The curve shortening problem associated to a density

Abstract

In $\mathbb{R}^n$ with a density $e^\psi$, we study the mean curvature flow associated to the density ($\psi$-mean curvature flow or $\psi$MCF) of a hypersurface. The main results concern with the description of the evolution under $\psi$MCF of a closed embedded curve in the plane with a radial density, and with a statement of subconvergence to a $\psi$-minimal closed curve in a surface under some general circumstances.