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.
Categories: math.DG