### Abstract

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 S^{3}× R, S^{2}× R^{2} or R^{4}. 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.

Original language | English (US) |
---|---|

Article number | 141 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 57 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2018 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Calculus of Variations and Partial Differential Equations*,

*57*(5), [141]. https://doi.org/10.1007/s00526-018-1415-x

**Gradient shrinking Ricci solitons of half harmonic Weyl curvature.** / Wu, Jia Yong; Wu, Peng; Wylie, William.

Research output: Contribution to journal › Article

*Calculus of Variations and Partial Differential Equations*, vol. 57, no. 5, 141. https://doi.org/10.1007/s00526-018-1415-x

}

TY - JOUR

T1 - Gradient shrinking Ricci solitons of half harmonic Weyl curvature

AU - Wu, Jia Yong

AU - Wu, Peng

AU - Wylie, William

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.

UR - http://www.scopus.com/inward/record.url?scp=85052629689&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052629689&partnerID=8YFLogxK

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 -