p given by (2.4) and that G → 0 at infinity, as observed in (2.9). Now, we identify existence of a solution to problem (1.9) with nonparabolicity of the ambient manifold. Let us first set up some notation that we are going to use in the rest of the paper. With respect to a bounded open subset Ω ⊂ M with smooth boundary, we denote by O a generic reference point taken inside Ω. Theorem 2.13. Let (M, g) be a complete noncompact Riemannian manifold with Ric ≥ 0, and let Ω ⊂ M be a bounded open subset with smooth boundary. Then, there exists a solution to problem (1.9) if and only if (M, g) is nonparabolic. Proof. The proof is an easy adaptation of arguments already presented in [46]. By Theorem 2.6 the existence of a solution to problem (1.9) implies nonparabolicity of M , since the restriction of u to M \ B(O, R) with Ω ⊂ B(O, R) clearly satisfies condition (2.5). Conversely, assume that M is nonparabolic, and consider an increasing sequence of radii {Ri }i∈N such that Ω ⊂ B(O, R1 ) and Ri → ∞. Let, for any i ∈ N, ui be the solution to the following problem:   ∆u = 0 in B(O, Ri ) \ Ω (2.11) u = 1 on ∂Ω   u = 0 on ∂B(O, Ri ).

Let now G be the minimal positive Green’s function, and consider the function G(O, ·). Due to the Maximum Principle for harmonic functions and boundary conditions in problem (2.11) we have G(O, q) . (2.12) 0 ≤ uRi (q) ≤ min∂Ω G(O, ·)

for q ∈ B(O, Ri ). Let then K be a compact set contained in M \ Ω. We can clearly suppose without loss of generality that K is contained in B(O, Ri )\Ω for any i. Then, (2.12) and Lemma


13

2.3 give that ui converges up to a subsequence to a harmonic function u on K. We can clearly extend by continuity u to 1 on ∂Ω. Again by (2.12), and by uniform convergence, 0 ≤ u(q) ≤

G(O, q) . min∂Ω G(O, ·)

on M \ Ω. Since, by (2.9), G(O, q) → 0 as d(O, q) → ∞, so does u, completing the proof.

We close this section with the following easy lemma, which shows that we can control function u by the minimal Green’s function G. Lemma 2.14. Let (M, g) be a nonparabolic Riemannian manifold with Ric ≥ 0, with minimal Green’s function G. Let u be a solution to (1.9) for some open and bounded Ω with smooth boundary, and let O ∈ Ω. Then, there exist constants C1 = C1 (M, Ω) > 0 and C2 = C2 (M, Ω) > 0 such that C1 G(O, q) ≤ u(q) ≤ C2 G(O, q) (2.13) on M \ Ω. In particular C1 d(O, q)2−n ≤ u(q).

(2.14)

on M \ Ω. Proof. Just set 0 < C1 < 1/ max∂Ω G(O, ·), and C2 > 1/ min∂Ω G(O, ·). Maximum principle and vanishing at infinity of u and G gives the claim. Inequality (2.14) is obtained just by combining the lower estimate on u by (2.13) with (2.10). 2.5. The exterior problem on parabolic manifolds. The following inequalities, proved in [47, Theorem 2.6], can be interpreted as a version for parabolic manifolds of the Li-Yau inequalities recalled in Theorem 2.11. We point out that we are always dealing with Green’s functions obtained by the Li-Tam’s construction. Theorem 2.15. Let (M, g) be a parabolic manifold with Ric ≥ 0, and let p ∈ M . Let G be a Green’s function. Then, for any fixed r0 > 0 and for any q with d(p, q) > 2r0 there holds ˆ d(p,q) r dr + C2 (2.15) −G(p, q) ≤ C1 |B(p, r)| r0 for some constants C and C1 depending only on n, r0 and the choice of G. Moreover, for any R > r0 , there holds ˆ R r dr + C4 ≤ sup [−G(p, ·)], C3 r0 |B(p, r)| ∂B(p,R)

(2.16)

for some constants C3 and C4 depending only on n, r0 and the choice of G When (M, g) is parabolic, Li-Tam proved in [46, Lemma 1.2] that the exterior problem (1.15) admits a solution. The construction of such a solution ψ, combined with Yau’s inequality and Theorem 2.15, readily implies a uniform gradient bound on ψ. Theorem 2.16. Let (M, g) be a parabolic Riemannian manifold with Ric ≥ 0, and let Ω ⊂ M be a bounded and open subset with smooth boundary. Then, there exists a solution to problem (1.15). Moreover, |Dψ| is uniformly bounded in M \ Ω. Remark 2.17. Recall from Subsection 2.3 that if (M, g) is a cylinder then M \ Ω can have two connected components. If this is the case, it will be understood that we consider problem (1.15) on a connected component of M \ Ω. All the proofs will work unchanged in this case.

14


Proof of Theorem 2.16. Let p ∈ Ω. Let U ⋐ Ω be an open neighborhood of p. Define the compact set K = Ω \ U . Consider, for a sequence B(O, Ri ) of geodesic balls containing Ω a corresponding sequence of positive Green’s functions Gi (O, ·) of B(O, Ri ) with pole in O such that Gi (O, x) = 0 for x ∈ ∂B(p, Ri ). We then consider the sequence of functions defined in B(O, Ri ) \ {p} defined by fi (q) = sup Gi (O, x) − Gi (O, q). x∈K

Then, the construction of a Green’s function in [45] imply that there exists a Green’s function G on M such that fi converges to −G(O, ·) uniformly on compact subsets of M \{O} (compare with the discussions around Lemma 1.2 in [46]). Observing that fi = supK Gi (O, ·) on ∂B(O, Ri ), we set Ci = sup Gi (O, x) x∈K

and consider the solution ψi to the problem    ∆ψ = 0 ψ=0   ψ = Ci

in M \ Ω on ∂Ω on B(O, Ri ).

Since supK Gi (O, ·) > sup∂Ω Gi (O, ·), the Maximum Principle immediately gives fi − sup fi ≤ ψi ≤ fi

(2.17)

∂Ω

on B(O, Ri ) \ Ω. Since fi converges, the second inequality in (2.17) and Lemma 2.3 show that ψi converges uniformly on compact subsets of M \ Ω to a harmonic function ψ, that we can clearly extend to 0 on ∂Ω. Moreover, since fi (q) converges to −G(O, q) and −G(O, qj ) → +∞ along a sequence of points qj such that d(O, qj ) → ∞, by (2.16), we get, using the first inequality in (2.17), that ψ(qj ) → +∞. In particular, since by [23, Lemma 3.40] ψ must admit a limit at infinity, ψ(q) → +∞ as d(O, q) → ∞. Therefore, ψ is a solution to problem (1.15). Observe that, again by (2.17), ψ ≤ −G(p, ·). Inequality (2.3) then yields |Dψ|(q) ≤ C

−G(O, q) ψ ≤C d(O, q) d(O, q)

(2.18)

for some constant C and any q outside some big geodesic ball B(O, r0 ). Combining now (2.15) with Yau’s lower bound on the growth of geodesic balls, saying that |B(O, r)| ≥ Cr for any r ≥ 1 and for some constant C, we also have d(O, q) + C2 −G(O, q) ≤ C1 d(O, q) d(O, q) for q with d(O, q) > 2r0 and constants C1 and C2 . Plugging it in (2.18), this shows that |Dψ| is uniformly bounded, as claimed. 3. Proof of the Monotonicity-Rigidity Theorem for nonparabolic manifolds 3.1. The conformal setting. Let (M, g) be a nonparabolic Riemannian manifold with nonnegative Ricci curvature. Let Ω ⊂ M be a bounded and open set with smooth boundary, and let u be the solution to problem (1.9). We introduce, in M \ Ω the metric 2

g˜ = u n−2 g.

(3.1)


15

The expression for g˜ is formally the same as in [2] and [1]. Let us explain why such a conformal change of metric makes sense also in the current setting. Our model geometry is that of a truncated metric cone (M \ Ω, g) ∼ (3.2) = r0 , +∞) × ∂Ω , dr ⊗ dr + Cr 2 g∂Ω , for some positive constant r0 and C. We also assume that ∂Ω is a smooth closed sub-manifold with Ric∂Ω ≥ (n − 2)g∂Ω . Such a curvature assumption on ∂Ω is equivalent to suppose that (M \ Ω, g) has nonnegative Ricci curvature. In this model setting, up to a suitable choice of C in (3.2), the solution to problem (1.9) is u(r) = r 2−n . With this specific u the metric g˜ becomes g˜ = dρ ⊗ dρ + g∂Ω , with ρ = log r, that is, the cylindrical metric gcyl . In parallel, as Theorem 1.3 gives a characterization of the truncated metric cone (3.2), so the conformal Theorem 3.2 below characterizes metric cylinders. We are now going to describe (M \ Ω, g˜). Let ϕ = − log u,

(3.3)

2f

then g˜ = e− n−2 g. We recall the following relations between the two metrics. As before, D is the Levi-Civita connection of (M, g). Moreover, we denote by DD the Hessian. We denote by ∇, the Levi-Civita connection of the metric g˜, by ∇∇ its Hessian. and we put the subscript g˜ on any other quantity induced by g. We have, for a smooth function w 1 ∂α w∂β ϕ + ∂β w∂α ϕ − hDw, Dϕigαβ , ∇α ∇β w = Dα Dβ w + n−2 where by h·, ·i we denote the scalar product induced by g. In particular, ∆g˜ ϕ = 0.

(3.4)

Moreover, the Ricci tensor Ricg˜ of g˜ and the Ricci tensor Ric of g satisfy 2 dϕ ⊗ dϕ |∇ϕ|g g Ricg˜ = Ric +∇α ∇β ϕ − + . n−2 n−2 Finally, by (3.4) and (3.5) problem (1.9) becomes  ∆g˜ ϕ = 0 in M \ Ω    2   |∇ϕ|g˜  dϕ ⊗ dϕ Ricg˜ −∇∇ϕ + = g˜ + Ric in M \ Ω n−2 n−2    ϕ=0 on ∂Ω    ϕ(q) → +∞ as d(O, q) → +∞.

(3.5)

(3.6)

The classical Bochner identity applied to ϕ in (M \ Ω, g˜) , combined with the first two equations of the above system, immediately yields the following identity i h (3.7) ∆g˜ |∇ϕ|2g˜ − h∇|∇ϕ|g˜ , ∇ϕig˜ = 2 Ric(∇ϕ, ∇ϕ) + |∇∇ϕ|2g˜ ,

where Ric is the Ricci tensor of the background metric g. Such a relation is at the heart of this work. As a first application, we have the following fundamental relation between the splitting of (M \ Ω, g˜) as a cylinder and the splitting (M \ Ω, g) as a cone.

Lemma 3.1. Let (M, g) be a complete nonparabolic Riemannian manifold with Ric ≥ 0, let Ω ⊂ M be a bounded and open subset with smooth boundary and let g˜ and ϕ be defined by (3.1) and (3.3). Assume that ∇|∇ϕ| = 0 on {ϕ ≥ s0 } for some s ∈ [0, ∞). Then the Riemannian manifold ({ϕ ≥ s0 }, g˜) is isometric to the Riemannian product [s0 , ∞) × {ϕ = s0 }, dρ ⊗ dρ + g˜{|ϕ=s0 } .

16


In particular, {ϕ = s0 } is a connected totally geodesic submanifold. Accordingly, for t0 = e−s0 , the Riemannian manifold ({u ≤ t0 }, g˜) has Euclidean volume growth and it is isometric to the truncated cone ! 1 2 n−1 |{u = t0 }| r with r0 = g{u=t0 } , . r0 , +∞) × {u = t0 } , dr ⊗ dr + n−1 r0 AVR(g)|S | In particular, ∂Ω is a connected totally umbilic submanifold with constant mean curvature.

Proof. Let us first observe that plugging ∇|∇ϕ|g˜ = 0 in (3.7) readily implies, since Ric ≥ 0, that ∇∇ϕ ≡ 0

in {ϕ > s0 } = {u < t0 }.

(3.8)

for s0 = − log t0 . The isometry with a Riemannian product then follows from [2, Theorem 4.2 (i)], proving the first claim. Observe that ∂Ω is connected as a consequence of Proposition 2.10. Recalling now the formula 1 ∇α ∇β w = Dα Dβ w + ∂α w ∂β ϕ + ∂β w ∂α ϕ − hDw, Dϕi gαβ , n−2 holding for every C 2 function w, we have in particular that condition (3.8) translates into Dα Dβ u n Dα uDβ u 1 Du 2 + − gαβ . (3.9) 0 = ∇α ∇β ϕ = − u n−2 u2 n−2 u Observing now that

2 − n−2

Dα Dβ u

=

we deduce from (3.9) that

2 Dα Dβ u n Dα uDβ u 2 − n−2 u + − n−2 u n−2 u2

Du 2 − 2 2 u n−2 gαβ . Dα Dβ u = (n − 2)2 u 2 In particular, we have that DD u−2/(n−2) is proportional to the metric. Let f = u− n−2 . By a standard result of Riemannian geometry (see e.g. [12, Theorem 1.1] for a complete proof, or [13, Section 1]) this fact implies that the potential f , and thus u, depends only on the (signed) distance r from {u = t} and that, up to a multiplicative factor in front of f , the metric g can be expressed as 2 g = dr ⊗ dr + f ′ (r) gΣ , Σ = {u = t}.

2 − n−2

Moreover, the associated Ricci tensor is given by Ric = −(n − 1)

2 f ′′′ dr ⊗ dr + RicΣ − (n − 2)(f ′′ )2 + f ′ f ′′′ gΣ . ′ f

(3.10)

The second information obtained by plugging ∇|∇ϕ|g˜ = 0 in the Bochner identity (3.7) is Ric (∇ϕ, ∇ϕ) = 0

in {ϕ > s0 } = {u < t0 },

that in particular implies Ric(Du, Du) = 0 in the same set. Using now expression (3.10) and recalling that Du is orthogonal to Σ, we obtain that f ′′′ (r) = 0. This yields r 2 gΣ , g = dr ⊗ dr + r0


17

where r0 > 0 is such that Σ = {r = r0 }. Finally, if O is the tip of the above cone, since n−1 ˆ n−1 q ˆ R R 1 n−1 Σ dσ = |Σ|, |∂B(O, R)| = det gij dϑ . . . dϑ = r0 r0 ∂B(O,R)

∂Σ

we can explicitly express r0 in terms of AVR(g) as in (1.13) by |∂B(O, R)| |Σ| = n−1 n−1 . n−1 n−1 R→+∞ R |S | r0 |S |

AVR(g) = lim

The first identity is a well known characterization of the AVR(g) in terms of areas of geodesic balls instead of volumes, and follows easily by the standard proof of the Bishop-Gromov Theorem (see e.g. [60]). The proof of the second claim is completed. We now briefly record some of the main relations between the two equivalent settings. We omit the computations, since they are straightforward and completely analogous to those carried out in [2] and [1]. First, observe that |Du|

n−1 . u n−2 Let H and Hg˜ be the mean curvatures of the level sets of u, that coincide with those of ϕ, respectively in the Riemannian manifolds (M \ Ω, g) and (M \ Ω, g˜), with respect, respectively, to the unit normal vectors −Du/|Du| and ∇ϕ/|∇ϕ|g˜ . We have, exploiting respectively g-harmonicity and g˜-harmonicity of u and ϕ ∇∇ϕ(∇ϕ, ∇ϕ) DDu(Du, Du) , Hg˜ = − . (3.11) H= 3 |Du| |∇ϕ|3g˜

|∇ϕ|g˜ =

These quantities are related as follows Hg˜ = u

1 − n−2

H−

n−1 n−2

|Du| . u

Letting dσg˜ and dµg˜ denote respectively the surface and the volume measure naturally induced by g˜ on M \ Ω, we have n−1 n dσg˜ = u n−2 dσ, dµg˜ = u n−2 dµ. (3.12) Finally, for every β ≥ 0, define the conformal analogue of the Uβ as the function Φβ : [0, +∞) −→ R mapping ˆ dσg˜ . (3.13) |∇ϕ|β+1 Φβ (s) = g˜ {ϕ=s}

The functions Uβ and Φβ , and their derivatives are related to each other as follows. Uβ ′ −tUβ (t)

= Φβ (− log t), = Φ′β (− log t).

(3.14)

The following is the conformal version of the Monotonicity-Rigidity Theorem. Theorem 3.2. Let (M, g) be a complete nonparabolic manifold with Ric ≥ 0. Let Ω ⊂ M be a bounded and open subset with smooth boundary, and let g˜, ϕ and Φβ be defined respectively as in (3.1), (3.27) and (3.13). Then, for every β ≥ (n − 2)/(n − 1), the function Φβ is differentiable with derivative ˆ dΦβ (3.15) (s) = − β |∇ϕ|βg˜ Hg˜ dσg , ds {ϕ=s}

18


where Hg is the mean curvature of the level set {ϕ = s} computed with respect to the unit normal vector field νg˜ = ∇ϕ/|∇ϕ|g˜ . Moreover, for every s ≥ 0, the derivative fulfills ˆ |∇ϕ|β−2 Ric(∇ϕ, ∇ϕ) + ∇∇ϕ 2 + (β − 2) ∇|∇ϕ|g˜ 2 g˜ dΦβ g˜ g˜ (s) = − β es dµg˜ (3.16) ϕ ds e {ϕ≥s}

In particular, Φ′β is alway nonpositive. Moreover, ( dΦβ / ds)(s0 ) = 0 for some s0 ≥ 1 and some β ≥ (n − 2)(n − 1) if and only if {ϕ ≥ s0 } is isometric to the Riemannian product [s0 , ∞) × {ϕ = s0 }, dρ ⊗ dρ + g˜{|ϕ=s0 } . In particular, {ϕ = s0 } is a connected totally geodesic submanifold. Observe that (3.16) is nonpositive because of the refined Kato’s inequality 2 n ∇∇ϕ ≥ ∇|∇ϕ|g˜ 2 . g˜ g˜ n−1

(3.17)

Notice also the striking analogy between the above statement and Theorem 1.5. We are now going to show how to recover the Monotonicity-Rigidity Theorem for nonparabolic manifolds from its conformal version. Proof of Theorem 1.3 after Theorem 3.2. Deducing formulas (1.11) and (1.12) from formulas (3.15) and (3.16) is just a matter of lengthy but straightforward computations carried out using the relations between g and g˜ recalled above. We sketch the main steps. First, compute ( ) Du 3 DDu 2 n(n − 1) Du 4 4 n − n−2 2 (3.18) |∇∇ϕ|g˜ = u u + (n − 2)2 u − 2 n − 2 u H , where H is defined as in (3.11), and ( ) 2 2 Du 3 Du 4 D|Du| 2 4 n − 1 n − 1 − ∇|∇ϕ| = u n−2 g˜ g˜ u + n − 2 u − 2n − 2 u H .

(3.19)

By (3.18) and (3.19), we can write |∇∇ϕ|2g˜

2 4 + (β − 2) ∇|∇ϕ|g˜ g˜ = u− n−2

2 n ) D|Du| |DDu|2 − ( n−1 u2 D|Du| 2 n − 1 2 Du 4 n−1 (3.20) + β− u + n−2 u n−2 ) n − 1 Du 3 −2 H . n−2 u

(

Now, considering a orthonormal frame as {e1 , . . . , en−1 , Du/|Du|}, where the first n − 1 vectors are tangent to the level sets of u, we can decompose X D|Du| 2 D|Du| 2 Du 2 2 n−1 = H + , ej . u u u j=1


19

Plugging the above decomposition in (3.20), we obtain, with the aid of some algebra, h 2 i 4 − n−2 2 DDu 2 − n D|Du| 2 |∇∇ϕ|g˜ + (β − 2) ∇|∇ϕ|g˜ g˜ = u n−1 T D |Du| 2 h 2 + β − n−2 H− |Du| n−1 + β−

where

n−1 X T D |Du| 2 = j=1

n−2 n−1

D|Du| , ej u

2

n−1 n−2

i2 |D log u| ,

.

The monotonicity formula (1.12) now follows easily. On the other hand, (1.11) follows from (3.15) by a direct computation. Assume now that Uβ′ (t0 ) = 0 for some t0 ∈ (0, 1] and some β ≥ 1 (recall Remark 1.4). Then, by (3.14), Φ′β (− log t0 ) = 0. Then, through Kato’s inequality (3.17), (3.16) implies that ∇|∇ϕ| = 0 in {ϕ ≥ − log t0 }. The rigidity statement in Theorem 1.3 then follows from g˜ g˜ Lemma 3.1. 3.2. Proof of Theorem 3.2. In what follows, we are always referring to a background nonparabolic Riemannian manifold (M, g) with Ric ≥ 0 endowed, outside a bounded and open subset Ω with smooth boundary, with the conformal metric g˜ defined in (3.1). In particular, with the same notations of the previous subsection, (M \ Ω, g˜, ϕ) is a solution to (3.6). A first fundamental ingredient for the proof of the (conformal) Monotonicity-Rigidity Theorem is the uniform boundedness of the function Φβ . We are going to prove it by first showing that |∇ϕ|g˜ is uniformly bounded in M \ Ω. Although not necessary for our aim, we actually provide a sharp upper bound for such a function, together with a rigidity statement when this bound is attained. The proof is substantially the one proposed for [2, Lemma 3.5] in the Euclidean case. Proposition 3.3 (A sharp bound for |∇ϕ|g˜ ). The inequality |∇ϕ|g˜(q) ≤ sup|∇ϕ|g˜

(3.21)

∂Ω

holds for any q ∈ M \ Ω. Moreover, if equality is attained for some q ∈ M \ Ω, then the Riemannian manifold (M \ Ω, g˜) is isometric to [0, ∞) × ∂Ω, dρ ⊗ dρ + g˜∂Ω .

Proof. We first show that there exists a constant C = C(M, Ω) such that |∇ϕ|g˜ ≤ C in M \ Ω. Fixing, a reference point O inside Ω, and letting d(O, ·) be the distance from this point with respect to the metric g, we have, by (2.3), that u |Du|(q) ≤ C d(O, q)

outside some ball containing Ω. Then, 1

u 2−n (q) ≤ CC1 , |∇ϕ|g˜ (q) = n−1 (q) ≤ C d(O, q) u n−2 |Du|

where in the last inequality we used (2.14). Consider now the auxiliary function wα = |∇ϕ|2g˜e−αϕ , for some constant α > 0. Observe that by the just proved upper bound on |∇ϕ|g˜ we have

20


wα (q) → 0 as d(O, q) → ∞ for any α > 0. Moreover, a direct computation combined with (3.7) shows that wα satisfies the identity i h ∆g˜ wα − (1 − 2α) h∇wα , ∇ϕig˜ − α(1 − α)|∇ϕ|g2˜ wα = 2 e−αϕ Ric(∇ϕ, ∇ϕ) + |∇∇ϕ|2g˜ .

In particular, choosing 0 < α < 1, the Maximum Principle yields (3.21) for any q ∈ M \ Ω. Assume now that the maximum value of |∇ϕ|g˜ is attained at some interior point q ∈ M \ Ω. Then, by the Strong Maximum Principle |∇ϕ|2g˜ must be constant, since it satisfies ∆g˜ |∇ϕ|2g˜ − h∇|∇ϕ|g˜ , ∇ϕig˜ ≥ 0. Lemma 3.1 then yields the desired conclusion.

Remark 3.4. The above results, in terms of (M \ Ω, g) and u, says that |Du| n−1 n−2

(q) ≤ sup

|Du| n−1

∂Ω u n−2 u for any q ∈ M \ Ω, with equality attained only if (M \ Ω, g) is isometric to a truncated metric cone. This is exactly the content of Colding’s [21, Theorem 3.2].

A direct consequence is that Φβ is bounded. Corollary 3.5. The function Φβ : [0, ∞) → R defined in (3.13) is bounded as ˆ β Φβ (s) ≤ sup|∇ϕ|g˜ |∇ϕ|g˜ dσg˜ . ∂Ω

∂Ω

for any s ∈ [0, ∞). Proof. Just observe that a simple application of the Divergence Theorem combined with the g˜-harmonicity of ϕ gives the constancy in s of the function ˆ |∇ϕ|g˜ dσg˜ . s 7→ {ϕ=s}

Then, by this observation and the bound (3.21), we have ˆ ˆ β β |∇ϕ|g˜ |∇ϕ|g˜ dσg˜ ≤ sup|∇ϕ|g˜ |∇ϕ| dσg˜ Φβ (s) = {ϕ=s}

∂Ω

∂Ω

for any s ∈ [0, ∞), as claimed.

A fundamental tool to write the derivative of Φβ in terms of a nonpositive integral as in (3.16) is the following Bochner-type identity. Lemma 3.6 (Bochner-type identity). On any point where |∇ϕ|g˜ 6= 0, the following identity holds for any β ≥ 0 h E D i 2 ∇|∇ϕ| 2 , (3.22) Ric(∇ϕ, ∇ϕ) + |∇∇ϕ| + (β − 2) ∆g˜ |∇ϕ|βg˜ − ∇|∇ϕ|βg˜ , ∇ϕ = β|∇ϕ|β−2 g˜ g˜ g˜ g˜ g˜

where the Ric is the Ricci tensor of the background metric g.

Proof. By a direct computation one gets 2 β−2 β β 2 ∆g˜ |∇ϕ|g˜ + β(β − 2) ∇|∇ϕ|g˜ g˜ , ∆g˜|∇ϕ|g˜ = |∇ϕ|g˜ 2

that, combined with (3.7), leads to (3.22).


21

We prove now an integral identity that will enable us to link the derivative of Φβ to the Bochner type formula above. As already said in Remark 1.4, we just consider the range β ≥ 1. Lemma 3.7 (An integral identity). Let 0 ≤ s < S < ∞, and let β ≥ 1. Then the following identity holds. D D E E ˆ ∇|∇ϕ|β , ∇ϕ ˆ ∇|∇ϕ|β , ∇ϕ g˜ |∇ϕ|g˜ g˜ |∇ϕ|g˜ dσ − dσg˜ g ˜ eS es {ϕ=S}

{ϕ=s}

=β

h

ˆ |∇ϕ|β−2 Ric(∇ϕ, ∇ϕ) + |∇∇ϕ|2 + (β − 2) ∇|∇ϕ| 2 g˜ g˜ g˜ g˜ eϕ

i

dµg˜ ,

{s≤ϕ≤S}

(3.23) where the Ricci tensor is referred to the background metric g. Proof. Let us drop the subscript g˜, all quantities appearing in this proof are referred to the metric g˜, except for Ric that is the Ricci tensor of the background metric g. We consider the vector field ∇|∇ϕ|β X= , eϕ that is well defined on any point with |∇ϕ| = 6 0, and thus H n−1 -almost everywhere, in virtue of the regularity theory for elliptic equations recalled above. Moreover, by the Bochner-type identity (3.22), we have that on all of these points h 2 i 2 β−2 β β Ric(∇ϕ, ∇ϕ) + |∇∇ϕ| + (β − 2) ∇|∇ϕ| |∇ϕ| ∆|∇ϕ| − h∇|∇ϕ| , ∇ϕi = β . divX = eϕ eϕ (3.24) Let us consider, in {s ≤ ϕ ≤ S}, a tubular neighborhood of the critical set Bε [Crit(ϕ)]. Let the function δ be the distance from Crit(ϕ). Then, by [4, Theorem 2.5], for almost any ε > 0 the set Bε [Crit(ϕ)] is (n − 1)-Lipschitz rectifiable with respect to σg˜ and the measure σg˜ (Bε [Crit(ϕ)]) is finite, recalling that σg˜ is abolutely continuous with respect to the (n − 1)-dimensional Hausdorff measure on (M, g) due to (3.12). Then, the Divergence Theorem applies and yields D E E D ∇ϕ ˆ ∇ϕ ˆ ˆ ∇|∇ϕ|β , |∇ϕ| ∇|∇ϕ|β , |∇ϕ| divX dµ = dσ − dσ eS es {s≤ϕ≤S}\Bεj [Crit(ϕ)]

{ϕ=S}\Bεj (Crit(ϕ))

{ϕ=s}\Bεj [Crit(ϕ)]

+

ˆ

E D ∇|∇ϕ|β , νεj eϕ

dσ,

∂Bεj [Crit(ϕ)]

for a sequence εj → 0+ , where νεj is the exterior normal to ∂Bεj [Crit(ϕ)], that is well defined almost everwhere on such a set due to rectifiability. By Kato’s inequality 2 |∇∇ϕ|2 ≥ ∇|∇ϕ| ,

and identity (3.24), divX is nonnegative. Thus, by the Monotone Convergence Theorem, we have ˆ ˆ divX dµ. divX dµ = lim {s≤ϕ≤S}

ε→0+ {s≤ϕ≤S}\Bε [Crit(ϕ)]

22


Moreover, observe that the vector field ∇|∇ϕ|β is bounded in {s ≤ ϕ ≤ S} for β ≥ 1 again by Kato’s inequality ∇|∇ϕ|β = β|∇ϕ|β−1 ∇|∇ϕ| ≤ β|∇ϕ|β−1 |∇∇ϕ|,

and thus the Dominated Convergence Theorem gives D E E D ∇ϕ ˆ ∇|∇ϕ|β , ∇ϕ ˆ ∇|∇ϕ|β , |∇ϕ| |∇ϕ| dσ = dσ, lim s s + e e εj →0 {ϕ=s}

{ϕ=s}\Bεj [Crit(ϕ)]

and analogously for {ϕ = S}. We are then left to show that E D ˆ ∇|∇ϕ|β , νε lim dσ = 0. eϕ εj →0+

(3.25)

∂Bεj [Crit(ϕ)]

To see this, we use the estimate on the tubular neighbourhoods of the critical set given in [16, Theorem 1.17]. Namely, for any η > 0 and any ε > 0 there exists a constant Cη > 0 such that |Bε (Crit(ϕ))| ≤ Cη ε2−η .

(3.26)

Let ρ be the distance function from the closed set Crit(ϕ). Then, by [54, Theorem 3.1], ρ is a viscosity solution of |∇ρ| = 1 outside Crit(ϕ), and in particular |∇ρ| = 1 almost everywhere outside Crit(ϕ). Therefore, we can apply the co-area formula to get, by (3.26), ˆεˆ 0

dσ dτ ≤ Cη ε2−η .

∂Bτ [Crit(ϕ)]

By the Mean Value Theorem, we can find a sequence εj → 0+ such that ˆ dσ ≤ Cη ε1−η , j ∂Bεj [Crit(ϕ)]

and then, up to relabeling, the limit in (3.25) holds true, completing the proof.

Theorem 3.8 (Monotonicity of e−s Φ′β (s)). Let β ≥ 1. The function Φβ defined in (3.13) is differentiable for any s ≥ 0, and its derivative satisfies + ˆ ˆ * ∇ϕ β ′ |∇ϕ|g˜ Hg˜ dσg˜ . (3.27) dσg˜ = −β ∇|∇ϕ|g˜ , Φβ (s) = |∇ϕ|g˜ {ϕ=s}

{ϕ=s}

In particular, for any 0 ≤ s < S < ∞, we have ˆ 2 β−2 2 −S ′ −s ′ Ric(∇ϕ, ∇ϕ) + |∇∇ϕ|g˜ + (β − 2) ∇|∇ϕ|g˜ e−ϕ dµg˜ , e Φβ (S) − e Φβ (s) = β |∇ϕ|g˜ g˜

{s≤ϕ≤S}

(3.28)

where the Ricci tensor is referred to the background metric g. Proof. Let us drop the subscript g˜. Let s0 ∈ [0, ∞), and s ≥ s0 . Consider ˆ ˆ ∇ϕ ∇ϕ β β Φβ (s) − Φβ (s0 ) = |∇ϕ| ∇ϕ, dσ − |∇ϕ| ∇ϕ, dσ. |∇ϕ| |∇ϕ| {ϕ=s}

{ϕ=s0 }


23

Ruling out the critical set of ϕ as shown in the proof of the above Lemma 3.7, we can apply the Divergence Theorem to get ˆ D ˆ E β ∇|∇ϕ|β , ∇ϕ dµ, div(|∇ϕ| ∇ϕ) dµ = Φβ (s) − Φβ (s0 ) = {s0 ≤ϕ≤s}

{s0 ≤ϕ≤s}

where the last identity is due to harmonicity of ϕ. By co-area formula, the above quantity can also be written as ˆs ˆ β ∇ϕ Φβ (s) − Φβ (s0 ) = ∇|∇ϕ| , dσ dτ. (3.29) |∇ϕ| {ϕ=τ } s0

By the Fundamental Theorem of Calculus, showing the continuity of the function I mapping ˆ β ∇ϕ dσ τ 7→ ∇|∇ϕ| , |∇ϕ| {ϕ=τ } suffices to show the differentiability if Φβ , together with the first identity in (3.27). The second one follows from the first just by a direct computation involving (3.11). In order to see the continuity of I at a time τ0 , consider, for τ > τ0 , the difference ˆ ˆ β ∇ϕ β ∇ϕ −τ −τ0 −τ −τ0 e I(τ ) − e I(τ0 ) = e dσ − e dσ. ∇|∇ϕ| , ∇|∇ϕ| , |∇ϕ| |∇ϕ| {ϕ=τ0 } {ϕ=τ } Then, by (3.23), we have e−τ I(τ ) − e−τ0 I(τ0 ) = β

h

ˆ |∇ϕ|β−2 Ric(∇ϕ, ∇ϕ) + |∇∇ϕ|2 + (β − 2) ∇|∇ϕ| 2 eϕ

i

dµ.

{τ0 ≤ϕ≤τ }

In particular, by the Dominated Convergence Theorem the above integral vanishes as τ → τ0+ , yielding the right continuity of e−τ I(τ ) at τ0 . By a completely analogous argument with τ < τ0 we get that e−τ I(τ ) is actually continuous at τ0 , and clearly so is I, ending the proof. We are now in position to complete the proof of Theorem 3.2. Conclusion of the proof of Theorem 3.2. We are going to pass to the limit as S → ∞ in (3.28). The following argument is due to Colding and Minicozzi, [26]. We first prove that the derivative of Ψβ has a sign. By (3.28), we have, for β ≥ 1 and for any 0 ≤ s < S < ∞, Φ′β (S) ≥ e(S−s) Φ′β (s), that, by integrating in S, implies Φβ (S) ≥ e(S−s) − 1 Φ′β (s) + Φβ (s).

(3.30)

for any 0 ≤ s < S < ∞. Assume now, by contradiction, that Φ′β (s) > 0 for some s ∈ [0, ∞). Then, passing to the limit as S → +∞ in (3.30), we would get Φβ (S) → +∞, against the boundedness of Φ provided in Corollary 3.5. Thus, Φ′β (s) ≥ 0 for any s ∈ [0, ∞). Therefore Φβ is a nonincreasing, differentiable bounded function on [0, ∞), and in particular, Φ′β (S) → 0 as S → ∞. Passing to the limit as S → ∞ in (3.28) finally gives the monotonicity formula (3.16). Assume now that Φ′β (s0 ) = 0 for some s0 ∈ [0, ∞). Then, by (3.16), and Kato’s inequality ∇|∇ϕ| = 0. We conclude by Lemma 3.1.

24


4. Proof of the Willmore-type inequalities In this section, we are going to prove the Willmore inequality on manifolds with nonnegative Ricci curvature and Euclidean volume growth. As sketched in the Introduction, the proof makes use of the global features of the Monotonicity-Rigidity Theorem, that is, it compares the behavior of Uβ in the large with Uβ (1). The well known asymptotics of u and of its gradient in the Euclidean case were the crucial tool to compute the limits of Uβ in [2], that accordingly gave the sharp lower bound on the Willmore energy. On manifolds with nonnegative Ricci curvature and Euclidean volume growth, the work of Colding-Minicozzi [23] actually implies that the asymptotic behavior of the potential is completely analogous to that in Rn . However, as we will clarify in Remark 4.7, there is no hope to get an Euclidean-like pointwise behavior of Du in the general case. Nevertheless, using techniques developed in the celebrated [13], we are able to achieve asymptotic integral estimates for the gradient that in turn will let us conclude the proof of our Willmore-type inequalities. As in Section 2, we denote by D the Levi-Civita connection on the manifold considered, by DD the Hessian and by ∆ the Laplacian. Moreover, we let r(x) = d(O, x), where O ∈ Ω. 4.1. Manifolds with Euclidean volume growth. With respect to the value assumed by AVR(g), we define manifolds with Euclidean volume growth and sub-Euclidean volume growth as follows. Definition 4.1 (Volume growth). Let (M, g) be an n- dimensional complete noncompact Riemannian manifold with Ric ≥ 0. Then we say that it has Euclidean volume growth if AVR(g) > 0, sub-Euclidean volume growth otherwise. If follows at once that manifolds with Euclidean volume growth satisfy (2.6), and thus from Varopoulos’ characterization manifolds with Euclidean volume growth are nonparabolic. Let us also recall the well known fact that in a noncompact complete Riemannian manifold (M, g) with Ric ≥ 0 also the function (0, ∞) ∋ r 7−→ ϑ(r) =

|∂B(p, r)| r n−1 |Sn−1 |

is monotone nonincreasing for any p ∈ M , and the asymptotic volume ratio satisfies AVR(g) = lim

r→+∞

|∂B(p, r)| . r n−1 |Sn−1 |

(4.1)

In the remainder of this section we are repeatedly going to use both the area and the volume formulations of the Bishop-Gromov Theorem without always mentioning them. Let us now derive some rough estimates for the electrostatic potential u and its gradient Du holding on manifolds with nonnegative Ricci curvature and Euclidean volume growth, to be refined later. Proposition 4.2. Let (M, g) be an n-dimensional complete noncompact Riemannian manifold with Ric ≥ 0 and Euclidean volume growth. Then, it is nonparabolic and the solution u to problem (1.9) for some bounded and open Ω with smooth boundary satisfies C1 r 2−n (x) ≤ u(x) ≤ C2 r 2−n (x)

(4.2)

on M \ Ω for some positive constants C1 and C2 depending on M and Ω. Moreover, if Ω ⊂ B(O, R), it holds |Du|(x) ≤ C3 r 1−n (x), (4.3) on M \ B(O, 2R) with C3 = C3 (M, Ω) > 0.


25

Proof. The first inequality in (4.2) is just (2.14). To obtain the second one, observe first that the monotonicity in the Bishop-Gromov Theorem implies, for any p ∈ M and any r ∈ (0, ∞), that |B(p, r)| ≥ n|Sn−1 |AVR(g) r n .

Then, the second inequality in the Li-Yau estimate (2.11), combined with the second inequality in (2.13) completes the proof of (4.2). Finally, inequality (4.3) is achieved just by plugging the upper estimate on u given by (4.2) into (2.2). 4.2. Asymptotics for u. The behavior at infinity of u can be deduced just by perusing [23], but we prefer to give an easier proof that uses also some refinements given in [48]. Let us first recall the following asymptotic behaviour of G (see [23, Theorem 0.1], or [48, Theorem 1.1] for a completely different proof), G(O, x) 1 = . 2−n AVR(g) r(x)→∞ r(x) lim

(4.4)

We define the electrostatic capacity of Ω as Cap(Ω) =

ˆ

|Du| dσ

∂Ω

(n − 2)|Sn−1 |

.

Lemma 4.3. Let (M, g) be a Riemannian manifold with Ric ≥ 0 and Euclidean volume growth, and let u be a solution to problem (1.9). Then Cap(Ω) u(x) = , 2−n r(x) AVR(g) r(x)→∞ lim

(4.5)

Proof. By [48, Theorem 1.2], that slightly extends [23, Theorem 0.3], we have that outside some large ball B(O, R) containing Ω ˆ G ∂u u=− dσ + v, (4.6) n−1 (n − 2)|S | B(O,R) ∂ν where G is the Green’s function with pole in O, v is a harmonic function defined in M \ B(O, R) satisfying G |v| ≤ D (4.7) r for some constant D > 0 and ν is the exterior unit normal to the boundary of B(O, R). We point out that the Green’s function considered in [48] is, in our notation, G/[(n − 2)|Sn−1 |]. By the Divergence Theorem and the harmonicity of u, we infer that ˆ ˆ ˆ ∂u ∂u ∆u dµ = 0, dσ + dσ = ∂Ω ∂ν B(O,R)\Ω ∂B(O,R) ∂ν where we denote by ν the exterior unit normal to the boundary of B(O, R) \ Ω. Since, on ∂Ω, ν = Du/|Du|, we get, by the above identity, that ˆ ˆ ∂u dσ |Du| dσ B(O,R) ∂ν ∂Ω = = Cap(Ω). (4.8) − (n − 2)|Sn−1 | (n − 2)|Sn−1 | Dividing both sides of (4.6) by r 2−n and passing to the limit as r → ∞ taking into account (4.4), (4.8) and (4.7), we get the claim.

26


As a straightforward corollary of the above lemma we compute the rescaled area of big geodesic balls of M . Corollary 4.4. Let (M, g) be a noncompact Riemannian manifold with Ric ≥ 0 and Euclidean volume growth, and let u be a solution to (1.9). Then n−1 ˆ n−1 n−1 Cap(Ω) n−2 n−2 u lim dσ = S AVR(g) (4.9) Ri →∞ ∂B(O,Ri ) AVR(g)

Proof. It follows from Lemma 4.3 and (4.1).

From the above asymptotics for u we now derive integral asymptotics for Du. They are achieved through a variation of the methods used in [13, Section 4]. Similar estimates have been widely considered in Colding and coauthors’ literature (see e.g. [21] [23], [22], [25]). We are providing here a complete proof. In the computations below, we are taking first and second derivatives of the distance function x 7→ d(O, x). This is justified by standard approximation arguments, that we omit. Proposition 4.5 (Integral asymptotics for Du). Let (M, g) be a Riemannian manifold with Ric ≥ 0 and Euclidean volume growth, and let u be a solution to problem (1.9). Then, for any k>1 2 ˆ Du − Cap(Ω) Dr 2−n dµ AVR(g) lim

R→∞

AR,kR

R2−2n |AR,kR |

= 0,

(4.10)

where, for R > 0, AR,kR = B(0, kR) \ B(O, R). Proof. Integration by parts combined with the harmonicity of u yields the following identity 2 ˆ ˆ Cap(Ω) 2−n Cap(Ω) 2−n Du − Cap(Ω) r 2−n dµ = − ∆r r u− dµ AVR(g) AVR(g) AR,kR AVR(g) AR,kR ˆ Cap(Ω) 2−n Cap(Ω) 2−n + u− r Dr , ν dσ. Du − AVR(g) AVR(g) ∂AR,kR

(4.11) Let us estimate separately the integrals on the right hand side of the above identity. Let ε > 0. Then, by (4.5), u Cap(Ω) r 2−n − AVR(g) < ε. for r big enough. We have, for R big enough ˆ ˆ Cap(Ω) Cap(Ω) Cap(Ω) 2−n Cap(Ω) 2−n 2−n 2−n u dµ ∆r r ∆r u− dµ ≤ − r 2−n − AVR(g) r AVR(g) AVR(g) AVR(g) AR,kR AR,kR ˆ Cap(Ω) 2−n 2−n dµ ≤ εR ∆r AR,kR AVR(g) ˆ Cap(Ω) 2−n ∆r dµ, = εR2−n AR,kR AVR(g) (4.12)


27

where in the last identity we used ∆r 2−n ≥ 0, as already shown along the proof of Lemma 2.10. Integrating by parts ∆r 2−n we obtain ˆ ∆r 2−n = (2 − n) (kR)1−n |{r = kR}| − R1−n |{r = R}| . AR,kR

In particular, by Euclidean volume growth, the above quantity is uniformly bounded in R. We have thus proved that the first summand on the right hand side of (4.11), for R large enough, is bounded as follows ˆ Cap(Ω) 2−n Cap(Ω) 2−n ∆r r (4.13) u− dµ ≤ C1 εR2−n . AR,kR AVR(g) AVR(g) Let us turn our attention to the second integral in the right hand side of (4.11). We have, proceeding as in (4.12), ˆ Cap(Ω) 2−n u − Cap(Ω) r 2−n Dr , ν dσ Du− AVR(g) AVR(g) {∂AR,kR } ˆ Cap(Ω) 1−n 2−n r dσ. ≤ εR |Du| + (n − 2) AVR(g) ∂AR,kR (4.14) Recall now that in Proposition 4.2 we proved that |Du| ≤ C2 r 1−n , for some positive constant C2 independent of r. Thus, by Euclidean volume growth, the integral on the right hand side of (4.14) is uniformly bounded in R and so ˆ Cap(Ω) 2−n Cap(Ω) 2−n u− (4.15) r Dr , ν dσ ≤ C3 εR2−n Du − ∂AR,kR AVR(g) AVR(g)

for some C3 independent of R. Finally, by (4.11), (4.13) and (4.15), we obtain, for R big enough, the estimate 2 ˆ Du − Cap(Ω) Dr 2−n dµ AVR(g) AR,kR R2−n ≤ C4 ε ≤ C5 εR2−2n , |AR,kR | |AR,kR |

for some positive constants C4 and C5 independent of R. In the last inequality we used the Euclidean volume growth assumption. Our claimed (4.10) is thus proved. From the above Proposition we easily deduce the integral asymptotic behaviour of |Du| on geodesic spheres. Corollary 4.6. Let (M, g) be a Riemannian manifold with Ric ≥ 0 and Euclidean volume growth. Let u be a solution to problem (1.9). Then there exists a sequence of positive real numbers Ri with Ri → ∞ such that ˆ Cap(Ω) 1−n lim r (4.16) |Du| − (n − 2) dσ = 0. Ri →∞ AVR(g) {r=Ri }

28


Proof. Let us first observe that, by means of Hölder inequality, we can deduce from the L2 asymptotics (4.10) an analogous L1 behaviour. Namely, for any ε > 0 we have 2 ˆ 1/2 ˆ Du − Cap(Ω) Dr 2−n dµ Du − Cap(Ω) Dr 2−n dµ   AVR(g) AVR(g) AR,kR  AR,kR  ≤   ≤ε 1−n 2−2n   R |AR,kR | R |AR,kR |

for any R large enough. By coarea formula, the above L1 estimate gives, for R large enough, ˆkR ˆ Cap(Ω) 2−n Dr Du − dσ ds AVR(g)

R {r=s}

R1−n |{AR,kR }|

≤ ε.

Thus, by the Mean Value Theorem, there exists rR ∈ (R, kR) such that ˆ Cap(Ω) 2−n Dr dσ Du − AVR(g) {r=rR } ≤ Cε R−n |{AR,kR }| for some constant C independent of R. The Euclidean volume growth of the annulus AR,kR as R increases then implies the existence of a sequence Ri → ∞ as in the statement. Remark 4.7. The integral asymptotic for |Du| given by Corollary 4.6 cannot, in general, be improved to a pointwise asymptotic expansion at infinity on a manifold with nonnegative Ricci curvature and Euclidean volume growth. Indeed, the validity of such a formula would imply |Du| = 6 0 outside some big ball B(O, R), and, in turn, M \ B(O, R) would be diffeomorphic to ∂B(O, R) × [R, ∞). This would imply that M has finite topological type. However, Menguy provided in [55] examples of manifolds of dimension n ≥ 4 with Ric ≥ 0 and AVR(g) > 0 with infinite topological type. n−1 ´ 4.3. Completion of the proof. Let us recall that Uβ (t) = t−β ( n−2 ) {u=t} |Du|1+β dσ. We have now collected all the elements to compute its limit as t → 0+ . Proposition 4.8. Let M be a Riemannian manifold with Ric ≥ 0 and Euclidean volume growth. Let β ≥ 0. Then the following formula for the the limit of Uβ (t) holds. β

β

lim Uβ (t) = Cap(Ω)1− n−2 AVR(g) n−2 (n − 2)β+1 |Sn−1 |.

t→0+

(4.17)

Proof. We multiply and divide inside the integral in (4.16) for u(n−1)/(n−2) . By the asymptotics of u given by Lemma 4.3, we obtain 1 ˆ Du n−1 AVR(g) n−2 n−2 dσ = 0. (4.18) u lim n−1 − (n − 2) Ri →∞ {r=Ri } Cap(Ω) u n−2 Let us now recall the following basic interpolation inequality kf kLp (X) ≤ kf kϑL1 (X) kf k1−ϑ Lq (X) ,

holding for any f ∈ L1 (X) ∩ Lq (X), where X is a measure space and the numbers p and q satisfy 1 < p < q < ∞ and 1/p = ϑ + (1 − ϑ)/q. We apply such estimate to f equal to the integrand


29

p = 1 + β, q > 1 + β, and with respect to the measure u(n−1)/(n−2) dσ. We get 1/(1+β) 1 1+β Du n−2 n−1 AVR(g) u n−2 dσ  n−1 − (n − 2) Cap(Ω) {r=Ri } u n−2 !ϑ 1 ˆ n−2 n−1 Du AVR(g) u n−2 dσ × n−1 − (n − 2) ≤ n−2 Cap(Ω) {r=Ri } u

in (4.18),  ˆ 

×

1 n−2 Du n−1 − (n − 2) AVR(g) n−2 Cap(Ω) {r=Ri } u

ˆ

q n−1 u n−2 dσ

!(1−ϑ)/q

(4.19)

.

It is easy to see to see , due to the uniform bound on |Du|/u(n−1)/(n−2) = |∇ϕ|g˜ given in (3.21) ´ n−1 and to the boundedness of {r=Ri } u n−2 dσ, that follows from Corollary 4.4, that the second integral on the right hand side of (4.19) is bounded in Ri . Thus, by (4.19) and (4.6), we deduce that 1 ˆ Du AVR(g) n−2 1+β n−1 u n−2 dσ = 0. lim − (n − 2) n−1 Ri →∞ {r=Ri } u n−2 Cap(Ω) The above limit in particular implies that 1+β ˆ ˆ Du 1+β n−1 n−1 AVR(g) n−2 1+β n−2 dσ = (n − 2) u n−2 dσ, lim u lim n−1 Ri →∞ {r=Ri } Ri →∞ {r=Ri } u n−2 Cap(Ω) that, combined with (4.9), gives ˆ Du 1+β n−1 β β n−1 u n−2 dσ = (n − 2)1+β Cap(Ω)1− n−2 AVR(g) n−2 |Sn−1 |. lim Ri →∞ {r=Ri } u n−2 Moreover, by the asymptotic behavior of u, we have ˆ ˆ Du 1+β n−1 Du 1+β n−1 n−2 dσ = lim lim u u n−2 dσ = lim Uβ (ti ), n−1 n−1 Ri →∞ {r=Ri } u n−2 ti →0+ {u=ti } u n−2 ti →0+ and we have thus proved our claim for some sequence ti → 0+ . However, by the boundedness and monotonicity of Uβ the whole limit as t → 0+ exists, and it coincides with the just computed one. Let us briefly discuss the sub-Euclidean volume growth case. Remark 4.9. If (M, g) has sub-Euclidean volume growth, that is, if |B(p, r)| =0 r→∞ rn for any p ∈ M , it is easy to realize that limt→0+ Uβ = 0 for any β ≥ 0. Indeed, by (2.13), (2.11) and (2.3), one easily obtains 1 |B(p, r)| n−2 |Du| n−1 ≤ C rn u n−2 outside some ball containing Ω for some C = C(M, Ω). This implies that ˆ Du β n−1 |Du| dσ = 0, lim Uβ (t) = lim t→0+ t→0+ {u=t} u n−2 lim

30


´ where we used the constancy in t of {u=t} |Du| dσ. This computation clearly shows that Uβ is useless for obtaining a Willmore inequality on manifolds with sub-Euclidean volume growth, and it supports the perception that the infimum of the Willmore-type functional is zero on these manifolds. This actually happens for example on noncompact Riemannian manifolds with metrics g = dρ ⊗ dρ + Cρ2α g|Σ outside some compact set K, where Σ is a compact submanifold, C > 0 and 0 < α < 1. Indeed, this is readily checked by computing the Willmore-type functional on large level sets {ρ = r}. Actually, this is true also for any noncompact complete 3-manifold with Ric ≥ 0, as a corollary to Theorem 1.9, proved below. Proof of Theorem 1.1. With Theorem 1.3 and Proposition 4.8 at hand, Theorem 1.1 follows exactly as in the Euclidean proof recalled in the Introduction. Precisely, let β = n − 2. Then, (4.17) reads lim Un−2 (t) = AVR(g)(n − 2)n−1 |Sn−1 |. t→0+

Moreover, by the nonnegativity of expression (1.11) proved in Theorem 1.3, and the Hölder inequality, 1/(n−1) (n−2)/(n−1) ˆ ˆ ˆ ˆ (n − 1) n−1 n−2 n−1 n−1 , H dσ |Du| dσ |Du| H dσ ≤ |Du| dσ ≤ (n − 2) ∂Ω ∂Ω ∂Ω ∂Ω that gives ˆ

n−1

|Du|

dσ ≤ (n − 2)

∂Ω

AVR(g)(n − 2)

|S

n−1

ˆ

∂Ω

Finally, n−1

n−1

H n−1 n − 1 dσ.

| = lim Un−2 (t) ≤ Un−2 (1) = t→0+

ˆ

|Du|n−1 dσ ∂Ω n−1

≤ (n − 2)

ˆ

∂Ω

H n−1 n − 1 dσ

completes the proof of the Willmore-type inequality. The rigidity statement when equality is attained follows straightforwardly from the rigidity part of Theorem 1.3. 4.4. Application to ALE manifolds. We can improve our Willmore-type inequality if (M, g) satisfies a quadratic curvature decay condition, showing that, in this case, the lower bound AVR(g) |Sn−1 | on the Willmore functional is actually an infimum. Let us recall the following well known definition. Definition 4.10 (Quadratic curvature decay). A complete noncompact Riemannian manifold (M, g) has quadratic curvature decay if there exists a point p ∈ M and a constant C = C(M, p) such that |Riem|(q) ≤ Cd(p, q)2 , where by Riem we denote the Riemann curvature tensor of (M, g). When this assumption is added on a Riemannian manifold with Ric ≥ 0 and Euclidean volume growth (M, g), [23, Proposition 4.1] gives the following asymptotic behaviour of the gradient and the Hessian of the minimal Green’s function G.


31

Theorem 4.11. Let (M, g) be a Riemannian manifold with Ric ≥ 0, Euclidean volume growth and quadratic curvature decay, and let G be its minimal Green’s function. Then (n − 2) |Dx G|(p, x) = 1−n (p, x) AVR(g) d(p,x)→∞ d 2 2−n 1 2/(2−n) (p, x) − 2 lim DDx G g(x) = 0. AVR(g) d(p,x)→∞ lim

(4.20) (4.21)

Observe that arguing as in Remark 4.7, one realizes that (4.20) actually implies that Riemannian manifolds satisfying the assumptions of the above Theorem have finite topological type. Theorem 4.11 enables us to prove that the Willmore-type functional of large level sets of the Green’s function approach AVR(g)|Sn−1 |. This fact, combined with our Willmore-type inequality (1.6), easily yields the following refinement. Theorem 4.12. Let (M, g) be a Riemannian manifold with Ric ≥ 0, Euclidean volume growth and quadratic curvature decay. Then     ˆ H n−1 dσ Ω ⊂ M bounded and smooth = AVR(g)|Sn−1 |. (4.22) inf   n−1 ∂Ω

Moreover, the infimum is attained at some bounded and smooth Ω ⊂ M if and only if (M \ Ω, g) is isometric to 1 n−1 |∂Ω| 2 . r0 , +∞) × ∂Ω , dr ⊗ dr + (r/r0 ) g∂Ω , with r0 = AVR(g)|Sn−1 |

Proof. Let p ∈ M be fixed, and let G be the minimal Green’s function of (M, g). Let us denote again by G the function q 7→ G(p, q). In light of Theorem 1.1, it suffices to prove that ˆ H n−1 lim dσ = AVR(g)|Sn−1 |. (4.23) t→+∞ n−1 {G2/(2−n) =t}

To see this, consider, at a point x, orthonormal vectors {e1 , . . . , en−1 } tangent to the level set {G = G(x)}. Then, letting r(x) = d(p, x), we have, by (4.21), 1 n−1 2 X 2−n 1 2−n . (4.24) DD G lim (ei , ei )(x) = 2(n − 1) AVR(g) r(x)→∞ i=1

The mean curvature of the level sets of G2/(2−n) is clearly computed as 2 Pn−1 DD G 2−n (ei , ei ) i=1 , HG2/(2−n) = |DG2/(2−n) | that, combined with (4.24), (4.20) and (4.4) gives lim r(x)HG2/(2−n) (x) = (n − 1).

r(x)→∞

Combining it again with (4.4) and Euclidean volume growth gives (4.23), in fact completing the proof.

32


We now particularize Theorem 4.12 to Asymptotically Locally Euclidean (ALE) manifolds, proving Corollary 1.2. We adopt the following definition, that is a sort of extension of the one considered in the celebrated [6], where striking relations between curvature decay conditions and behaviour at infinity of manifolds are drawn and sensibly weaker than the one used by Joyce in the classical reference [39]. Definition 4.13 (ALE manifolds). We say that a complete noncompact Riemannian manifold (M, g) is ALE (of order τ ) if there exist a compact set K ⊂ M , a ball B ⊂ Rn , a diffemorphism Ψ : M \ K → Rn \ B, a subgroup Γ < SO(n) acting freely on Rn \ B and a number τ > 0 such that −1 (4.25) (Ψ ◦ π)∗ g (z) = gRn + O(|z|)−τ (4.26) ∂i [(Ψ−1 ◦ π)∗ g] (z) = O(|z|)−τ −1 (4.27) ∂i ∂j [(Ψ−1 ◦ π)∗ g] (z) = O(|z|)−τ −2 , where π is the natural projection Rn → Rn /Γ, z ∈ Rn \ B and i, j = 1, . . . , n.

Proof of Corollary 1.2. Condition (4.25), (4.26), (4.27) readily imply that ALE manifolds have Euclidean volume growth and quadratic curvature decay. Moreover, condition (4.25) and a direct computation give that AVR(g) =

|Sn−1 /Γ| 1 = . n−1 |S | card Γ

(4.28)

The characterization (1.7) then follows immediately from (4.22). Assume now that the infimum of the Willmore functional is attained at some Ω ⊂ M . Then, by the rigidity part in Theorem 4.12, M \ Ω is isometric to a truncated cone over ∂Ω. However, by (4.25), (M, g) is also C 0 -close at infinity to a metric cone of cross section Sn−1 /Γ. Since the cross sections of a cone are all homothetic to each other, ∂Ω is homothetic to Sn−1 /Γ, that is, they are diffeomorphic and g∂Ω = λ2 gSn−1 /Γ for some positive constant λ. This fact, together with (4.28) in the rigidity part of Theorem 1.1 imply that 1 n−1 |∂Ω| 2 with r0 = r0 , +∞) × ∂Ω , dr ⊗ dr + (λr/r0 ) gSn−1 /Γ , . |Sn−1 /Γ| In particular,

Rλ n−1 n−1 |S /Γ|. r0 Combining it with (4.28) shows that λ = r0 , proving the isometry with (1.8) and completing the proof. |∂B(O, R)| =

5. Proof of the Enhanced Kasue’s Theorem The proof of Theorem 1.5 is completely analogous to the proof of Theorem 3.2, and this is why this section will be rather sketchy. We recall the definition, for β ≥ 0, of the function Ψβ : [0, ∞) → R satisfying ˆ Ψβ (s) =

|Dψ|β+1 dσ,

{ψ=s}

where ψ is a solution to problem (1.15) for some bounded Ω ⊂ M with smooth boundary and β ≥ 0. Combining the uniform bound on |Dψ| given in Theorem 2.16 with the constancy of Ψ0 ,


33

we obtain, as in Corollary 3.5, that Ψβ is uniformly bounded in s for any β ≥ 0. We record this fact for future reference. Lemma 5.1. Let (M, g) be a parabolic manifold with Ric ≥ 0, and let ψ be a solution to problem (1.15). Then, the function Ψβ defined in (1.16) is uniformly bounded for any β ≥ 0. Proof of Theorem 1.5. As for Theorem 3.2, we limit ourselves to prove Theorem 1.5 just for β ≥ 1, compare with Remark 1.4. Let us first extend the Bochner’s identity to |Dψ|β , obtaining, analogously to Lemma 3.6 h i 2 (5.1) ∆|Dψ|β = β|Dψ|β−2 |DDψ|2 + (β − 2) D|Dψ| + Ric(Dψ, Dψ) .

Observe that the right hand side of (5.1) is nonnegative if β ≥ (n − 2)/(n − 1), by means of the refined Kato’s inequality (3.17) for harmonic functions. Applying the Divergence Theorem and the co-area formula as done to obtain (3.29), we get ˆs ˆ β Dψ Ψβ (s) − Ψβ (s0 ) = D|Dψ| , dσ dτ. (5.2) |Dψ| {ψ=τ } s0

Exactly as in Lemma 3.7, we can use the Divergence Theorem and (5.1) to obtain, for any 0 ≤ τ0 < τ ˆ ˆ β Dψ β Dψ dσ − D|Dψ| , dσ D|Dψ| , |Dψ| |Dψ| {ψ=τ0 } {ψ=τ } ˆ i h 2 |Dψ|β−2 |DDψ|2 + (β − 2) D|Dψ| + Ric(Dψ, Dψ) dµ. =β {τ0 ≤ψ≤τ }

This, together with (5.2), allows to prove, as in the proof of Theorem 3.8, that Φβ is differentiable, that ˆ ˆ β Dψ ′ |Dψ|β H dσ, Ψβ (s) = D|Dψ| , dσ = −β |Dψ| {ψ=s} {ψ=s} and that, for any S ≥ s ≥ 0, ˆ 2 2 ′ ′ |Dψ|β−2 Ric(Dψ, Dψ) + DDψ + (β − 2) D|Dψ| dµ ≥ 0. Ψβ (S) − Ψβ (s) = β {s≤ψ≤S}

(5.3) Since Φβ is bounded by Lemma 5.1, we can argue as in the conclusion of the proof of Theorem 3.2 to pass to the limit as S → +∞ in (5.3) and obtain the monotonicity formula (1.17). The rigidity part of the statement is obtained exactly as for that of Theorem 3.2. The Enhanced Kasue’s Theorem 1.6 now follows at once. Proof of Theorem 1.6 and Corollary 1.7. Assume first that (M, g) is nonparabolic. Then, just by comparing (1.11) with (1.12) at t = 1, we get ˆ ˆ (n − 1) 1 |Du|β dσ, H |Du|β dσ ≤ sup H∂Ω Uβ (0) + Uβ′ (0) = (n − 2) β ∂Ω ∂Ω ∂Ω that is (1.18). If (M, g) is parabolic, then inequality (1.19) is proved in a completely analogous way. Since Uβ > 0, combining (1.18) and (1.19) shows that that H∂Ω ≤ 0 on ∂Ω if and only if (M, g) is parabolic and Ψ′β = 0. The rigidity statement in Theorem 1.5 then gives Corollary 1.7. Corollary 1.7 can also be interpreted as a rigidity statement when equality is attained in (1.6) if AVR(g) = 0. The following is then a direct consequence of Theorem 1.1 and Corollary 1.7.

34


Corollary 5.2. Let (M, g) be a complete noncompact Riemannian manifold with Ric ≥ 0 of dimension n ≥ 3. If Ω ⊂ M is a bounded subset with smooth boundary, then ˆ H n−1 n−1 |. (5.4) n − 1 dσ ≥ AVR(g)|S ∂Ω

If AVR(g) > 0, then equality in (5.4) holds if and only if (M \ Ω, g) is isometric to 1 n−1 |∂Ω| 2 r0 , +∞) × ∂Ω , dr ⊗ dr + (r/r0 ) g∂Ω , with r0 = . AVR(g)|Sn−1 |

In particular, ∂Ω is a connected submanifold with constant mean curvature. If AVR(g) = 0 , equality holds in (5.4) if and only if (M \ Ω, g) is isometric to a Riemannian product ([0, +∞) × ∂Ω, dr ⊗ dr + g∂Ω ). In particular, ∂Ω is a connected totally geodesic submanifold of (M, g). 6. Proof of the isoperimetric inequality for 3-manifolds As already discussed in the Introduction, we show here how our Willmore inequality (1.6) improves a result stated by Huisken in [35], characterizing, on 3-manifolds, the infimum of the Willmore energy in terms of the infimum of the isoperimetric ratio. Namely, we prove Theorem 1.9. Remark 6.1. Observe that Theorem 1.9 is obvious if (M, g) has cylindrical ends, that is, if there exists a bounded subset Ω ⊂ M with smooth boundary such that (M \ Ω, g) is isometric to half a cylinder, as in the rigidity statement of Theorems 3.2 and 1.5. Indeed, since on such an end any cross-section is minimal, the infimum of the Willmore functional is clearly zero, and, since any cross-section has the same surface area, considering an increasing sequence of regions enclosed by cross-sections shows that the infimum of the isoperimetric ratio is zero too. Remark 6.2. We point out that in the 3-dimensional case Theorem 1.9 extends Theorem 4.12 to any complete manifold manifold with Ric ≥ 0, with no restrictions on the volume growth or other assumptions on the curvatures. On the other hand, in dimension n = 3 the topology of nonnegatively Ricci curved manifolds is completely understood, see [50]. Let us briefly present the heuristic argument to deduce an isoperimetric inequality from Willmore’s through the mean curvature flow. Let Ω be a smooth and bounded set with meanconvex boundary. Let {Ωt }, with t ∈ [0, T ), be a smooth mean curvature flow starting from Ω. Suppose, in addition, that lim |Ωt | = 0. (6.1) t→T

Consider, for some constant C > 0 to be defined later, the isoperimetric difference D(t) = |∂Ωt |3/2 − C|Ωt |.

(6.2)

Taking derivatives in t, and using standard formulas (see for example [38, Theorem 3.2]), one finds ˆ ˆ d 3 1/2 2 H dσ, H dσ + C D(t) = − |∂Ωt | dt 2 ∂Ωt ∂Ωt that, through Hölder inequality, gives 1/2 " 1/2 # ˆ ˆ 3 d C− . H2 dσ H2 dσ D(t) ≤ |∂Ωt | dt 2 ∂Ωt ∂Ωt


35

Thus, if we choose C such that 1/2 ˆ 3 2 C≤ H dσ 2 ∂Ω for any bounded and smooth Ω ⊂ M , D is monotone nonincreasing in t. This implies that D(0) = |∂Ω|3/2 − C|Ω| ≥ lim D(t) ≥ 0, t→T

where we have also used (6.1). The above comparison in particular gives the (possibly not sharp) isoperimetric inequality |∂Ω|3/2 ≥ C. |Ω| 6.1. Tools from the theory of mean curvature flow. We are first concerned with justifying the above computations. The main steps consist in applying results of Schulze and White. We assume, for the time being, that the boundary ∂Ω of the bounded and smooth set Ω we are considering is mean-convex, that we understand as smooth with H∂Ω > 0. We will see later how to deal with more general cases. Since the mean curvature flow in general develops singularities, we consider the weak mean curvature flow in the sense of [30]. Then, a special case of the regularity theorem [69, Theorem 1.1] gives Theorem 6.3 (White’s regularity theorem). Let (M, g) be a Riemannian manifold of dimension 3. Let Ω ⊂ M be a bounded mean-convex set, and let {Ωt }t∈[0,T ) be its weak mean curvature flow. Then the boundary of Ωt is smooth for almost any t ∈ [0, T ). We point out that the maximal time T can be infinite on a general Riemannian manifold. We are going to combine the above regularity result with the following special case of [65, Proposition 7.2], that is a weak version of the monotonicity of the isoperimetric ratio. It can be checked, indeed, that the computations performed to obtain such a result do not involve the geometry of the underlying manifold. Theorem 6.4 (Schulze). Let (M, g) be a Riemannian manifold of dimension 3. Let Ω ⊂ M be a bounded mean-convex set, and let {Ωt }t∈[0,T ) be its weak mean curvature flow. Assume there exists a constant C = C(M ) ≥ 0 such that 1/2 ˆ 3 2 C≤ H dσ 2 ∂Ωt for almost any t ∈ [0, T ). Then the isoperimetric difference defined in (6.2) is monotone nonincreasing for any t ∈ [0, T ) and C as above. Remark 6.5. A different approach to deal with the singularities could be the mean curvature flow with surgery, recently developed by Brendle and Huisken in [10] and [11]. In this regard, it could be interesting understanding whether the monotonicity of the isoperimetric survives the surgeries. The following is a complete description of the long time behaviour of the weak mean curvature flow in Riemannian manifolds, given by [69, Theorem 11.1]. Theorem 6.6 (Long time behaviour of MCF). Let (M, g) be a Riemannian manifold. Let Ω ⊂ M be a bounded mean-convex set, let {Ωt }t∈[0,T ) be its weak mean curvature flow. If |Ωt | and |∂Ωt | do not vanish at finite time, then Ωt converges smoothly to a subset K, and the boundary of any connected component of K is a stable minimal submanifold.

36


As a consequence, if (M, g) contains no bounded subsets with minimal boundary, the weak MCF of a bounded set with mean-convex boundary is going to vanish. In particular, Kasue’s result Corollary 1.7, combined with Theorem 6.6, gives the following corollary. Corollary 6.7. Let (M, g) be a Riemannian manifold with Ric ≥ 0 and no cylindrical ends, let Ω ⊂ M be a bounded and mean-convex set, and let {Ωt }t∈[0,T ) be its weak mean curvature flow. Then T is finite and |Ωt | and |∂Ωt | tend to 0 as t → T − . In order to prove the isoperimetric inequality for possibly not mean-convex Ω, we are going to consider the minimizing hull Ω∗ (see [36, Section 1]). By [66], ∂Ω enjoys C 1,1 regularity, and by the minimizing property, its weak mean curvature H∂Ω∗ is nonnegative. We will actually actually flow Ω∗ by mean curvature. To this aim, we will invoke [37, Lemma 2.6], where the authors show that C 1 bounded hypersurfaces with nonnegative variational mean curvature can be approximated in C 1,β ∩ W 2,p , for any β ∈ (0, 1) and p ∈ [1, ∞) by smooth submanifolds with strictly positive mean-curvature. Interestingly, the approximating sequence is built through an appropriate notion of mean curvature flow starting from such a C 1 hypersurface. Although presented in Rn , the proof given in [37] can be easily adapted to go through the case of a general ambient Riemannian manifold, and we refer the reader to [74, Lemma 4.4] for the details of this extension. Such a result has also been pointed out in [43, Lemma 4.2] in the ambient setting of a Kottler-Schwarzschild manifold, and used in the proof of [65, Corollary 1.2], where the ambient manifold was a Cartan-Hadamard 3-manifold. We include here the precise statement. Lemma 6.8 (Huisken-Ilmanen’s approximation lemma). Let (M, g) be a Riemannian manifold, and let F : Σ ֒→ M be a C 1 closed immersed hypersurface. Assume Σ has nonnegative weak mean curvature, that is, there exists a nonnegative function H defined almost everywhere on Σ such that ˆ ˆ H hX, νi dσ divΣ X dσ = Σ

Σ

for any compactly supported vector field X of M . Assume also that Σ is not minimal, that is, there exists a subset K ⊂ Σ of positive measure such that H > 0 on K. Then, F is of class C 1,β ∩ W 2,p and there exists a family of smooth immersions F (·, ε) : Σ ֒→ M , with ε ∈ (0, ε0 ) such that d F (p, ε) = −HΣε (p, ε)ν(p, ε) dε for any ε ∈ (0, ε0 ), where HΣε is the mean curvature of Σε and ν its outer unit normal, and lim F (p, ε) = F (p)

ε→0+

locally uniformly in C 1,β ∩ W 2,p . Moreover, HΣε > 0 for any ε ∈ (0, ε0 ). Remark 6.9. Observe that approximating a hypersurface Σ with HΣ ≥ 0 by a sequence of smooth hypersurfaces Σε with HΣε > 0 is straightforward if Σ is C 2 . Indeed, by the classical short time existence theorem for the MCF (see e.g. [53]) there exists a solution to the MCF Σε defined for times ε ∈ (0, ε0 ), whose mean curvature satisfies (see e.g. [38, Theorem 3.2]) the following linear parabolic equation, ∂ H = ∆H + H |h|2 + Ric(ν, ν) , ∂ε where h is the second fundamental form of the evolving hypersurface and Ric is the Ricci tensor of the ambient manifold. Then, a standard maximum principle for parabolic equations (see e.g. Theorem 7 and subsequent remarks in [61]) shows that HΣε > 0 for any ε ∈ (0, ε0 ), unless HΣ is constantly null. We can finally prove Theorem 1.9.


37

Proof of Theorem 1.9. Observe first, that, in light of Remark 6.1, we can suppose that (M, g) has no cylindrical ends. We argue as in [65, proof of Corollary 1.2]. Let us first suppose that the boundary of ∂Ω is strictly mean-convex, that is, H∂Ω > 0. Let {Ωt }t∈[0,T ) be a mean curvature flow starting from Ω. Then, by Theorem 6.3, for almost any t ∈ [0, T ) the boundary ∂Ωt is a smooth submanifold. Let C be defined as ) ( ˆ 1/2 3 2 Ω ⊂ M bounded set with smooth boundary . H dσ C = inf 2 ∂Ω

Observe that C is possibly zero. Then, by Theorem 6.4, for such a choice of C the isoperimetric difference D(t) defined in (6.2) is monotone nonincreasing for any t ∈ [0, T ). Moreover, by Corollary 6.7 D tends to 0 as t → T − . This gives the inequality |∂Ω|3/2 ≥ C. |Ω|

for any Ω with positive mean curvature. If Ω is not mean-convex, take the minimizing hull Ω∗ of Ω (see [36, Section 1] for details). By [66] (compare also with the comprehensive [36, Theorem 1.3]) ∂Ω∗ is a C 1,1 hypersurface. Observe that, by the minimizing property, |∂Ω∗ | ≤ |∂Ω|, while trivially |Ω∗ | ≥ |Ω|, and thus proving a lower bound on the isoperimetric ratio for Ω∗ readily implies that the same lower bound holds for Ω. Moreover, again by the minimizing property, H∂Ω∗ ≥ 0 (see also [36, (1.15)]). By Lemma 6.8, we find a sequence of smooth hypersurfaces Σε with HΣε > 0 approximating ∂Ω∗ locally uniformly in C 1,β for any β ∈ (0, 1). Arguing as above we thus obtain the isoperimetric inequality |∂Σε |3/2 ≥ C, |Σε | that, through letting ε → 0+ , gives the isoperimetric inequality for Ω∗ , and, in turn, for any bounded Ω with smooth boundary. Combining it with our Willmore inequality (1.6), we get ˆ 2 H dσ |∂Ω|3 ∂Ω ≥ AVR(g), (6.3) inf ≥ inf 16π 36π|Ω|2 where the infima are taken over any bounded Ω with smooth boundary. By the Bishop-Gromov Theorem, we can find, for any ε > 0, a radius Rε such that |∂B(p, Rε )|3 ≤ AVR(g) + ε. 36π|B(p, Rε )|2 Observe that we can suppose ∂B(p, Rε ) to be smooth. Indeed, otherwise, it suffices to consider in place of B(p, Rε ) a smooth set whose perimeter and volume approximate |∂B(p, Rε )| and |B(p, Rε )|. This can be done by standard tools, see e.g. [51, Remark 13.2]. This proves, together with (6.3), that ( ) |∂Ω|3 inf Ω ⊂ M bounded and smooth = AVR(g). 36π|Ω|2

Combining it again with (6.3), this proves (1.20). We assume now that (1.21) holds with the equality sign for a smooth and bounded Ω ⊂ M . We can clearly suppose that AVR(g) > 0. By the minimizing property, Ω∗ satisfies the same equality (recall that we actually proved the isoperimetric inequality for minimizing hulls). We claim that it also holds for the region enclosed by any approximating Σε as above. Indeed, by

38


Lemma 6.8, this sequence is a smooth mean curvature flow for any ε ∈ (0, ε0 ), and then, by the monotonicity of the isoperimetric difference, for a fixed ε1 ∈ (0, ε0 ) we have D(ε) ≥ D(ε1 ) ≥ 0 for any ε ∈ (0, ε1 ). Since Σε converges to ∂Ω∗ as ε → 0+ , and on Ω∗ the isoperimetric difference is 0, D(ε) → 0+ as ε → 0+ , and thus D(ε1 ) = 0. By the arbitrariness of ε1 , the region enclosed by Σε satisfies equality in the isoperimetric inequality for any ε ∈ (0, ε0 ), as claimed. Consider, for any ε ∈ (0, ε0 ), a weak mean curvature flow {Σtε } of Σε , with t ∈ [0, T ). Then, the isoperimetric difference D is constantly 0 for t ∈ [0, T ). By White’s regularity result Theorem 6.3, for almost any t ∈ [0, T ) the derivative of D satisfies !1/2  !1/2  ˆ ˆ d  ≤ 0. C − 3 H2 dσ 0= H2 dσ D(t) ≤ Σtε dt 2 Σtε Σtε

This implies that Σtε minimizes the Willmore energy for almost any t ∈ [0, T ), and then, by the rigidity part of Theorem 1.1, outside of the region enclosed by Σtε there is an isometry with 1/2 |Σtε | rt , +∞) × Σtε , dr ⊗ dr + (r/rt )2 gΣtε , where rt = . AVR(g)|S2 |

Since, in particular, Σtε must be homothetic to Σε , a computation similar to the one performed in the proof of Corollary 1.2 in Section 5 shows that the above metric can be written as 1/2 |Σtε | AVR(g)|S2 | 2 where rt = . r gΣε , rt , +∞) × Σε , dr ⊗ dr + |Σε | AVR(g)|S2 | Since rt → 0 as t → T − , we deduce the isometry AVR(g)|S2 | 2 r gΣε , (M \ {p}, g) ∼ = 0, +∞) × Σε , dr ⊗ dr + |Σε | for some p ∈ M . In particular, the surface area of the geodesic balls centered at p decays as r 2 AVR(g)|S2 |, and, since g is smooth at p, this implies that AVR(g) = 1. By Bishop-Gromov, we infer that (M, g) is isometric to (R3 , gR3 ) and Σε is isometric to a sphere. Letting ε → 0+ , we infer that Ω∗ is a ball. This implies that Ω = Ω∗ , since, otherwise, the mean curvature of ∂Ω∗ would be null on the points not belonging to ∂Ω (compare also with [36, (1.15)]), leading to a contradiction. We have thus shown that Ω is a ball, completing the proof.

Appendix: comparison with the monotonicity formulas by Colding and Minicozzi In this section, we provide a comparison between our monotonicity formulas and those obtained by Colding and by Colding-Minicozzi in [21] and [24], respectively. To start with, let u be a solution of (1.9) in a nonparabolic Riemannian manifold (M, g) with Ric ≥ 0, for a bounded subset Ω ⊂ M with smooth boundary, and set 1

b = u− n−2 .

(6.4)

Note that b = 1 on ∂Ω and that b → +∞ at infinity. Associated with the level sets of b, consider the family of functions {Aβ }, where Aβ : [1, +∞) → [0, +∞) is defined for every β ≥ 0 as ˆ 1 |Db|β+1 dσ. Aβ (r) = n−1 r {b=r}

Now, replacing u be a minimal Green’s function G(O, ·), for some pole O ∈ M , the above defined Aβ is exactly the quantity considered in [24, formula (1.1)]. Note that in Colding’s setting, the level sets {b = r} are considered for every r > 0, since b(q) → 0 as d(O, q) → 0.


39

Our aim is to see how the monotonicity of our family of functions {Φβ } translates in terms of the family {Aβ }. First of all, it is straightforward from (3.1)–(3.3) and (6.4) that ϕ

b = e n−2 ,

|Db| =

|∇ϕ|g˜ , n−2

dσ = bn−1 dσg˜ ,

(6.5)

and in turn that s Φβ (s) = (n − 2)β+1 Aβ e n−2 ,

for every s ≥ 0.

(6.6)

We look at the derivative (3.15) of Φβ and at its equivalent expression (3.16). In particular, the volume integral (3.16) contains the following terms. n−2 2 Ric(∇ϕ, ∇ϕ) = Ric(Db2 , Db2 ), (6.7) 2 and ∇∇ϕ 2 + (β − 2) ∇|∇ϕ|g˜ 2 = g˜ g˜

n−2 2

2 ∆b2 2 g DD b2 − n 2 + (β − 2) DT |Db| i2 2 2 h H − (n − 1) D log b + (β − 2) Db (6.8)

which have been expressed in terms of the function b and of the metric g via some computations (compare with the proof of (1.12)) . Differentiating both sides of (6.6) and writing expression (3.16) in terms of b and g through formulas (6.5), (6.7) and (6.8), we obtain dAβ (n − 2)−β dΦβ (r) = (n − 2) log r (6.9) dr r ds ˆ β ∆b2 2 |Db|β−2 Ric(Db2 , Db2 ) + DDb2 − g = − r n−3 4 n {b>r} 2 + (β − 2) DT |Db| 2 2 h i2 2−2n + (β − 2) Db H − (n − 1) D log b b dµ ≤ 0 .

Setting b = 2 in the above formula, we obtain exactly the integrand of the right hand side of [21, (2.106)], that, arguing as in the conclusion of the present Theorem 3.2, leads to the monotonicity of A2 . For a general β ≥ (n − 2)/(n − 1), in [26, Theorem 1.3] the monotonicity of Aβ is inferred grouping the terms in (6.9) in a different way. Observe indeed that for β < 2 the volume integral in (6.9) does not evidently carry a sign. On the other hand, (6.8) combined with Kato’s inequality immediately show the nonnegativity of the expression. We close this appendix by showing how our methods can be applied also to obtain the Monotonicity-Rigidity Theorem for the Green’s function, obtaining a new (conformal) proof of Colding-Minicozzi’s [26, Theorem 1.3]. Indeed, let (M, g) be a nonparabolic Riemannian manifold with Ric ≥ 0, let G be its minimal Green’s function and consider the new metric on M \ {O} 2

g˜ = G(O, ·) n−2 g, for some point O ∈ M . Set ϕ = − log G(O, ·).

40


Then, we have that the triple M, g˜, ϕ satisfies the system  ∆g˜ ϕ = 0      |∇ϕ|2g˜  dϕ ⊗ dϕ Ricg˜ −∇∇ϕ + = g˜ + Ric n − 2 n − 2    ϕ(q) → +∞    ϕ(q) → −∞

in M \ {O} in M \ {O} as d(O, q) → +∞ as d(O, q) → 0.

We denote by d the distance with respect to g. Define the function Φβ : R 7→ R given by ˆ dσg˜ . |∇ϕ|β+1 Φβ (s) = g˜ {ϕ=s}

All the theory developed in Section 3 holds with trivial modification for Φβ as above, and immediately yields a conformal Monotonicity-Rigidity Theorem for the Green’s function. Theorem 6.10. Let (M, g) be a nonparabolic Riemannian manifold with Ric ≥ 0. Let G be its minimal Green’s function. Then, with the notations above, we have ˆ |∇ϕ|β−2 Ric(∇ϕ, ∇ϕ) + ∇∇ϕ 2 + (β − 2) ∇|∇ϕ|g˜ 2 g˜ dΦβ g˜ g˜ (s) = − β es dµg˜ ϕ ds e {ϕ≥s}

Φ′β

is alway nonpositive. Moreover, ( dΦβ / ds)(s0 ) = 0 for some s0 ∈ R and In particular, some β ≥ (n − 2)/(n − 1) if and only if {ϕ ≥ s0 } is isometric to the Riemannian product [s0 , ∞) × {ϕ = s0 }, dρ ⊗ dρ + g˜{|ϕ=s0 } .

The above Theorem clearly translates in terms of (M, g) and G, exactly as Theorem 1.3 was deduced from Theorem 3.2.

Acknowledgements. The author are grateful to C. Arezzo, A. Carlotto, A. Farina, G. Huisken, L. Mari, D. Peralta-Salas and F. Schulze for useful comments and discussions during the preparation of the paper. The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (GNAMPA), which is part of the Istituto Nazionale di Alta Matematica (INdAM). The paper was partially completed during the authors’ attendance to the program “Geometry and relativity” organized by the Erwin Schroedinger International Institute for Mathematics and Physics (ESI). References [1] V. Agostiniani and L. Mazzieri. Monotonicity formulas in potential theory. arXiv:1606.02489v3. [2] V. Agostiniani and L. Mazzieri. Riemannian aspects of potential theory. J. Math. Pures Appl., 104(3):561 – 586, 2015. [3] V. Agostiniani and L. Mazzieri. On the geometry of the level sets of bounded static potentials. Commun. Math. Phys., 355:261 – 301, 2017. [4] G. Alberti, S. Bianchini, and G. Crippa. Structure of level sets and sard-type properties of lipschitz maps. (SISSA;51/2011/M), 2011. [5] F. Almgren. Optimal isoperimetric inequalities. Indiana Univ. Math. J., 35:451–547, 1986. [6] S. Bando, A. Kasue, and H. Nakajima. On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Inventiones mathematicae, 97(2):313–350, 1989. [7] S. Borghini, G. Mascellani, and L. Mazzieri. Some sphere theorem on linear potential theory. arXiv:1705.09940. [8] S. Borghini and L. Mazzieri. On the mass of static metrics with positive cosmological constant: II. arXiv:1711.07024. [9] S. Borghini and L. Mazzieri. On the mass of static metrics with positive cosmological constant: I. Classical Quantum Gravity, 35(12):125001, 43, 2018.


41

[10] S. Brendle and G. Huisken. Mean curvature flow with surgery of mean convex surfaces in R3 . Invent. Math., 203(2):615–654, 2016. [11] S. Brendle and G. Huisken. Mean curvature flow with surgery of mean convex surfaces in three-manifolds. J. Eur. Math. Soc. (JEMS), 20(9):2239–2257, 2018. [12] G. Catino, C. Mantegazza, and L. Mazzieri. On the global structure of conformal gradient solitons with nonnegative Ricci tensor. Commun. Contemp. Math., 14(6):1250045, 12, 2012. [13] J. Cheeger and T. H. Colding. Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. of Math. (2), 144(1):189–237, 1996. [14] J. Cheeger and D. Gromoll. The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differential Geometry, 6:119–128, 1971/72. [15] J. Cheeger, M. Gromov, and M. Taylor. Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom., 17(1):15–53, 1982. [16] J. Cheeger, A. Naber, and D. Valtorta. Critical sets of elliptic equations. Comm. Pure Appl. Math., 68(2):173– 209, 2015. [17] B.-Y. Chen. On a theorem of Fenchel-Borsuk-Willmore-Chern-Lashof. Mathematische Annalen, 194(1):19–26, 1971. [18] G. Chen. Classification of gravitational instantons with faster than quadratic curvature decay. PhD thesis, Stony Brook University, 2017. [19] O. Chodosh, M. Eichmair, and A. Volkmann. Isoperimetric structure of asymptotically conical manifolds. J. Differential Geom., 105(1):1–19, 01 2017. [20] B. Chow, P. Lu, and L. Ni. Hamilton’s Ricci flow, volume 77 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2006. [21] T. H. Colding. New monotonicity formulas for Ricci curvature and applications. I. Acta Mathematica, 209(2):229–263, 2012. [22] T. H Colding and W. P. Minicozzi. Harmonic functions with polynomial growth. J. Differential Geom., 45:1–77, 1997. [23] T. H. Colding and W. P. Minicozzi. Large scale behaviour of kernels of shroedinger operators. American Journal of Mathematics, 119:1355–1398, 1997. [24] T. H. Colding and W. P. Minicozzi. Monotonicity and its analytic and geometric implications. Proceedings of the National Academy of Sciences, 110(48):19233–19236, 2013. [25] T. H Colding and W. P. Minicozzi. On uniqueness of tangent cones for einstein manifolds. Invent. math., 196:515–588, 2014. [26] T. H. Colding and W. P. Minicozzi. Ricci curvature and monotonicity for harmonic functions. Calculus of Variations and Partial Differential Equations, 49(3):1045–1059, 2014. ´ Norm. Sup. (4), [27] C. B. Croke. Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. Ecole 13(4):419–435, 1980. [28] C. B. Croke. A sharp four-dimensional isoperimetric inequality. Comment. Math. Helv., 59(2):187–192, 1984. [29] T. Eguchi and A. J. Hanson. Self-dual solutions to Euclidean gravity. Ann. Physics, 120(1):82–106, 1979. [30] L. C. Evans and J. Spruck. Motion of level sets by mean curvature. I. J. Differential Geom., 33(3):635–681, 1991. [31] A. Farina, L. Mari, and E. Valdinoci. Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds. Comm. Partial Differential Equations, 38(10):1818–1862, 2013. [32] M. Fogagnolo, L. Mazzieri, and A. Pinamonti. Geometric aspects of p-capacitary potentials. To appear on Ann. Inst. H. Poincaré Anal. Non Linéaire. https://doi.org/10.1016/j.anihpc.2018.11.005. [33] A. Grigor’yan. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. (N.S.), 36(2):135–249, 1999. [34] S. W. Hawking. Gravitational instantons. Phys. Lett. A, 60(2):81–83, 1977. [35] G. Huisken. An isoperimetric concept for the mass in general relativity. Video, available at https://video.ias.edu/node/234. [36] G. Huisken and T. Ilmanen. The inverse mean curvature flow and the riemannian penrose inequality. J. Differential Geom., 59(3):353–437, 11 2001. [37] G. Huisken and T. Ilmanen. Higher regularity of the inverse mean curvature flow. J. Differential Geom., 80(3):433–451, 11 2008. [38] G. Huisken and A. Polden. Geometric evolution equations for hypersurfaces. In Calculus of variations and geometric evolution problems (Cetraro, 1996), volume 1713 of Lecture Notes in Math., pages 45–84. Springer, Berlin, 1999.

42


[39] D. D. Joyce. Compact manifolds with special holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000. [40] A. Kasue. Ricci curvature, geodesics and some geometric properties of riemannian manifolds with boundary. J. Math. Soc. Japan, 35(1):117–131, 01 1983. [41] P. B. Kronheimer. The construction of ALE spaces as hyper-K¨ ahler quotients. J. Differential Geom., 29(3):665–683, 1989. [42] P. B. Kronheimer. A Torelli-type theorem for gravitational instantons. J. Differential Geom., 29(3):685–697, 1989. [43] H. Li and Y. Wei. On inverse mean curvature flow in Schwarzschild space and Kottler space. Calc. Var. Partial Differential Equations, 56(3):Art. 62, 21, 2017. [44] P. Li. Lectures on harmonic function. Lectures at UCI. [45] P. Li and L.-F. Tam. Symmetric Green’s functions on complete manifolds. Amer. J. Math., 109(6):1129–1154, 1987. [46] P. Li and L.-F. Tam. Harmonic functions and the structure of complete manifolds. J. Differential Geom., 35(2):359–383, 1992. [47] P. Li and L.-F. Tam. Green’s functions, harmonic functions, and volume comparison. J. Differential Geom., 41(2):277–318, 1995. [48] P. Li, L.-F. Tam, and J. Wang. Sharp bounds for the Green’s function and the heat kernel. Math. Res. Lett., 4(4):589–602, 1997. [49] P. Li and S.-T. Yau. On the parabolic kernel of the Schr¨ odinger operator. Acta Math., 156(3-4):153–201, 1986. [50] G. Liu. 3-manifolds with nonnegative Ricci curvature. Invent. Math., 193(2):367–375, 2013. [51] F. Maggi. Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2012. [52] B. Malgrange. Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution. Ann. Inst. Fourier, Grenoble, 6:271–355, 1955–1956. [53] C. Mantegazza. Lecture notes on mean curvature flow, volume 290 of Progress in Mathematics. Birkh¨ auser/Springer Basel AG, Basel, 2011. [54] C. Mantegazza and A. C. Mennucci. Hamilton-Jacobi equations and distance functions on Riemannian manifolds. Appl. Math. Optim., 47(1):1–25, 2003. [55] X. Menguy. Noncollapsing examples with positive Ricci curvature and infinite topological type. Geom. Funct. Anal., 10(3):600–627, 2000. [56] V. Minerbe. A mass for ALF manifolds. Comm. Math. Phys., 289(3):925–955, 2009. ´ Norm. Supér. (4), [57] V. Minerbe. On the asymptotic geometry of gravitational instantons. Ann. Sci. Ec. 43(6):883–924, 2010. [58] V. Minerbe. Rigidity for multi-Taub-NUT metrics. J. Reine Angew. Math., 656:47–58, 2011. [59] A. Mondino and E. Spadaro. On an isoperimetric-isodiametric inequality. Anal. PDE, 10(1):95–126, 2017. [60] P. Petersen. Riemannian geometry, volume 171 of Graduate Texts in Mathematics. Springer, New York, second edition, 2006. [61] M. H. Protter and H. F. Weinberger. Maximum principles in differential equations. Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. [62] M. Reiris. On Ricci curvature and volume growth in dimension three. J. Differential Geom., 99(2):313–357, 2015. [63] R. Schoen and S.-T. Yau. Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu, Translated from the Chinese by Ding and S. Y. Cheng, With a preface translated from the Chinese by Kaising Tso. [64] F. Schulze. Optimal isoperimetric inequalities for surfaces in any codimension in cartan-hadamard manifolds. arXiv:1802.00226. [65] F. Schulze. Nonlinear evolution by mean curvature and isoperimetric inequalities. J. Differential Geom., 79(2):197–241, 06 2008. [66] P. Sternberg, W. P. Ziemer, and G. Williams. C1,1-regularity of constrained area minimizing hypersurfaces. Journal of Differential Equations, 94(1):83 – 94, 1991. [67] P. Topping. Mean curvature flow and geometric inequalities. J. Reine Angew. Math., 503:47–61, 1998. [68] N. Th. Varopoulos. Green’s functions on positively curved manifolds. II. J. Funct. Anal., 49(2):170–176, 1982. [69] B. White. The size of the singular set in mean curvature flow of mean-convex sets. Journal of the American Mathematical Society, 13(3):665–695, 2000.


43

[70] T. J. Willmore. Mean curvature of immersed surfaces. An. S ¸ ti. Univ. “All. I. Cuza” Ia¸si Sect¸. I a Mat. (N.S.), 14:99–103, 1968. [71] S.-T. Yau. Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math., 28:201–228, 1975. [72] S.-T. Yau. Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J., 25(7):659–670, 1976. [73] N. Yeganefar. On manifolds with quadratic curvature decay. Compositio Mathematica, 145(2):528–540, 2009. [74] H. Zhou. Inverse mean curvature flows in warped product manifolds. arXiv:1609.09665v5. ` degli Studi di Verona, strada Le Grazie 15, 37134 Verona, Italy V. Agostiniani, Universita E-mail address: [email protected] ` degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy M. Fogagnolo, Universita E-mail address: [email protected] ` degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy L. Mazzieri, Universita E-mail address: [email protected]