Frank Pacard

Ecole Polytechnique

Geodesics on surfaces and the Allen-Cahn equation

There is a strong parallel between the theory of minimal hypersurfaces and the solutions of the double-well phase transition Allen-Cahn equation in a n-dimensional compact manifold, when the phase transition parameter tends to 0. In this talk, I will report some recent progress on the relations between geodesics on surfaces and solutions of the Allen-Cahn equation with uniform bounds on the Morse Index and energy. Our results show that the situation in 2-dimensions is strikingly different from the situation in dimension 3. This is a joint work with Juncheng Wei and Yong Liu.