SOME EXTENSIONS OF THE MEAN CURVATURE FLOW IN RIEMANNIAN MANIFOLDS

doi:10.1016/S0252-9602(12)60203-3

Abstract

Abstract:

Given a family of smooth immersions of closed hypersurfaces in a locally sym-metric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curvature flow, certain subcritical quan-tities concerning the second fundamental form blow up. This result not only generalizes a result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of Le in the Euclidean case.

Key words: mean curvature flow, Riemannian submanifold, integral curvature, maximal existence time

CLC Number:

53C40

WU Jia-Yong. SOME EXTENSIONS OF THE MEAN CURVATURE FLOW IN RIEMANNIAN MANIFOLDS[J].Acta mathematica scientia,Series B, 2013, 33(1): 171-186.

Trendmd

References

[1] Chen J, He W. A note on singular time of mean curvature flow. Math Z, 2010, 266: 921–931

[2] Chen X -Z, Shen Y -B. A note on the mean curvature flow in Riemannian manifolds. Acta Math Sci, 2010, 30B: 1053–1064

[3] Chow B, Lu P, Ni L. Hamilton’s Ricci Flow. Lectures in Contemporary Mathematics 3. Beijing: Science Press; Providence RI: American Mathematical Society, 2006

[4] Cooper A A. A characterization of the singular time of the mean curvature flow. Proc Amer Math Soc, 2011, 139: 2933–2942

[5] Ecker K. On regularity for mean curvature flow of hypersurfaces. Calc Var Partial Differ Equ, 1995, 3: 107–126

[6] Enders J, M¨uller R, Topping P. On Type I Singularities in Ricci flow. Comm Anal Geom, 2011, 19: 905–922

[7] Hamilton R S. Three-manifolds with positive Ricci curvature. J Diff Geom, 1982, 17: 255–306

[8] Hoffman D, Spruck J. Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm Pure Appl Math, 1974, 27: 715–727; Erratum. Comm Pure Appl Math, 1975, 28: 765–766

[9] Huisken G. Flow by mean curvature of convex surfaces into spheres. J Diff Geom, 1984, 20: 237–266

[10] Huisken G. Contracting convex hyperserfaces in Riemannian manifolds by their mean curvature. Invent Math, 1986, 84: 463–480

[11] Huisken G, Sinestrari C. Mean curvature flow singularities for mean convex surfaces. Calc Var Partial Differ Equ, 1999, 8: 1–14

[12] Le N Q. Blow up of subcritical quantities at the first singular time of the mean curvature flow. Geom Dedicata, 2011, 151: 361–371

[13] Le N Q, Sesum N. On the extension of the mean curvature flow. Math Z, 2011, 267: 583–604

[14] Le N Q, Sesum N. The mean curvature at the first singular time of the mean curvature flow. Ann I H Poincar´e-AN, 2010, 27: 1441–1459

[15] Le N Q, Sesum N. Remarks on curvature behavior at the first singular time of the Ricci flow. Pacific J Math, 2012, 255: 155–175

[16] Liu K -F, Xu H -W, Ye F, Zhao E -T. The extension and convergence of mean curvature flow in higher codimension. arXiv: math.DG/1104.0971v1.

[17] Michael J H, Simon L M. Sobolev and mean-value inequalities on generalized submanifolds of Rn. Comm Pure Appl Math, 1973, 26: 361–379

[18] Perelman G. The entropy formula for the Ricci flow and its geometric applications. arXiv: math.DG/0211159.

[19] Sesum N. Curvature tensor under the Ricci flow. Amer J Math, 2005, 127: 1315–1324

[20] Tan Z, Wun G -C. On the heat flow equation of surfaces of constant mean curvature in higher dimensions. Acta Math Sci, 2011, 31B: 1741–1748

[21] Wang B. On the conditions to extend Ricci flow. Int Math Res Not IMRN, 2008, (8)

[22] Xu H -W, Ye F, Zhao E -T. Extend mean curvature flow with finite integral curvature. Asian J Math, 2011, 15: 549–556

[23] Xu H -W, Ye F, Zhao E -T. The extension for mean curvature flow with finite integral curvature in Riemannian manifolds. Sci China Math, 2011, 54: 2195–2204