Abstract
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.
Similar content being viewed by others
References
Apostolov V., Calderbank D., Gauduchon P., Tønnesen-Friedman C.: Hamiltonian 2-forms in Kähler geometry IV: Weakly Bochner–Flat Kähler manifolds. Commun. Anal. Geom. 16, 91–126 (2008)
Back, A.: Local Theory of Equivariant Einstein Metrics and Ricci Realizability on Kervaire Spheres, Preprint (1986)
Bando S.: Real analyticity of solutions of Hamilton’s equation. Math. Z. 195, 93–97 (1987)
Bérard Bergery L.: Sur des nouvelles variétés riemanniennes d’Einstein. Publication de l’Institut Elie Cartan, Nancy (1982)
Besse A.: Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 10. Springer, Berlin (1987)
Bryant, R.: Unpublished work
Cao H.D.: Existence of Gradient Ricci Solitons, Elliptic and Parabolic Methods in Geometry, pp. 1–16. A. K. Peters, Wellesley (1996)
Cao H.D.: Limits of solutions to the Kähler–Ricci flow. J. Differ. Geom. 45, 257–272 (1997)
Cao H.D.: Geometry of Ricci solitons. Chin. Ann. Math. 27B, 121–142 (2006)
Chave T., Valent G.: On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties. Nucl. Phys. B478, 758–778 (1996)
Chow B., Chu S.C., Glickenstein D., Guenther C., Isenberg J., Ivey T., Knopf D., Lu P., Luo F., Nei L.: The Ricci Flow: Techniques and Applications, Mathematical Surveys and Monographs, vol. 135. American Mathematical Society, Providence (2007)
Chow B., Lu P., Ni L.: Hamilton’s Ricci Flow, Graduate Studies in Mathematics, vol. 77. American Mathematical Society–Science Press, Providence (2006)
Clemens C.H., Griffiths P.A.: The intermediate Jacobian of the cubic threefold. Ann. Math. 95, 281–356 (1972)
Cvetič M., Gibbons G., Lü H., Pope C.: Ricci-flat metrics, harmonic forms and brane resolutions. Commun. Math. Phys. 232, 457–500 (2003)
Dancer A., Wang M.: Kähler–Einstein metrics of cohomogeneity one. Math. Ann. 312, 503–526 (1998)
Dancer A., Wang M.: Some new examples of non-Kähler Ricci solitons. Math. Res. Lett. 16, 349–363 (2009)
DeTurck D., Kazdan J.: Some regularity theorems in Riemannian geometry. Ann. Sci École Norm. Sup., 4e série, t. 14, 249–260 (1981)
Eminenti, M., La Nave, G., Mantegazza, C.: Ricci solitons—the equation point of view. arXiv:math/0607546
Eschenburg J., Wang M.: The initial value problem for cohomogeneity one Einstein metrics. J. Geom. Anal. 10, 109–137 (2000)
di Cerbo, L.F.: Generic properties of homogeneous Ricci solitons. arXiv:math.DG/07110465
Feldman M., Ilmanen T., Knopf D.: Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons. J. Differ. Geom. 65, 169–209 (2003)
Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order, 2nd edn. Springer, Berlin (1998)
Guan Z.D.: Quasi-Einstein metrics. Int. J. Math. 6, 371–379 (1995)
Hamilton R.S.: The formation of singularities in the Ricci flow. Surv. Differ. Geom. 2, 7–136 (1995)
Ivey T.: New examples of complete Ricci solitons. Proc. AMS 122, 241–245 (1994)
Koiso, N.: On rotationally symmetric Hamilton’s equation for Kähler–Einstein metrics. Advanced Studies in Pure Mathematics, vol. 18-I. Academic Press, Tokyo, pp. 327–337 (1990)
Koiso, N., Sakane, Y.: Non-homogeneous Kähler–Einstein metrics on compact complex manifolds. In: Curvature and Topology of Riemannian Manifolds. Springer Lecture Notes in Mathematics, vol. 1201. Springer, Berlin, pp. 165–179 (1986)
Koiso N., Sakane Y.: Non-homogeneous Kähler–Einstein metrics on compact complex manifolds II. Osaka Math. J. 25, 933–959 (1988)
Lauret J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319, 715–733 (2001)
Lawson, H.B. Jr.: Lectures on minimal submanifolds, Publish or Perish, Berkeley (1980)
Nadel A.M.: Multiplier ideal sheaves and Kähler–Einstein metrics of positive scalar curvature. Ann. Math. 132, 549–596 (1990)
Page D.: A compact rotating gravitational instanton. Phys. Lett. 79B, 235–238 (1979)
Pedersen H., Tønnesen-Friedman C., Valent G.: Quasi-Einstein Kähler metrics. Lett. Math. Phys. 50, 229–241 (2000)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
Podesta F., Spiro A.: Kähler manifolds with large isometry group. Osaka J. Math. 36, 805–833 (1999)
Podesta F., Spiro A.: Kähler–Ricci solitons on homogeneous toric bundles. J. Reine Angew. Math. 642, 109–127 (2010)
Sakane Y.: Examples of compact Einstein Kähler manifolds with positive Ricci tensor. Osaka Math. J. 23, 585–617 (1986)
Shi W.X.: Deforming the metric on complete Riemannian manifolds. J. Differ. Geom. 30, 223–301 (1989)
Shi W.X.: Ricci deformation of the metric on complete noncompact Riemannian manifolds. J. Differ. Geom. 30, 303–394 (1989)
Siu Y.-T.: The existence of Kähler–Einstein metrics on manifolds with positive anti-canonical line bundle and a suitable finite symmetry group. Ann. Math. 127, 585–627 (1988)
Tian G.: On Kähler–Einstein metrics on certain Kähler manifolds with c 1(M) > 0. Invent. Math. 89, 225–246 (1987)
Tian G.: Kähler–Einstein manifolds of positive scalar curvature, in Surveys in Differential Geometry, vol. VI: Essays on Einstein Manifolds. International Press, Boston (1999)
Tian G., Zhu X.: Uniqueness of Kähler–Ricci solitons. Acta Math. 184, 271–305 (2000)
Tian G., Zhu X.: A new holomorphic invariant and uniqueness of Kähler–Ricci solitons. Commun. Math. Helv. 77, 297–325 (2002)
Wang M.: Einstein metrics from symmetry and bundle constructions. Surv. Differ. Geom. 6, 287–325 (1999)
Wang J., Wang M.: Einstein metrics on S 2-bundles. Math. Ann. 310, 497–526 (1998)
Wang X.-J., Zhu X.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188, 87–103 (2004)
Yang, B.: A characterization of Koiso’s typed solitons. arXiv:math.DG/0802.0300
Yano K.: On harmonic and Killing vector fields. Ann. Math. 55, 38–45 (1952)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dancer, A.S., Wang, M.Y. On Ricci solitons of cohomogeneity one. Ann Glob Anal Geom 39, 259–292 (2011). https://doi.org/10.1007/s10455-010-9233-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-010-9233-1
Keywords
- Cohomogeneity one Ricci solitons
- Non-compact shrinking Kähler Ricci solitons
- Equidistant hypersurface families
- Coadjoint orbits