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

We show that there exists a universal positive constant $$\varepsilon _0 > 0$$ with the following property: let g be a positive Einstein metric on the four-sphere $$S^4$$ . If the Yamabe constant of the conformal class [g] satisfies $$\begin{aligned} Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}]) - \varepsilon _0\,, \end{aligned}$$ 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 $$S^4$$ .