Min-max construction of minimal hypersurfaces

Abstract: In the 1960s, Almgren developed a min-max theory to construct
closed minimal submanifolds in an arbitrary closed Riemannian manifold. The
regularity theory in the co-dimension 1 case was further developed by Pitts
and Schoen-Simon. In particular, by the combined works of Almgren, Pitts
and Schoen-Simon, in every closed Riemannian manifold M^n, n \geq 3, there
exists at least one closed, minimal hypersurface. Recently, the
Almgren-Pitts min-max theory has been further developed to show that
minimal hypersurfaces exist in abundance.

In addition to the Almgren-Pitts min-max theory, there is an alternative
PDE based approach for the min-max construction of minimal hypersurfaces.
This approach was introduced by Guaraco and further developed by Gaspar and
Guaraco. It is based on the study of the limiting behaviour of solutions to
the Allen-Cahn equation. In my talk, I will briefly describe the
Almgren-Pitts min-max theory and the Allen-Cahn min-max theory and discuss
the question to what extent these two theories agree.