FLAT $\phi $ CURVATURE FLOW OF CONVEX SETS

Abstract

For an arbitrary initial compact and convex subset $K_{0}$ of $\mathbb{R}% ^{n},$ and for an arbitrary norm $\phi $ on $\mathbb{R}^{n},$ we construct a flat $\phi $ curvature flow $K\left( t\right) $ such that $K\left( t\right) $ is compact and convex throughout the evolution. Previously and using similar methods, R. McCann had shown that flat $\phi $ curvature flow in the plane preserves convex, balanced sets. More recently, G. Bellettini, V. Caselles, A. Chambolle, and M. Novaga showed that flat $\phi $ curvature flow in $% \mathbb{R}^{n}$ preserves compact, convex sets. We also establish a new H o lder continuity estimate for the flow. Flat $\phi $ curvature flows, introduced by F. Almgren, J. Taylor, and L. Wang, model motion by $\phi $-weighted mean curvature. Under certain regularity assumptions, they coincide with smooth $\phi $-weighted mean curvature flows given by partial differential equations as long as the smooth flows exist.