An optimal gap theorem
type: Item
instance of: scholarly article
title: An optimal gap theorem
zbMATH Open document ID: 1253.32009
author: Lei Ni
published in: Inventiones Mathematicae
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1104.3185
review text: This paper studies complete non-compact Kähler manifolds with non-negative holomorphic bisectional curvature. The main theorem asserts that such a Kähler manifold \(M\) is flat if the average of scalar curvature \(S\) over the metric ball \(B_{p}(r)\) of radius \(r\) centered at a fixed point \(p\in M\) is a function of \(o(r^{-2})\). This theorem generalizes a earlier result in [the author and \textit{L.-F. Tam}, J. Differ. Geom. 64, No. 3, 457--524 (2003; Zbl 1088.32013)]. The proof involves the Cauchy problem of the Hodge-Laplacian heat equation for closed positive \((1,1)\)-forms. A `positive mass' type corollary is obtained by combining the main result and a monotonicity property, which says that a non-negative non-flat complete Kähler manifolds satisfies \[ \liminf_{r \to \infty} \frac{r^{2}}{\text{Vol}(B_{p}(r))}\int_{B_{p}(r)}Sd\mu >0 \] for any fixed \(p\).
reviewed by: Yuguang Zhang
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32Q30
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 6084024
zbMATH Keywords: complete non-comapct Kähler manifolds
zbMATH Keywords: non-negative holomorphic bisectional curvature
zbMATH Keywords: heat equation
MaRDI profile type: Publication
OpenAlex ID: W2072114120
arXiv ID: 1104.3185
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DOI: 10.1007/S00222-012-0375-6
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description: scientific article; zbMATH DE number 6084024