Volume estimate about shrinkers

Cheng, Xu; Zhou, Detang

doi:10.1090/S0002-9939-2012-11922-7

Volume estimate about shrinkers
HTML articles powered by AMS MathViewer

by Xu Cheng and Detang Zhou PDF

Proc. Amer. Math. Soc. 141 (2013), 687-696 Request permission

Abstract:

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space.

References

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, Recent progress on Ricci solitons, Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, Int. Press, Somerville, MA, 2010, pp. 1–38. MR 2648937
Huai-Dong Cao, Bing-Long Chen, and Xi-Ping Zhu, Recent developments on Hamilton’s Ricci flow, Surveys in differential geometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp. 47–112. MR 2488948, DOI 10.4310/SDG.2007.v12.n1.a3
Huai-Dong Cao and Detang Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no. 2, 175–185. MR 2732975
Bing-Long Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009), no. 2, 363–382. MR 2520796
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., DOI 10.4007/annals.2012.175.2.7
Tobias H. Colding and William P. Minicozzi II, Smooth compactness of self-shrinkers, Comment. Math. Helv. 87 (2012), no. 1, 463–475., DOI 10.4171/CMH/260
Qi Ding and Yuanlong Xin, Volume growth, eigenvalue and compactness for self-shrinkers. arXiv:1101.1411.
Klaus Ecker and Gerhard Huisken, Mean curvature evolution of entire graphs, Ann. of Math. (2) 130 (1989), no. 3, 453–471. MR 1025164, DOI 10.2307/1971452
Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255
Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
G. Perelman, The entropy formula for the Ricci flow and its geometric applications. arXiv: math.DG/0211159.
William Wylie, Complete shrinking Ricci solitons have finite fundamental group, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1803–1806. MR 2373611, DOI 10.1090/S0002-9939-07-09174-5
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