El flujo lagrangiano de la curvatura media

Resum

The mean curvature flow is possibly the most important geometric evolution equation of submanifolds in Geometric Analysis. More specifically, the MCF is an evolution process under which a submanifold is deformed in the direction of its mean curvature vector. In general, the MCF fails to exist after a finite time, giving rise to a singularity. The main objective of this thesis is to study a special class of solutions that preserve the shape of the evolving submanifolds: the self-similar solutions, they are those whose evolution is by homotheties of the ambient space, and the translating solitons, which are submanifolds evolving by translations of the ambient space with constant speed. In this thesis we obtain new families of examples and characterize them when we assume additional hypotheses related with natural variational problems in the Lagrangian setting, as well as, global results about the self-similar solutions which characterize the Clifford torus.