On the classification of gradient Ricci solitons

We show that the only shrinking gradient solitons with vanishing Weyl tensor and Ricci tensor satisfying a weak integral condition are quotients of the standard ones S n , S n 1 R and R n . This gives a new proof of the Hamilton‐Ivey‐Perelman classification of 3‐dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of H n , H n 1 R, R n , S n 1 R or S n . 53C25

[1]  Jeff Cheeger,et al.  Lower bounds on Ricci curvature and the almost rigidity of warped products , 1996 .

[2]  Compact gradient shrinking Ricci solitons with positive curvature operator , 2006, math/0601599.

[3]  On the long-time behavior of type-III Ricci flow solutions , 2005, math/0509639.

[4]  Brett L. Kotschwar.  On rotationally invariant shrinking Ricci solitons , 2008 .

[5]  R. Hamilton,et al.  The formations of singularities in the Ricci Flow , 1993 .

[6]  A. Lichnerowicz.  Variétés kählériennes à première classe de Chern non negative et variétés riemanniennes à courbure de Ricci généralisée non negative , 1971 .

[7]  Robert McOwen,et al.  Partial differential equations : methods and applications , 1996 .

[8]  W. Wylie.  Complete Shrinking Ricci Solitons have Finite Fundamental Group , 2007, 0704.0317.

[9]  Ricci solitons: the equation point of view , 2006, math/0607546.

[10]  T. Ivey.  Ricci solitons on compact three-manifolds , 1993 .

[11]  R. Hamilton.  Three-manifolds with positive Ricci curvature , 1982 .

[12]  Manifolds with positive curvature operators are space forms , 2006, math/0606187.

[13]  C. Mantegazza,et al.  Evolution of the Weyl Tensor under the Ricci Flow , 2009, 0910.4761.

[14]  H. Cao.  Recent Progress on Ricci Solitons , 2009, 0908.2006.

[15]  E. Calabi.  An extension of E. Hopf’s maximum principle with an application to Riemannian geometry , 1958 .

[16]  P. Baird,et al.  Three-dimensional Ricci solitons which project to surfaces , 2005, math/0510313.

[17]  B. Chow,et al.  Hamilton's Ricci Flow , 2018 .

[18]  R. Greene,et al.  $C^\infty $ approximations of convex, subharmonic, and plurisubharmonic functions , 1979 .

[19]  P. Petersen,et al.  RIGIDITY OF GRADIENT RICCI SOLITONS , 2007, 0710.3174.

[20]  Guofang Wei,et al.  Comparison geometry for the Bakry-Emery Ricci tensor , 2007, 0706.1120.

[21]  P. Lu,et al.  A note on uniformization of riemann surfaces by ricci flow , 2005, math/0505163.

[22]  Richard S. Hamilton,et al.  The Ricci flow on surfaces , 1986 .

[23]  William Wylie,et al.  On Gradient Ricci Solitons with Symmetry , 2007, 0710.3595.

[24]  R. Hamilton.  Four-manifolds with positive curvature operator , 1986 .

[25]  Frank Morgan,et al.  Manifolds with Density , 2005 .

[26]  Jorge Lauret,et al.  Ricci soliton homogeneous nilmanifolds , 2001 .

[27]  B. Chow,et al.  The Ricci Flow : An Introduction I , 2013 .

[28]  A. Naber.  Noncompact Shrinking 4-Solitons with Nonnegative Curvature , 2007, 0710.5579.

[29]  Haiwen Chen.  Pointwise 1/4-pinched 4-manifolds , 1991 .

[30]  C. Böhm,et al.  Nonnegatively Curved Manifolds with Finite Fundamental Groups Admit Metrics with Positive Ricci Curvature , 2007 .

[31]  A. Naber.  Noncompact shrinking four solitons with nonnegative curvature , 2010 .

[32]  H. W. Brinkmann.  Einstein spaces which are mapped conformally on each other , 1925 .

[33]  Xiaodong Cao,et al.  On Locally Conformally Flat Gradient Shrinking Ricci Solitons , 2008, 0807.0588.

[34]  Zhu-Hong Zhang,et al.  GRADIENT SHRINKING SOLITONS WITH VANISHING WEYL TENSOR , 2008, 0807.1582.

[35]  N. Wallach,et al.  On a classification of the gradient shrinking solitons , 2007, 0710.3194.

[36]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[37]  B. Chow,et al.  The Ricci flow on surfaces , 2004 .

[38]  R. McOwen.  Partial differential equations , 1995 .

[39]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.