On Ricci solitons of cohomogeneity one

We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansätze of cohomogeneity one to produce new explicit examples of complete Kähler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano Kähler–Einstein manifolds or over coadjoint orbits of a compactly connected semisimple Lie group.

[1]  G. Tian,et al.  Uniqueness of Kähler-Ricci solitons , 2000 .

[2]  G. Tian,et al.  A new holomorphic invariant and uniqueness of Kähler-Ricci solitons , 2002 .

[3]  T. Willmore.  EINSTEIN MANIFOLDS (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10) , 1987 .

[4]  M. Cvetič,et al.  Communications in Mathematical Physics Ricci-Flat Metrics , Harmonic Forms and Brane Resolutions , 2002 .

[5]  Mckenzie Y. Wang.  Einstein metrics from symmetry and bundle constructions , 2001 .

[6]  G. Tian.  On Kähler-Einstein metrics on certain Kähler manifolds withC1 (M)>0 , 1987 .

[7]  G. Valent,et al.  On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties , 1996 .

[8]  Toshikazu Sunada,et al.  Curvature and Topology of Riemannian Manifolds , 1986 .

[9]  Mckenzie Y. Wang,et al.  Some New Examples of Non-K\"ahler Ricci Solitons , 2008, 0806.4408.

[10]  J. Graver,et al.  Graduate studies in mathematics , 1993 .

[11]  Y. Siu.  The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group , 1988 .

[12]  D. Guan.  QUASI-EINSTEIN METRICS , 1995 .

[13]  H. Lawson.  Lectures On Minimal Submanifolds , 1980 .

[14]  Mckenzie Y. Wang,et al.  Einstein metrics on $S^{2}$-bundles , 1998 .

[15]  P. Griffiths,et al.  The intermediate Jacobian of the cubic threefold , 1972 .

[16]  Dennis DeTurck,et al.  Some regularity theorems in riemannian geometry , 1981 .

[17]  Xu-jia Wang,et al.  Kahler-Ricci solitons on toric manifolds with positive first Chern class , 2004 .

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

[19]  H. Pedersen,et al.  Quasi-Einstein Kähler Metrics , 1999 .

[20]  Mckenzie Y. Wang,et al.  The initial value problem for cohomogeneity one Einstein metrics , 2000 .

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

[22]  Wan-Xiong Shi.  Ricci deformation of the metric on complete noncompact Riemannian manifolds , 1989 .

[23]  Bo Yang.  A characterization of Koiso's typed solitons , 2008, 0802.0300.

[24]  L. D. Cerbo,et al.  Generic Properties of Homogeneous Ricci Solitons , 2007, 0711.0465.

[25]  Thomas A. Ivey,et al.  New examples of complete Ricci solitons , 1994 .

[26]  Peng Lu,et al.  The Ricci Flow: Techniques and Applications , 2007 .

[27]  A. Spiro,et al.  Kähler-Ricci solitons on homogeneous toric bundles , 2006, math/0604070.

[28]  B. Chow.  Elliptic and parabolic methods in geometry , 1996 .

[29]  D. Page.  A COMPACT ROTATING GRAVITATIONAL INSTANTON , 1978 .

[30]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[31]  Wan-Xiong Shi.  Deforming the metric on complete Riemannian manifolds , 1989 .

[32]  K. Yano.  On Harmonic and Killing Vector Fields , 1952 .

[33]  S. Bando.  Real analyticity of solutions of Hamilton's equation , 1987 .

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

[35]  N. Koiso,et al.  Nonhomogeneous Kähler-Einstein metrics on compact complex manifolds. II , 1988 .

[36]  H. Cao.  Geometry of Ricci Solitons* , 2006 .

[37]  A. Spiro,et al.  Kähler manifolds with large isometry group , 1997, dg-ga/9709003.

[38]  A. Nadel.  Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature , 1990 .

[39]  Mckenzie Y. Wang,et al.  Kähler-Einstein metrics of cohomogeneity one , 1998 .

[40]  N. Koiso.  On Rotationally Symmetric Hamilton’s Equation for Kähler–Einstein Metrics , 1990 .

[41]  P. Baird.  The Ricci flow: techniques and applications -Part I: Geometric aspects (Mathematical Surveys and Monographs 135) , 2008 .

[42]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

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

[44]  M. Feldman,et al.  Rotationally symmetric shrinking and ex - panding gradient Kahler - Ricci solitons , 2003 .

[45]  上海科学技術文献出版社.  数学年刊 = Chinese annals of mathematics , 1982 .