Einstein four-manifolds of three-nonnegative curvature operator



In this paper we prove that Einstein four-manifolds of 3-positive curvature operator are isometric to \((S^4, g_0)\) or \(({\mathbb {C}}P^2, g_{FS})\), and Einstein four-manifolds of 3-nonnegative curvature operator are isometric to \((S^4, g_0)\), \(({\mathbb {C}}P^2, g_{FS})\), or \((S^2\times S^2, g_0\oplus g_0)\), up to rescaling. We also prove that the first eigenvalue of the Laplace operator for Einstein four-manifolds with \(\mathrm {Ric}=g\) and nonnegative sectional curvature is bounded above by \(\frac{4}{3}+4^{\frac{1}{3}}\). The basic idea of the proofs is to construct an “integrated subharmonic function”, and the main ingredients of the proofs are curvature decompositions (in particular Berger decomposition), the Weitzenböck formula, and the refined Kato inequality. Along with the proofs, we also discover an alternative proof for the Weitzenböck formula using Berger decomposition, and an alternative proof for the refined Kato inequality using Derdziński’s argument.


Einstein four-manifolds k-positive curvature operator Positive sectional curvature Positive isotropic curvature Berger curvature decomposition Weitzenböck formula Refined Kato inequality First eigenvalue of the Laplace operator 

Mathematics Subject Classification

Primary 58E11 53C25 53C24 Secondary 58C40 



This paper is the first extension of the author’s Ph.D. thesis. He expresses his sincere gratitude to his advisors Professors Xianzhe Dai and Guofang Wei for their guidance, encouragement, and constant support. He thanks Professors Jeffrey Case, Jingrun Chen, Claude LeBrun, Yuan Yuan, and Wolfgang Ziller for helpful discussions. The rigidity results were proved in Spring 2013, and the first eigenvalue upper bound estimate was proved in Fall 2015. The paper was revised in 2017. The author was partially supported by an AMS-Simons postdoctoral travel grant, China Recruit Program for Global Young Talents, and NSFC (11701093).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Shanghai Center for Mathematical SciencesFudan UniversityShanghaiChina

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