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, t). Below, du is the R gradient of a function u, ‖u‖∞ is the sup norm, and Bs is the ball in R with radius s centered at the origin. Our gradient estimate is the following (see Section 2 for the sharp area estimate):