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

We show that there exists a universal positive constant ε 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 Y ( S 4 , [ g ] ) > 1 3 Y ( S 4 , [ g S ] ) - ε 0 , where g S denotes the standard round metric on S 4 , then, up to rescaling, g is isometric to g S . This is an extension of Gursky’s gap theorem for positive Einstein metrics on S 4 .