May 6, 2005: Maria Calle, CIMS
Ancient Solutions for Mean Curvature Flow
In the first part of the talk, I'll introduce mean curvature flow. A family
of surfaces in R3 (or, in general, ksubmanifolds in Rn) is said to move by
mean curvature flow, if its movement satisfies a particular parabolic PDE.
This evolution follows the steepest descent direction for the area, that is,
the surfaces decrease their area at the fastest possible rate. I present
some basic facts about mean curvature flow's solutions, such as a mean value
inequality and the definition of density at a point.
After that, I'll present a result about ancient solutions. An ancient
solution for mean curvature flow is a solution defined for all times t<0. I
give a bound on the dimension of the ambient space of an ancient solution,
depending on a bound on the density of the evolving submanifold.
