The Aleksandrov Reflection Method for Hypersurfaces Expanding by a Function of Curvatures

Let Γ_t be a compact embedded hypersurface \(\mathbb(R)^{n+1}\) which moves with speed determined at each point by a function F(k₁,...,k_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 k₁,...,k_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 Γ₀ and lying inside its convex hull. As a consequence, if Γ_t expands to reach points further and further away from Γ₀ then Γ_t converges to a round sphere after rescaling. This is proved without estimates on curvature, in contrast to earlier proofs of asymptotic roundness, which typically require strict parabolicity. Our method is the parabolic analogue of Aleksandrov's method of moving planes.
Aleksandrov's reflection method is also extended to treat generalized solutions of this evolution problem, that is, level sets of viscosity solutions to the corresponding geometric PDE. These generalized solutions have recently been shown in certain cases to develop a nonempty interior after the evolving hypersurface collides with itself or develops singularities. We shall prove that the same local Lipschitz bounds as in the embedded-hypersurface case hold for the inner and outer boundaries of such a "fattened" level set Γ_t.
As an application, we give some new results about 1/H flow for nonconvex hypersurfaces, which was recently investigated by Huisken and Ilmanen. Our solutions are limits of viscosity solutions, and are therefore solutions of a local problem, in \(\mathbb(R)^{n+1}\) only. In contrast, the evolving hypersurfaces given by Huisken and Ilmanen, which were used to prove a version of the Penrose conjecture, are solutions of a non-local problem, valid in general asymptotically flat Riemannian manifolds.
This is joint work with Bennett Chow.