Aleksandrov Reeection and Geometric Evolution of Hypersurfaces
نویسنده
چکیده
Consider a compact embedded hypersurface ? t in R n+1 which moves with speed determined at each point by a function F (1 ; : : : ; 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 1 ; : : : ; n : We shall show that for t > 0 the hypersurface ? t sat-isses local a priori Lipschitz bounds outside of a convex set determined by ? 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 ow of a smooth hypersurface may be generalized to the evolution of a closed set ? t described as the level set of a continuous function u t which satisses in the viscosity sense a degenerate parabolic PDE deened by F for 0 t < 1; ES], CGG]. It has recently been noted that this level-set ow, even when starting from a smooth hypersurface ? 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 ? t : As an application, we give some new results about 1=H ow 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 at Riemannian manifolds.
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