El problema del acortamiento de curvas asociado a una densidad

Resumen

In this Thesis we study the mean curvature flow associated to the density (psi-mean curvature flow or psiMCF) of a hypersurface in a Riemannian manifold with density. In the chapter two, the main results concern with the description of the evolution under psiMCF 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. In the chapter three, we define Type I singularities for the mean curvature flow associated to a density psi 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 psiMCF (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.