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

TL;DR

This paper extends Gursky's gap theorem by establishing a universal constant that characterizes when a positive Einstein metric on the four-sphere must be isometric to the standard round metric, based on the Yamabe constant proximity.

Contribution

The paper introduces a universal gap constant for positive Einstein metrics on S^4, strengthening the classification criteria based on the Yamabe constant.

Findings

Existence of a universal positive constant  > 0.

Metrics with Yamabe constant close to that of the round sphere are isometric to it.

Extension of Gursky's gap theorem to a broader class of metrics.

Abstract

We show that there exists a universal positive constant $\varepsilon_0 > 0$ with the following property: Let $g$ be a positive Einstein metric on $S^4$. If the Yamabe constant of the conformal class $[g]$ satisfies $$ Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}]) - \varepsilon_0 $$ 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.