An optimal gap theorem

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 ${\mathrm{liminf}}_{r\to \infty }\frac{r^2}{V_o(r)}{\int }_{B_o(r)}\mathcal{S}(y)d\mu (y)>0$ for any $o\in M$.

References in corpus (1)

• Poincaré-Lelong equation via the Hodge Laplace heat equation

Cited by in corpus (4)

• Poincaré-Lelong equation via the Hodge Laplace heat equation
• Kähler manifolds and cross quadratic bisectional curvature
• Existence of nonconstant CR-holomorphic functions of polynomial growth in Sasakian Manifolds
• Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature