A gap theorem for Ricci-flat 4-manifolds

TL;DR

This paper establishes a gap theorem showing that compact Ricci-flat 4-manifolds with sectional curvatures satisfying a specific ratio condition must be flat, extending to Ricci-flat Kähler surfaces with a similar holomorphic curvature condition.

Contribution

The paper proves new curvature gap theorems for Ricci-flat 4-manifolds and Ricci-flat Kähler surfaces, identifying precise bounds under which these manifolds must be flat.

Findings

If sectional curvatures satisfy $K_{max} \, \leq -c K_{min}$ with $c<\frac{2+\sqrt{6}}{4}$, then the manifold is flat.

A similar flatness result holds for Ricci-flat Kähler surfaces with holomorphic sectional curvatures satisfying a comparable ratio.

The results provide sharp thresholds for curvature ratios implying flatness in these geometric contexts.

Abstract

Let $(M,g)$ be a compact Ricci-flat 4-manifold. For $p \in M$ let $K_{max}(p)$ (respectively $K_{min}(p)$) denote the maximum (respectively the minimum) of sectional curvatures at $p$. We prove that if $$K_{max} (p) \le \ -c K_{min}(p)$$ for all $p \in M$, for some constant $c$ with $0 \leq c < \frac{2+\sqrt 6}{4}$, then $(M,g)$ is flat. We prove a similar result for compact Ricci-flat K\"ahler surfaces. Let $(M,g)$ be such a surface and for $p \in M$ let $H_{max}(p)$ (respectively $H_{min}(p)$) denote the maximum (respectively the minimum) of holomorphic sectional curvatures at $p$. If $$H_{max} (p) \le -c H_{min}(p)$$ for all $p \in M$, for some constant $c$ with $0 \leq c < \frac {1+\sqrt 3}{2}$, then $(M,g)$ is flat.