Consider a compact embedded hypersurface Γt in ℝn+1 which moves with speed determined at each point by a function F(κ1, . . . , κn, 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 κ1, . . . , κn. We shall show that for t > 0 the hypersurface Γt satisfies local a priori Lipschitz bounds outside of a convex set determined by Γ0 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 Γt described as the level set of a continuous function ut 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 Γ0, 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 Γt. 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 ℝn+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)|
|Number of pages||20|
|Journal||Communications in Analysis and Geometry|
|State||Published - Apr 1 2001|