Sharp estimates for mean curvature flow of graphs

Sharp Estimates for Mean Curvature Flow of Graphs

A one-parameter family of smooth hypersurfaces {Mt} ⊂ R flows by mean curvature if zt = H(z) = ∆Mtz , (0.1) where z are coordinates on R and H = −Hn is the mean curvature vector. In this note, we prove sharp gradient and area estimates for graphs flowing by mean curvature. Thus, each Mt is assumed to be the graph of a function u(·, t). So, if z = (x, y) with x ∈ R, then Mt is given by y = u(x, ...

Mean curvature flow of spacelike graphs

We prove the mean curvature flow of a spacelike graph in (Σ1 ×Σ2,g1 −g2) of a map f : Σ1 → Σ2 from a closed Riemannian manifold (Σ1,g1) with Ricci1 > 0 to a complete Riemannian manifold (Σ2,g2) with bounded curvature tensor and derivatives, and with K2 ≤ K1, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption K2 ≤ K1...

Weak Solutions for the Mean Curvature Flow of Planar Graphs

Lagrangian Mean Curvature Flow for Entire Lipschitz Graphs

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in R, we show that the parabolic equation (1.1) has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time t = 0. In particular, under the mean curvature...

Lagrangian mean curvature flow for entire Lipschitz graphs II

We prove longtime existence and estimates for smooth solutions to a fully nonlinear Lagrangian parabolic equation with locally C1,1 initial data u0 satisfying either (1) −(1+ η)In ≤ Du0 ≤ (1+ η)In for some positive dimensional constant η, (2) u0 is weakly convex everywhere, or (3) u0 verifies a large supercritical Lagrangian phase condition. Mathematics Subject Classification (2000) Primary 53C...