Lower volume growth estimates for self-shrinkers of mean curvature flow
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- by Haizhong Li and Yong Wei PDF
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Abstract:
We obtain a Calabi-Yau type volume growth estimate for complete noncompact self-shrinkers of the mean curvature flow. More precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth.References
- E. Calabi, On manifolds with non-negative Ricci-curvature II, Notices Amer. Math. Soc. 22 (1975), A205
- Huai-Dong Cao, Geometry of complete gradient shrinking Ricci solitons, Geometry and analysis. No. 1, Adv. Lect. Math. (ALM), vol. 17, Int. Press, Somerville, MA, 2011, pp. 227–246. MR 2882424
- Huai-Dong Cao and Haizhong Li, A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension, Calc. Var. Partial Differential Equations 46 (2013), no. 3-4, 879–889. MR 3018176, DOI 10.1007/s00526-012-0508-1
- Huai-Dong Cao and Detang Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no. 2, 175–185. MR 2732975
- Huai-Dong Cao and Xi-Ping Zhu, A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math. 10 (2006), no. 2, 165–492. MR 2233789, DOI 10.4310/AJM.2006.v10.n2.a2
- José A. Carrillo and Lei Ni, Sharp logarithmic Sobolev inequalities on gradient solitons and applications, Comm. Anal. Geom. 17 (2009), no. 4, 721–753. MR 3010626, DOI 10.4310/CAG.2009.v17.n4.a7
- Xu Cheng and Detang Zhou, Volume estimate about shrinkers, Proc. Amer. Math. Soc. 141 (2013), no. 2, 687–696. MR 2996973, DOI 10.1090/S0002-9939-2012-11922-7
- Tobias H. Colding and William P. Minicozzi II, Generic mean curvature flow I: generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755–833. MR 2993752, DOI 10.4007/annals.2012.175.2.7
- Qi Ding and Y. L. Xin, Volume growth, eigenvalue and compactness for self-shrinkers, Asian J. Math. 17 (2013), no. 3, 443–456. MR 3119795, DOI 10.4310/AJM.2013.v17.n3.a3
- Klaus Ecker, Logarithmic Sobolev inequalities on submanifolds of Euclidean space, J. Reine Angew. Math. 522 (2000), 105–118. MR 1758578, DOI 10.1515/crll.2000.033
- Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
- Peter Li, Lecture notes on geometric analysis, Lecture Notes Series, vol. 6, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. MR 1320504
- J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$, Comm. Pure Appl. Math. 26 (1973), 361–379. MR 344978, DOI 10.1002/cpa.3160260305
- Ovidiu Munteanu and Jiaping Wang, Analysis of weighted Laplacian and applications to Ricci solitons, Comm. Anal. Geom. 20 (2012), no. 1, 55–94. MR 2903101, DOI 10.4310/CAG.2012.v20.n1.a3
- G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math. DG/0211159
- R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; With a preface translated from the Chinese by Kaising Tso. MR 1333601
- Lu Wang, A Bernstein type theorem for self-similar shrinkers, Geom. Dedicata 151 (2011), 297–303. MR 2780753, DOI 10.1007/s10711-010-9535-2
- Shing Tung Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. MR 417452, DOI 10.1512/iumj.1976.25.25051
- Shi Jin Zhang, On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below, Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 5, 871–882. MR 2786449, DOI 10.1007/s10114-011-9527-7
Additional Information
- Haizhong Li
- Affiliation: Department of Mathematical Sciences, and Mathematical Sciences Center, Tsinghua University, 100084, Beijing, People’s Republic of China
- MR Author ID: 255846
- Email: hli@math.tsinghua.edu.cn
- Yong Wei
- Affiliation: Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, People’s Republic of China
- MR Author ID: 1036099
- ORCID: 0000-0002-9460-9217
- Email: wei-y09@mails.tsinghua.edu.cn
- Received by editor(s): May 18, 2012
- Received by editor(s) in revised form: September 18, 2012
- Published electronically: May 20, 2014
- Additional Notes: The authors were supported by NSFC No. 11271214 and Tsinghua University-K.U. Leuven Bilateral Scientific Cooperation Fund.
- Communicated by: Lei Ni
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 3237-3248
- MSC (2010): Primary 53C44; Secondary 53C42
- DOI: https://doi.org/10.1090/S0002-9939-2014-12037-5
- MathSciNet review: 3223379