On asymptotic behavior for singularities of the powers of mean curvature flow

Abstract

Let M ⁿ be a smooth, compact manifold without boundary, and F ₀: M ⁿ → R ⁿ⁺¹ a smooth immersion which is convex. The one-parameter families F(·, t): M ⁿ × [0, T) → R ⁿ⁺¹ of hypersurfaces M _t ⁿ = F(·, t)(M ⁿ) satisfy an initial value problem dt/dF(·, t) = −H ^k(·, t)ν(·, t), F(·, 0) = F ₀(·), where H is the mean curvature and ν(·, t) is the outer unit normal at F(·, t), such that −H ν = $\overrightarrow H $ is the mean curvature vector, and k > 0 is a constant. This problem is called H ^k-flow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type I and Type II. It is proved that for Type I singularity, the limiting hypersurface satisfies an elliptic equation; for Type II singularity, the limiting hypersurface must be a translating soliton.

Keywords

H ^k-Curvature flow / Type I singularities / Type II singularities

References

[1] Andrews B.. Contraction of convex hypersurfaces in Euclidian space. Calc. Var., 1994, 2: 151-171

[2] Andrews B.. Harnack inequalities for evolving hypersurfaces. Math. Z., 1994, 217: 179-197

[3] Andrews B.. Motion of hypersurfaces by Gauss curvature. Pacific J. Math., 2000, 195: 1-36

[4] Andrews B.. Classification of limiting shapes for isotropic curve flows. J. Amer. Math. Soc., 2003, 16: 443-459

[5] Andrews B.. Contraction of convex hypersurfaces by their affine normal. J. Diff. Geom., 1996, 43: 207-230

[6] Chou K. S., Zhu X. P.. A convexity theorem for a class of anisotropic flows of plane curves. Indiana Univ. Math. J., 1999, 48: 139-154

[7] Chow B.. Deforming convex hypersurfaces by the nth root of the Gaussian curvature. J. Diff. Geom., 1985, 22: 117-138

[8] Chow B.. Deforming hypersurfaces by the square root of the scalar curvature. Invent. Math., 1987, 87: 63-82

[9] Firey W. J.. On the shapes of worn stones. Mathematika, 1974, 21: 1-11

[10] Gage M. E.. Evolving plane curves by curvature in relative geometries. Duke Math. J., 1993, 72: 441-466

[11] Gage M. E., Hamilton R.. The heat equation shrinking convex plane curves. J. Diff. Geom., 1986, 23: 69-96

[12] Gerhardt C.. Flow of nonconvex hypersurfaces into sphere. J. Diff. Geom., 1990, 32: 299-314

[13] Grayson M.. The heat equation shrinks embedded closed curves to round points. J. Diff. Geom., 1987, 26: 285-314

[14] Hamilton R.. Harnack estimate for the mean curvature flow. J. Diff. Geom., 1995, 41: 215-226

[15] Huisken G.. Flow by meancurvature of convex surfaces into spheres. J. Diff. Geom., 1984, 20: 237-266

[16] Huisken G.. Asymptotic behavior for singularities of the mean curvature flow. J. Diff. Geom., 1990, 31: 285-299

[17] Huisken G., Polden A.. Geometric evolution equations for hypersurfaces, Calculus of Variations and Geometric Evolution Problems. CIME Lectures of Cetraro of 1996, Lecture Notes in Math., 1999, Berlin: Springer 45-84

[18] Huisken G., Sinestrari C.. Mean curvature flow singularities for mean convex surface. Calc. Var., 1999, 8: 1-14

[19] Huisken G., Sinestrari C.. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math., 1999, 183: 47-70

[20] Krylov N. V.. Nonlinear Elliptic and Parabolic Equations of the Second Order, 1987, Dordrecht: D. Reidel

[21] Li P., Tau S. T.. On the parabolic kernel of the Schrödinger operator. Acta Math., 1986, 156: 153-201

[22] Lieberman G. M.. Second Order Parabolic Diffrential Equations, 1996, Singapore: World Scientific

[23] Oaks J.. Singularities and self-intersections of curves evolving on surfaces. Indiana Univ. Math. J., 1994, 43: 959-981

[24] Schulze F.. Evolution of convex hypersurfaces by powers of the mean curvature. Math. Z., 2005, 251: 721-733

[25] Tso K.. Deforming a hypersurface by its Gauss-Kronecker curvature. Comm. Pure Appl. Math., 1985, 38: 867-882

[26] Urbas J. I. E.. Complete noncompact self-similar solutions of Gauss curvature flows I, positive powers. Math. Ann., 1998, 311: 251-274

[27] Wang J.. Harnack estimate for the H ^k-flow. Sci. Chin., Ser. A, 2007, 50: 1642-1650

[28] Wang, X. J., Convex solutions to the mean curvature flow. arXiv: amth.DG/0404326

[29] White B.. The nature of singularities in mean curvature flow of mean-convex sets. J. Amer. Math. Soc., 2003, 13: 665-695

[30] Zhu X. P.. Lectures on Mean Curvature Flows, AMS/IP Studies in Advanced Mathematics, 2002, Somervill, MA, A. M. S., Providence, RI: International Press