I’ll describe what is the mean curvature flow associated to a density and will give some account of my recent work with F. Viñado-Lereu.
First, in $\mathbb{R}^n$ with a density $e^\psi$, we study the mean curvature flow associated to the density ($\psi$MCF) of a hypersurface. The main results of this part of the talk 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.
Second, we define Type I singularities for the $\psi$MCF and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the $\psi$-curvature due to the density. We describe a family of curves whose evolution under $\psi$MCF (in a Riemannian surface of non-negative curvature with a density which is singular at a geodesic of the surface) produces only type I singularities and study the limits of their blow-ups.
These results and their proofs are collected in:
Miquel, Vicente; Viñado-Lereu, Francisco; "The curve shortening problem associated to a density". Calc. Var. Partial Differential Equations 55 (2016), no. 3, 55:61 and
Miquel, Vicente; Viñado-Lereu, Francisco; "Type I singularities in the curve shortening flow associated to a density” arXiv:1607.08402