Min-max for phase transitions and the existence of embedded minimal hypersurfaces

TL;DR

This paper extends regularity results for phase transition interfaces to finite Morse index cases and provides a PDE-based proof for the existence of embedded minimal hypersurfaces, connecting phase transitions with geometric analysis.

Contribution

It introduces a PDE approach to prove the existence of minimal hypersurfaces and extends regularity results to finite Morse index phase transition interfaces.

Findings

Extended regularity results to finite Morse index cases.

Provided a PDE-based proof of Almgren-Pitts theorem.

Compared min-max theories with new results.

Abstract

Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated theorem of Almgren-Pitts, on the existence of embedded minimal hypersurfaces in compact manifolds. We compare our results with other min-max theories.