An optimal gap theorem |
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Authors: | Email author" target="_blank">Lei?NiEmail author |
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Institution: | 1.Department of Mathematics,University of California at San Diego,La Jolla,USA |
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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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