Abstract
Suppose \((M^n, g, f)\) is a complete shrinking gradient Ricci soliton. Assume that \(|Ric|<\frac{n-2}{2\sqrt{n}}\), where \(n \ge 3\), then it has only one end. Similar results hold for the expanding gradient Ricci soliton.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Cao, H.D., Chen, B. L., Zhu, X. P.: Recent developments on Hamilton’s Ricci flow, Surveys in differential geometry. Vol. XII. Geometric flows, pp. 47-112
Cao, H.D.: Existence of Gradient Kähler-Ricci Solitons, Elliptic and Parabolic Methods in Geometry, Minneapolis (1994)
Cao, H.D., Chen, Q.: On locally conformally flat gradient steady Ricci solitons. Trans. Am. Math. Soc. 364(5), 2377–2391 (2012)
Cao, H.D., Chen, Q.: On Bach-flat gradient shrinking Ricci solitons. Duke Math. J. 162(6), 1149–1169 (2013)
Cao, H.D., Zhou, D.T.: On complete gradient shrinking Ricci solitons. J. Differ. Geom. 85(2), 175–185 (2010)
Cao, X.D., Wang, B., Zhang, Z.: On locally conformally flat gradient shrinking Ricci solitons. Commun. Contemp. Math. 13(2), 269–282 (2011)
Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differ. Geometry 6, 119-128 (1971/72)
Chen, B.L.: Strong uniqueness of the Ricci flow. J. Differ. Geom. 82(2), 363–382 (2009)
Chen, X.X., Wang, Y.Q.: On four-dimensional anti-self-dual gradient Ricci solitons. J. Geom. Anal. 25(2), 1335–1343 (2015)
Eminenti, M., LaNave, G., Mantegazza, C.: Ricci solitons: the equation point of view. Manuscripta Math. 127(3), 345–367 (2008)
Feldman, M., Ilmanen, T., Knopf, D.: Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons. J. Differ. Geom. 65(2), 169–209 (2003)
Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)
Koiso, N.: On rotationally symmetric Hamilton’s equation for Kähler-Einstein metrics. Recent topics in differential and analytic geometry, pp. 327-337
Kotschwar, B.: On rotationally invariant shrinking Ricci solitons. Pacific J. Math. 236(1), 73–88 (2008)
Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4), 153–201 (1986)
Munteanu, O., Sesum, N.: On gradient Ricci solitons. J. Geom. Anal. 23(2), 539–561 (2013)
Munteanu, O., Wang, J.P.: Smooth metric measure spaces with non-negative curvature. Comm. Anal. Geom. 19(3), 451–486 (2011)
Munteanu, O., Wang, J.P.: Analysis of weighted Laplacian and applications to Ricci solitons. Comm. Anal. Geom. 20(1), 55–94 (2012)
Munteanu, O., Wang, J.P.: Geometry of manifolds with densities. Adv. Math. 259, 269–305 (2014)
Munteanu, O., Wang, J.P.: Topology of Kähler Ricci solitons. J. Differ. Geom. 100(1), 109–128 (2015)
Munteanu, O., Wang, J.P.: Geometry of shrinking Ricci solitons. Compos. Math. 151(12), 2273–2300 (2015)
Munteanu, O., Wang, J.P.: Positively curved shrinking Ricci solitons are compact. J. Differ. Geom. 106(3), 499–505 (2017)
Munteanu, O., Wang, J.P.: Conical structure for shrinking Ricci solitons. J. Eur. Math. Soc. (JEMS) 19(11), 3377–3390 (2017)
Munteanu, O., Schulze, F., Wang, J.P.: Positive solutions to Schrödinger equations and geometric applications. J. Reine Angew. Math. 774, 185–217 (2021)
Naber, A.: Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math. 645, 125–153 (2010)
Ni, L., Wallach, N.: On a classification of gradient shrinking solitons. Math. Res. Lett. 15(5), 941–955 (2008)
Perelman, G.: Finite extincition time for the solutions to the Ricci flow on certain three-manifolds. arXiv: math/0307245
Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv: math/0303109
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159v1
Petersen, P., Wylie, W.: On the classification of gradient Ricci solitons. Geom. Topol. 14(4), 2277–2300 (2010)
Pigola, S., Rimoldi, M., Setti, A.: Remarks on non-compact gradient Ricci solitons. Math. Z. 268(3–4), 777–790 (2011)
Wang, X.J., Zhu, X.H.: Kähler-Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188(1), 87–103 (2004)
Wu, P., Wu, J. Y., Wylie, W.: Gradient shrinking Ricci solitons of half harmonic Weyl curvature, Calc. Var. Partial Differential Equations 57(5), Art. 141 (2018)
Zhang, Z.H.: Gradient shrinking solitons with vanishing Weyl tensor. Pacific J. Math. 242(1), 189–200 (2009)
Acknowledgements
The authors would like to thank Professor Xianzhe Dai, Professor Xi-nan Ma and Professor Yu Zheng for their constant encouragement. The second author also benefits a lot from the discussion with Professor BingWang. The second author is supported by the National Natural Science Foundation of China (Grant No. 11701516) and the Fundamental Research Funds of Zhejiang Sci-Tech University(Grant No. 2020Q043).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Qu, Y., Wu, G. When does gradient Ricci soliton have one end?. Ann Glob Anal Geom 62, 679–691 (2022). https://doi.org/10.1007/s10455-022-09868-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-022-09868-8