An optimal gap theorem

TL;DR

This paper proves an optimal gap theorem for Kähler manifolds with nonnegative bisectional curvature, showing flatness under certain scalar curvature decay conditions, and establishes a positive mass type result for non-flat manifolds.

Contribution

It introduces a novel approach using the Hodge-Laplace heat equation to establish an optimal gap theorem and a stronger positive mass type result for Kähler manifolds.

Findings

Manifolds are flat if scalar curvature averages decay as r^{-2}.

Established a positive mass type result for non-flat manifolds.

Used the Cauchy problem for the Hodge-Laplace heat equation in the proof.

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 K\"ahler 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 $o\in M$.