However, if the manifold M is complex-analytic and Kaihler, an important ... we provethat we can alwaysfind a holomorphic function with polynomial growth ...
Proc. Natl. Acad. Sci. USA Vol. 73, No. 4, p. 1008, April 1976
Mathematics
On the structure of complete simply-connected Kihler manifolds with nonpositive curvature (curvature decay/biholotnorphism/geometric characterization/complex Euclidean space)
YUM-TONG SIU* AND SHING-TUNG YAUt * Yale University, Department of Mathematics, New Haven, Connecticut 06520; and t Stanford University, Department of Mathematics, Stanford, California 94305
Communicated by S. S. Chern, February 9, 1976
complex), there exists no nontrivial bounded harmonic function (ref. 2). Their method was to apply J. Moser's Harnack inequality by observing that the Laplacian of the manifold is uniformly elliptic. They also observed that one cannot set c = 0 in the hypothesis of the theorem, because
ABSTRACT We prove that a complete simply-connected Kahier manifold with nonpositive sectional curvature is biholomorphic to the complex Euclidean space if the curvature is suitably small at infinity.
It is a well-known theorem of Cartan-Hadamard that.every complete simply-connected Riemannian manifold M with nonpositive curvature is diffeomorphic to an Euclidean space. However, if the manifold M is complex-analytic and Kaihler, an important question remains to be settled, namely, what the complex structure on M looks like. Concerning this question the following two statements are well-known: (i) If M has zero curvature everywhere, then M is biholomorphic to a complex Euclidean space. (ii) If the holomorphic sectional curvature of M is a negative constant, then M is biholomorphic to the open ball of the same dimension. In this note we announce a "stable" version of statement (i), namely, when the sectional curvature of M is sufficiently close to zero at infinity, we prove that M is still biholomorphic to a complex Euclidean space. This is a nontrivial geometric characterization of a complex Euclidean space. The precise statement of our theorem is the following: THEOREM. Let M be a complete simply-connected Kdihler manifold with nonpositive sectional curvature. Suppose for some fixed point 0 of M there are positive constants A and c such that all sectional curvatures at a point x of M are not less than -A
Idzl2 (1-
1Z12)3
is a Kahler metric on the open unit 1-disc which satisfies the hypothesis of the theorem. Our method is based on the L2 estimates of a of Hormander and Andreotti-Vesentini. We construct holomorphic functions on M similar to the linear functions on an Euclidean space and use them to map M into the complex Euclidean space of the same dimension. In the first step we only know that the map is nonsingular at one point. Then we use the growth property of these "linear" functions to prove certain uniqueness property to conclude that the map is nonsingular everywhere. (This depends on the estimate of the growth of the volume of complex subvarieties in M.) In order to complete the proof of the theorem we have to prove that the above map is proper. In this step we use the above-mentioned uniqueness property to prove that every holomorphic function that has polynomial growth is actually a polynomial of the coordinates of the above map. Then we prove that we can always find a holomorphic function with polynomial growth which grows fast enough at infinity. Putting these two intermediate results together, we show that the above map is proper. Details of the proof of the theorem will appear elsewhere.
d(O,x)2+c
where d(O,x) is the distance from x to 0. Then M is biholomorphic to the complex Euclidean space of the same dimension. It should be noted that this theorem was first conjectured by R. Greene and H. Wu (ref. 1). Prior to this, they had already proved that under the assumption of the theorem and within the general category of Riemannian manifolds (not necessarily
1. Greene, R. & Wu, H. (1976) Analysis on Noncompact Kahler Manifolds, Lecture Notes of the 1975 Amer. Math. Soc. Summer Institute, in press. 2. Greene, R. & Wu, H. (1971) "Curvature and complex analysis," Bull. Am. Math. Soc. 77, 1045-1049.
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