Almost Complex Manifolds and Hyperbolicity
Copyright policy )Almost Complex Manifolds and Hyperbolicity
One of the sufficient conditions for a complex manifold to be (complete) hyperbolic (that is, its intrinsic pseudo-distance is a (complete) distance) is that it has a (complete) Hermitian metric with holomorphic sectional curvature bounded above by a negative constant. Such a concept of hyperbolicity can be extended to almost complex manifolds. In this paper, the author shows that the above result on hyperbolicity also holds for the almost complex case. In fact, he proves that if an almost complex manifold \(M\) admits a (complete) Hermitian metric with holomorphic sectional curvature bounded above by \(-1\), then \(M\) is (complete) hyperbolic. Moreover, as an application, he shows that every point of an almost complex manifold has a complete hyperbolic neighborhood. In real dimension 4, this fact was proved by \textit{R. Debalme} and \textit{S. Ivashkovich} [Int. J. Math. 12, 211-221 (2001; Zbl 1110.32306)] by a completely different method.
- University of California, Berkeley United States
- University of California, San Francisco United States
hyperbolicity, General geometric structures on manifolds (almost complex, almost product structures, etc.), Hermitian metrics, holomorphic sectional curvature, almost complex manifolds, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, distance, Other complex differential geometry
hyperbolicity, General geometric structures on manifolds (almost complex, almost product structures, etc.), Hermitian metrics, holomorphic sectional curvature, almost complex manifolds, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, distance, Other complex differential geometry
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