Mean curvature flow with surgeries of two–convex hypersurfaces

We consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow. It is well known that the solution exists up to a finite singular time at which the curvature becomes unbounded. The purpose of this paper is to define a flow after singularities by a new approach based on a surgery procedure. Compared with the notions of weak solutions existing in the literature, the flow with surgeries has the advantage that it keeps track of the changes of topology of the evolving surface and thus can be applied to classify all geometries that are possible for the initial manifold. Our construction is inspired by the procedure originally introduced by Hamilton for the Ricci flow, and then employed by Perelman in the proof of Thurston's geometrization conjecture. In this paper we consider initial hypersurfaces which have dimension at least three and are two-convex, that is, such that the sum of the two smallest principal curvatures is nonnegative everywhere. Under these assumptions, we construct a flow with surgeries which has uniformly bounded curvature until the evolving manifold is split in finitely many components with known topology. As a corollary, we obtain a classification up to diffeomorphism of the hypersurfaces under consideration.

[1]  A. Fraser.  Minimal disks and two-convex hypersurfaces , 2002 .

[2]  G. Huisken,et al.  Convexity estimates for mean curvature flow and singularities of mean convex surfaces , 1999 .

[3]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .

[4]  Ji-Ping Sha.  p-convex Riemannian manifolds , 1986 .

[5]  Yukio Matsumoto.  An Introduction to Morse Theory , 2001 .

[6]  R. Hamilton,et al.  The formations of singularities in the Ricci Flow , 1993 .

[7]  B. White.  THE NATURE OF SINGULARITIES IN MEAN CURVATURE FLOW OF MEAN-CONVEX SETS , 2002 .

[8]  J. H. Michael,et al.  Sobolev and mean‐value inequalities on generalized submanifolds of Rn , 1973 .

[9]  G. Huisken.  Flow by mean curvature of convex surfaces into spheres , 1984 .

[10]  R. Hamilton.  Convex hypersurfaces with pinched second fundamental form , 1994 .

[11]  Kenneth A. Brakke,et al.  The motion of a surface by its mean curvature , 2015 .

[12]  S. Angenent,et al.  Degenerate neckpinches in mean curvature flow. , 1997 .

[13]  L. Evans,et al.  Motion of level sets by mean curvature. II , 1992 .

[14]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[15]  R. Hamilton.  Four-manifolds with positive curvature operator , 1986 .

[16]  Yun-Gang Chen,et al.  Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations , 1989 .

[17]  Y. Giga,et al.  Mean curvature flow through singularities for surfaces of rotation , 1991 .

[18]  Gerhard Huisken,et al.  Mean curvature flow singularities for mean convex surfaces , 1999 .

[19]  Singularity structure in mean curvature flow of mean-convex sets , 2003, math/0310242.

[20]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[21]  Klaus Ecker,et al.  Regularity Theory for Mean Curvature Flow , 2003 .

[22]  R. Hamilton.  Four-manifolds with positive isotropic curvature , 1997 .

[23]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[24]  Gerhard Huisken,et al.  Local and global behaviour of hypersurfaces moving by mean curvature , 1993 .

[25]  M. Noronha,et al.  LOW CODIMENSIONAL SUBMANIFOLDS OF EUCLIDEAN SPACE WITH NONNEGATIVE ISOTROPIC CURVATURE , 1996 .

[26]  H. Lawson.  Local Rigidity Theorems for Minimal Hypersurfaces , 1969 .

[27]  R. Hamilton.  Harnack estimate for the mean curvature flow , 1995 .

[28]  The topology of hypersurfaces moving by mean curvature , 1995 .

[29]  G. Huisken,et al.  Interior estimates for hypersurfaces moving by mean curvature , 1991 .