A gap theorem for Ricci-flat 4-manifolds

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 $0\le c<\frac{2+\sqrt{6}}{4}$, then $(M,g)$ is flat. We prove a similar result for compact Ricci-flat Kähler 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 $0\le c<\frac{1+\sqrt{3}}{2}$, then $(M,g)$ is flat.

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