Abstract
We show that expanding Kähler–Ricci solitons which have positive holomorphic bisectional curvature and are \(C^{2}\)-asymptotic to a conical Kähler manifold at infinity must be the U(n)-rotationally symmetric expanding solitons constructed by Cao.
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Notes
We thank Yasha Eliashberg for explaining to us the proof of Lemma 3.1.
References
Bando, S.: On the classification of three-dimensional compact Kaehler manifolds of nonnegative bisectional curvature. J. Differ. Geom. 19(2), 283–297 (1984)
Bernstein, J., Mettler, T.: Two-dimensional gradient Ricci solitons revisited. Int. Math. Res. Not. 2015(1), 78–98 (2015)
Brendle, S.: Rotational symmetry of self-similar solutions to the Ricci flow. Invent. Math. 194(3), 731–764 (2013)
Brendle, S.: Rotational symmetry of Ricci solitons in higher dimensions. J. Differ. Geom. 97(2), 191–214 (2014)
Bryant, R.: Ricci flow solitons in dimension three with SO(3)-symmetries. http://www.math.duke.edu/~bryant/3DRotSymRicciSolitons.pdf (2005)
Cao, H.-D.: Existence of gradient Kähler-Ricci solitons, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), pp. 1–16. A K Peters, Wellesley (1996)
Cao, H.-D.: Limits of solutions to the Kähler–Ricci flow. J. Differ. Geom. 45(2), 257–272 (1997)
Cao, H.-D.: Recent progress on Ricci solitons, Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, pp. 1–38. Int. Press, Somerville (2010)
Chen, C.-W., Deruelle, A.: Structure at infinity of expanding gradient Ricci soliton. Asian J. Math. (2015, to appear). arXiv:1108.1468
Chen, B.-L.: Strong uniqueness of the Ricci flow. J. Differ. Geom. 82(2), 363–382 (2009)
Chodosh, O.: Expanding Ricci solitons asymptotic to cones. Calc. Var. Partial Differ. Equ. 51(1–2), 1–15 (2014)
Cabezas-Rivas, E., Wilking, B.: How to produce a Ricci flow via Cheeger–Gromoll exhaustion. JEMS. (2015, to appear). arXiv:1107.0606
Chau, A., Tam, L.-F.: On the complex structure of Kähler manifolds with nonnegative curvature. J. Differ. Geom. 73(3), 491–530 (2006)
Chau, A., Tam, L.-F.: A survey on the Kähler-Ricci flow and Yau’s uniformization conjecture, Surveys in differential geometry, vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, pp. 21–46. Int. Press, Somerville (2008)
Chen, B.-L., Zhu, X.-P.: Uniqueness of the Ricci flow on complete noncompact manifolds. J. Differ. Geom. 74(1), 119–154 (2006)
Dancer, A., Wang, M.: On Ricci solitons of cohomogeneity one. Ann. Global Anal. Geom. 39(3), 259–292 (2011)
Feldman, M., Ilmanen, T., Knopf, D.: Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons. J. Differ. Geom. 65(2), 169–209 (2003)
Futaki, A., Wang, M.-T.: Constructing Kähler–Ricci solitons from Sasaki-Einstein manifolds. Asian J. Math. 15(1), 33–52 (2011)
Giesen, G., Topping, P.M.: Existence of Ricci flows of incomplete surfaces. Commun. Partial Differ. Equ. 36(10), 1860–1880 (2011)
Hamilton, R.S.: The formation of singularities in the Ricci flow, Surveys in differential geometry, vol. II (Cambridge, MA, 1993), pp. 7–136. Int. Press, Cambridge (1995)
Kotschwar, B.: A note on the uniqueness of complete, positively-curved expanding Ricci solitons in 2-D. http://math.la.asu.edu/kotschwar/pub/kotschwar_expsol_11_01.pdf (2006)
Kotschwar, B., Wang, L.: Rigidity of asymptotically conical shrinking gradient Ricci solitons. J. Differ. Geom. 100(1), 55–108 (2015)
Mok, N.: The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature. J. Differ. Geom. 27(2), 179–214 (1988)
Mori, S.: Projective manifolds with ample tangent bundles. Ann. Math. (2) 110(3), 593–606 (1979)
Ni, L.: Ancient solutions to Kähler–Ricci flow. Math. Res. Lett. 12(5–6), 633–653 (2005)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159 (2002)
Shi, W.-X.: Ricci Flow and the Uniformization on complete noncompact Kähler manifolds. J. Differ. Geom. 45(1), 94–220 (1997)
Schulze, F., Simon, M.: Expanding solitons with non-negative curvature operator coming out of cones. Math. Z. 275(1–2), 625–639 (2013)
Siu, Y.-T., Yau, S.-T.: Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59(2), 189–204 (1980)
Acknowledgments
The first named author was supported in part by a National Science Foundation Graduate Research Fellowship DGE-1147470. He would like to thank Simon Brendle for many discussions concerning his soliton uniqueness results, as well as for his support and encouragement. The second named author would like to thank Richard Schoen, Simon Brendle and Yanir Rubinstein who aroused his interest concerning topics related to this paper when he was a graduate student at Stanford. He would also like to thank Nicos Kapouleas for discussions which motivated him to consider this problem. The authors are also grateful to the anonymous referee for several remarks improving the exposition.
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Chodosh, O., Fong, F.TH. Rotational symmetry of conical Kähler–Ricci solitons. Math. Ann. 364, 777–792 (2016). https://doi.org/10.1007/s00208-015-1240-x
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DOI: https://doi.org/10.1007/s00208-015-1240-x