TY - JOUR

T1 - Gradient shrinking Ricci solitons of half harmonic Weyl curvature

AU - Wu, Jia Yong

AU - Wu, Peng

AU - Wylie, William

N1 - Funding Information:
Acknowledgements Part of the work was done when the first author was visiting the Department of Mathematics at Cornell University, he greatly thanks Professor Xiaodong Cao for his help and Department of Mathematics for their hospitality. The second author thanks Professors Jeffrey Case, Yuanqi Wang and Yuan Yuan for helpful discussions. We thank the anonymous referee for many valuable suggestions. The first author was partially supported by the China Scholarship Council (201208310431), NSFC (11671141) and the Natural Science Foundation of Shanghai (17ZR1412800). The second author was partially supported by an AMS-Simons postdoc travel grant, China Recruit Program for Global Young Talents, and NSFC (11701093). The third author was partially supported by a grant from the Simons Foundation (355608, William Wylie) and the NSF (1654034).
Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - Gradient Ricci solitons and metrics with half harmonic Weyl curvature are two natural generalizations of Einstein metrics on four-manifolds. In this paper we prove that if a metric has structures of both gradient shrinking Ricci soliton and half harmonic Weyl curvature, then except for three examples, it has to be an Einstein metric with positive scalar curvature. Precisely, we prove that a four-dimensional gradient shrinking Ricci soliton with δW±= 0 is either Einstein, or a finite quotient of S3× R, S2× R2 or R4. We also prove that a four-dimensional gradient Ricci soliton with constant scalar curvature is either Kähler–Einstein, or a finite quotient of M× C, where M is a Riemann surface. The method of our proof is to construct a weighted subharmonic function using curvature decompositions and the Weitzenböck formula for half Weyl curvature, and the method was motivated by previous work (Gursky and LeBrun in Ann Glob Anal Geom 17:315–328, 1999; Wu in Einstein four-manifolds of three-nonnegative curvature operator 2013; Trans Am Math Soc 369:1079–1096, 2017; Yang in Invent Math 142:435–450, 2000) on the rigidity of Einstein four-manifolds with positive sectional curvature, and previous work (Cao and Chen in Trans Am Math Soc 364:2377–2391, 2012; Duke Math J 162:1003–1204, 2013; Catino in Math Ann 35:629–635, 2013) on the rigidity of gradient Ricci solitons.

AB - Gradient Ricci solitons and metrics with half harmonic Weyl curvature are two natural generalizations of Einstein metrics on four-manifolds. In this paper we prove that if a metric has structures of both gradient shrinking Ricci soliton and half harmonic Weyl curvature, then except for three examples, it has to be an Einstein metric with positive scalar curvature. Precisely, we prove that a four-dimensional gradient shrinking Ricci soliton with δW±= 0 is either Einstein, or a finite quotient of S3× R, S2× R2 or R4. We also prove that a four-dimensional gradient Ricci soliton with constant scalar curvature is either Kähler–Einstein, or a finite quotient of M× C, where M is a Riemann surface. The method of our proof is to construct a weighted subharmonic function using curvature decompositions and the Weitzenböck formula for half Weyl curvature, and the method was motivated by previous work (Gursky and LeBrun in Ann Glob Anal Geom 17:315–328, 1999; Wu in Einstein four-manifolds of three-nonnegative curvature operator 2013; Trans Am Math Soc 369:1079–1096, 2017; Yang in Invent Math 142:435–450, 2000) on the rigidity of Einstein four-manifolds with positive sectional curvature, and previous work (Cao and Chen in Trans Am Math Soc 364:2377–2391, 2012; Duke Math J 162:1003–1204, 2013; Catino in Math Ann 35:629–635, 2013) on the rigidity of gradient Ricci solitons.

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U2 - 10.1007/s00526-018-1415-x

DO - 10.1007/s00526-018-1415-x

M3 - Article

AN - SCOPUS:85052629689

VL - 57

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 5

M1 - 141

ER -