Mathematische Annalen

, Volume 373, Issue 3–4, pp 1329–1339 | Cite as

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

  • Kazuo AkutagawaEmail author
  • Hisaaki Endo
  • Harish Seshadri


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 \(S^4\).



The authors would like to thank Anda Degeratu and Rafe Mazzeo for valuable discussions on the eta invariant, and Shouhei Honda for helpful discussions on convergence results of Riemannian manifolds with bounded Ricci curvature. They would also like to thank Matthew Gursky and Claude LeBrun for useful advice, and Gilles Carron and the anonymous referee for crucial comments.


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Authors and Affiliations

  1. 1.Department of MathematicsTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of MathematicsChuo UniversityTokyoJapan
  3. 3.Mathematics DepartmentIndian Institute of ScienceBangaloreIndia

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