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An optimal gap theorem

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An Erratum to this article was published on 18 October 2013

Abstract

By solving the Cauchy problem for the Hodge-Laplace heat equation for d-closed, positive (1,1)-forms, we prove an optimal gap theorem for Kähler manifolds with nonnegative bisectional curvature which asserts that the manifold is flat if the average of the scalar curvature over balls of radius r centered at any fixed point o is a function of o(r −2). Furthermore via a relative monotonicity estimate we obtain a stronger statement, namely a ‘positive mass’ type result, asserting that if (M,g) is not flat, then \(\liminf_{r\to\infty} \frac {r^{2}}{V_{o}(r)}\int_{B_{o}(r)}\mathcal{S}(y)\, d\mu(y)>0\) for any o∈M.

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Authors and Affiliations

  1. Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093, USA

    Lei Ni

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  1. Lei Ni
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Correspondence to Lei Ni.

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The author was supported in part by NSF grant DMS-1105549.

An erratum to this article is available at http://dx.doi.org/10.1007/s00222-013-0487-7.

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Ni, L. An optimal gap theorem. Invent. math. 189, 737–761 (2012). https://doi.org/10.1007/s00222-012-0375-6

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  • DOI: https://doi.org/10.1007/s00222-012-0375-6

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