## Abstract

Consider a compact embedded hypersurface Γ_{t} in ℝ^{n+1} which moves with speed determined at each point by a function F(κ_{1}, . . . , κ_{n}, t) of its principal curvatures, for 0 ≤ t < T. We assume the problem is degenerate parabolic, that is, that F(. , t) is nondecreasing in each of the principal curvatures κ_{1}, . . . , κ_{n}. We shall show that for t > 0 the hypersurface Γ_{t} satisfies local a priori Lipschitz bounds outside of a convex set determined by Γ_{0} and lying inside its convex hull. Our method is the parabolic analogue of Aleksandrov's method of moving planes [A1], [A2], [A3], [A4], [AVo]. The flow of a smooth hypersurface may be generalized to the evolution of a closed set Γ_{t} described as the level set of a continuous function u_{t} which satisfies in the viscosity sense a degenerate parabolic PDE defined by F for 0 ≤ t < ∞, [ES], [CGG]. It has recently been noted that this level-set flow, even when starting from a smooth hypersurface Γ_{0}, may develop a nonempty interior after the evolving hypersurface collides with itself or develops singularities [BP], [AIC], [AVe], [K]. We shall prove that the same local Lipschitz bounds as in the hypersurface case hold for the inner and outer boundaries of Γ_{t}. As an application, we give some new results about 1/H flow for non-star-shaped hypersurfaces, which was recently investigated by Huisken and Ilmanen [HI]. We prove existence and asymptotic roundness, in the Lipschitz sense, for "extended" viscosity solutions in ℝ^{n+1}. In contrast, the evolving hypersurfaces given in [HI], which were used to prove a version of the Penrose conjecture, are solutions of a non-local variational problem, valid in general asymptotically flat Riemannian manifolds.

Original language | English (US) |
---|---|

Pages (from-to) | 261-280 |

Number of pages | 20 |

Journal | Communications in Analysis and Geometry |

Volume | 9 |

Issue number | 2 |

State | Published - Apr 1 2001 |