Gradient estimates for inverse curvature flows in hyperbolic space

Generalized inverse mean curvature flows in spacetime

Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine necessary conditions on flows of two-surfaces in spacetime under which the Hawking quasilocal mass is monotone. We focus on a subclass of such flows which we call uniformly expanding, which can be considered for null as well as for spacelike dir...

Interior Curvature Estimates and the Asymptotic Plateau Problem in Hyperbolic Space

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in H satisfying f(κ) = σ ∈ (0, 1) with a prescribed asymptotic boundary Γ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if Γ is (Euclidean) starshaped, the solution is uni...

0 Se p 20 12 INTERIOR CURVATURE ESTIMATES AND THE ASYMPTOTIC PLATEAU PROBLEM IN HYPERBOLIC SPACE

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in H n+1 satisfying f(κ) = σ ∈ (0, 1) with a prescribed asymptotic boundary Γ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if Γ is (Euclidean) starshaped, the solution is...

A Priori Estimates for Lagrangian Mean Curvature Flows

Page 1 INTERIOR GRADIENT ESTIMATES FOR MEAN CURVATURE EQUATIONS

In this paper we give a simple proof for the interior gradient estimate for curvature and Hessian equations. We also derive a Liouville type result for these equations. §0. Introduction The interior gradient estimate for the prescribed mean curvature equation has been extensively studied, see [9] and the references therein. For high order mean curvature equations it has also been obtained in [1...