Sharp dimension estimates of holomorphic functions and rigidity

Chen, Bing-Long; Yin, Le; Zhu, Xia

doi:10.1090/s0002-9947-05-04105-x

Cited by 17 publications

(5 citation statements)

References 25 publications

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“…Suppose the holomorphic sectional curvature is positive at one point, then there exists ǫ > 0 depending only on M such that for any integer d Remark. Similar results were proved by Chen, Fu, Le and Zhu [8].…”

Section: Remark the Inequality Dim(osupporting

confidence: 88%

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Three Circle Theorems on Kähler manifolds and applications

Liu¹

2013

Preprint

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The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions are derived. Among applications, we derive sharp dimension estimates (with rigidity) of holomorphic functions with polynomial growth when the holomorphic sectional curvature is nonnegative. When the bisectional curvature is nonnegative, this was due to Ni. Also we study holomorphic functions with polynomial growth near infinity. On a complete noncompact Kähler manifold with nonnegative bisectional curvature, we prove any holomorphic function with polynomial growth is homogenous at infinity. This result is closely related with Yau's conjecture on the finite generation of holomorphic functions (see page 3 − 4 for detailed explanation).We also generalize the three circle theorem to Kähler manifolds with holomorphic sectional curvature lower bound. These inequalities are sharp in general. As applications, we establish sharp dimension estimates for holomorphic functions with certain growth conditions when the holomorphic sectional curvature is asymptotically nonnegative.Finally observe that Hitchin constructed some complex manifolds which admit Kähler metric with positive holomorphic sectional curvature, yet do not even admit Kähler metrics with nonnegative Ricci curvature. Thus the set of complete Kähler manifolds with nonnegative holomorphic sectional curvature is strictly wider than those with nonnegative bisectional curvature. r→∞ Vol(B(p,r)) r 2n > 0. In [45], Ni proved a remarkable theorem which confirmed conjecture 3 by assuming the maximal volume growth condition:

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Section: Remark the Inequality Dim(osupporting

confidence: 88%

“…By using same technique in [45], Chen, Fu, Le and Zhu [8] removed the maximal volume growth condition in theorem 4. Therefore, conjecture 3 was solved completely.…”

Section: Theorem 2 Let M Be a Complete Kähler Manifold Then M Satisfi...mentioning

confidence: 99%

Three Circle Theorems on Kähler manifolds and applications

Liu¹

2013

Preprint

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“…Conjecture 3 was confirmed by Ni [23] with the assumption that M has maximal volume growth. Later, by using Ni's method, Chen-Fu-Le-Zhu [6] removed the extra condition. See also [17] for a different proof.…”

Section: Conjecture 3 Let M N Be a Complete Noncompact Kähler Manifol...mentioning

confidence: 99%

“…A multiplicity estimate by Ni [23](see also [6]): Theorem 9. Let M n be a complete noncompact Kähler manifold with nonnegative bisectional curvature.…”

Section: Definitionmentioning

confidence: 99%

Gromov-Hausdorff limits of Kähler manifolds and the finite generation conjecture

Liu

2016

Ann. Math.

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We study the uniformization conjecture of Yau by using the Gromov-Haudorff convergence. As a consequence, we confirm Yau's finite generation conjecture. More precisely, on a complete noncompact Kähler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely generated. During the course of the proof, we prove if M n is a complete noncompact Kähler manifold with nonnegative bisectional curvature and maximal volume growth, then M is biholomorphic to an affine algebraic variety. We also confirm a conjecture of Ni on the existence of polynomial growth holomorphic functions on Kähler manifolds with nonnegative bisectional curvature.

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“…for any positive integer d. If the equality holds for some d, M is isometric and biholomorphic to C n . Later Chen, Fu, Le, Zhu [6] removed the maximal volume growth condition by using the same technique in [17]. See also [14] for a different proof.…”

Section: Here R Is the Distance From A Fixed Point P On M; M F (R) Is...mentioning

confidence: 99%

On the volume growth of Kähler manifolds with nonnegative bisectional curvature

Liu

2016

J. Differential Geom.

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Abstract. Let M be a complete Kähler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and M admits a nonconstant holomorphic function with polynomial growth, we prove M must be of maximal volume growth. This confirms a conjecture of Ni in [17]. There are two essential ingredients in the proof: the Cheeger-Colding theory [2]-[5] on Gromov-Hausdorff convergence of manifolds; the three circle theorem for holomorphic functions in [14].

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Sharp dimension estimates of holomorphic functions and rigidity

Cited by 17 publications

References 25 publications

Three Circle Theorems on Kähler manifolds and applications

Three Circle Theorems on Kähler manifolds and applications

Gromov-Hausdorff limits of Kähler manifolds and the finite generation conjecture

On the volume growth of Kähler manifolds with nonnegative bisectional curvature

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