Surfaces and interfaces in manifolds: geometric and analytic aspects – INTERFACE

The main goal of the proposal is to characterize the global geometry of minimal and constant mean curvature surfaces in general manifolds. We wish to understand embedded minimal and CMC surfaces in all homogeneous manifolds (we expect a unified and global theory in all this spaces) and more generally in any Riemannian manifold.

We also propose to study of a new class of higher codimensional constant mean curvature submanifolds (in the sense of Almgren).

The strong links between minimal and CMC surfaces on the one hand and solutions to some phase transition models on the other hand, has recently opened a completely new field of investigation. Ideas and tools from the theory of minimal and CMC surfaces have proven to be very fruitful and have lead to many new interesting examples both in the theory of nonlinear PDE (Allen-Cahn type equation) and in the theory of overdetermined PDEs (free boundary type problems), which were completely unexpected. We plan to study and classify these new solutions.