Asymptotic-behavior for singularities of the mean-curvature flow

is satisfied. Here H(p,ή is the mean curvature vector of the hypersurface Mt at F(/?, t). We saw in [7] that (1) is a quasilinear parabolic system with a smooth solution at least on some short time interval. Moreover, it was shown that for convex initial data Mo the surfaces Mt contract smoothly to a single point in finite time and become spherical at the end of the contraction. Here we want to study the singularities of (1) which can occur for nonconvex initial data. Our aim is to characterize the asymptotic behavior of Mt near a singularity using rescaling techniques. These methods have been used in the theory of minimal surfaces and more recently in the study of semilinear heat equations [3], [4]. An important tool of this approach is a monotonicity formula, which we establish in §3. Assuming then a natural upper bound for the growth rate of the curvature we show that after appropriate rescaling near the singularity the surfaces Mt approach a selfsimilar solution of (1). In §4 we consider surfaces Mt9 n > 2, of positive mean curvature and show that in this case the only compact selfsimilar solutions of (1) are spheres. Finally, in §5 we study the model-problem of a rotationally symmetric shrinking neck. We prove that the natural growth rate estimate is valid in this case and that the rescaled solution asymptotically approaches a cylinder.