Asymptotic behavior for singularities of the mean curvature flow.

*(English)*Zbl 0694.53005This paper deals with the motion of a surface by its mean curvature vector. The context here is a one parameter family \(F_ 1=F(,t)\) with each \(F_ 1\) a smooth immersion into \(R^{n+1}\) of the fixed n- dimensional manifold M. Motion by the mean curvature vector is described by the differential equation
\[
\frac{d}{dt}F(p,t)=H(p,t),
\]
where H(p,t) is the mean curvature vector of the hypersurface \(M_ i=F_ i(M)\) at F(p,t). Typically, initial data \(F(p,0)=F_ 0(p)\) is also imposed. In this paper, \(M=M_ 0\) is assumed to be a compact manifold without boundary, and it is assumed that \(M_ 0\) bounds a non-convex region. In a previous paper, the author considered the situation when \(M_ 0\) is assumed to bound a convex region, and regularity results were obtained. In the non-convex case considered here singularities are expected to develop and the behavior of the surfaces as they approach such singularities is investigated.

The main result is that if a singularity develops at time T, and if a bound of the form \(\max \{| A|^ 2/M_ i\}\leq const/(T-t)\) holds, where A is the second fundamental form, then there is a rescaled sequence near the singularity which converges to a limiting hypersurface and such limiting hypersurfaces satisfy the equation \(H=<x,\nu >\), where x is the position vector, H is the mean curvature, and \(\nu\) is the unit normal such that the mean curvature vector is given by \(H=-H\nu\). Important for the proof is a general monotonicity formula obtained for surfaces moving along their curvature vector. It is shown by an example that the curvature bound above does, in fact, hold for a non-trivial family of surfaces.

The main result is that if a singularity develops at time T, and if a bound of the form \(\max \{| A|^ 2/M_ i\}\leq const/(T-t)\) holds, where A is the second fundamental form, then there is a rescaled sequence near the singularity which converges to a limiting hypersurface and such limiting hypersurfaces satisfy the equation \(H=<x,\nu >\), where x is the position vector, H is the mean curvature, and \(\nu\) is the unit normal such that the mean curvature vector is given by \(H=-H\nu\). Important for the proof is a general monotonicity formula obtained for surfaces moving along their curvature vector. It is shown by an example that the curvature bound above does, in fact, hold for a non-trivial family of surfaces.

Reviewer: H.Parks

##### MSC:

53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |

53C40 | Global submanifolds |