A gap theorem for positive Einstein metrics on the four-sphere

Abstract

We show that there exists a universal positive constant ${\epsilon }_0>0$ with the following property: Let $g$ be a positive Einstein metric on $S^4$. If the Yamabe constant of the conformal class $[g]$ satisfies $$Y(S^4,[g])>\frac{1}{\sqrt{3}}Y(S^4,[g_{\mathbb{S}}])-{\epsilon }_0$$ where $g_{\mathbb{S}}$ denotes the standard round metric on $S^4$, then, up to rescaling, $g$ is isometric to $g_{\mathbb{S}}$. This is an extension of Gursky's gap theorem for positive Einstein metrics on the four-sphere.

9 pages