Abstract
Given a nontrivial 3-dimensional steady gradient Ricci soliton, if the scalar curvature has decay order between \(-b\) and \(-a\) for some \(a\in (0,1], b\ge a\), then the umbilical ratio of the level set of the potential function lies in the class \(O(r^{6a-\frac{8a^2}{b}})\cap O(r^{2b-4a})\).
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References
Brendle, S.: Rotational symmetry of self-similar solutions to the Ricci flow. Invent. Math. 194, 731–764 (2013)
Cao, H.-D., Chen, B.-L., Zhu, X.-P.: Recent developments on Hamilton’s Ricci flow, volume XII of Surv. Diff. Geom. Int. Press. (2008)
Cao, H.-D., He, C.: Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three. arXiv: 1412.2714, to appear in Comm. Anal. Geom
Chen, B.-L.: Strong uniqueness of the Ricci flow. J. Differ. Geom. 82, 363–382 (2009)
Chen, C.-W.: On the regularity of the Ricci flow. National Taiwan University and University of Grenoble, Ph.D. Thesis (2011)
Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci flow, volume 77 of Graduate Studies in Mathematics. Am. Math. Soc. (2006)
Catino, G., Mastrolia, P., Monticelli, D.D.: Classification of expanding and steady Ricci solitons with integral curvature decay. Geom. Topol. 20(5), 2665–2685 (2016)
Deruelle, A.: Steady gradient soliton with curvature in \({L}^1\). Commun. Anal. Geom. 20(1), 31–53 (2012)
Daskalopoulos, P., Hamilton, R., Sesum, N.: Classification of ancient compact solutions to the Ricci flow on surfaces. J. Differ. Geom. 91(2), 171–214 (2012)
Deng, Y., Zhu, X.: Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature. Math. Z 279, 211–226 (2015)
Deng, Y., Zhu, X.: 3d steady Gradient Ricci Solitons with linear curvature decay. arXiv:1612.05713v1 (2016)
Guo, H.: Area growth rate of the level surface of the potential function on the \(3\)-dimensional steady gradient Ricci soliton. Proc. Am. Math. Soc. 137(6), 2093–2097 (2009)
Guo, H.: Remarks on noncompact steady gradient Ricci solitons. Math. Ann. 345, 883–894 (2009)
Guo, H.: Evolution equation of the Gauss curvature under hypersurface flows and its applications. Acta Math. Sin. Engl. Ser. 26(7), 1299–1308 (2010)
Huisken, G.: Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84, 463–480 (1986)
Munteanu, O., Sung, C.-J., Wang, J.: Poisson equation on complete manifolds. arXiv:1701.02865 (2017)
Munteanu, O., Wang, J.: Smooth metric measure spaces with non-negative curvature. Commun. Anal. Geom. 19(3), 451–486 (2011)
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 the gradient Ricci solitons. Math. Res. Lett. 15(5), 941–955 (2008)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159v1 (2002)
Wu, P.: On the potential function of gradient steady Ricci solitons. J. Geom. Anal. 23(1), 221–228 (2013)
Acknowledgements
The first author would like to thank Prof. Pengfei Guan for helpful discussions during the workshop for Besson’s 60th birthday. The second author is supported by the MOST research Grant 106-2115-M-018-002-MY2.
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Chen, CW., Lee, KW. Level set flow in 3D steady gradient Ricci solitons. manuscripta math. 158, 223–234 (2019). https://doi.org/10.1007/s00229-018-1020-5
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DOI: https://doi.org/10.1007/s00229-018-1020-5