On the second inner variation of the Allen-Cahn Functional and its applications

TL;DR

This paper investigates the second inner variation of the Allen-Cahn functional, revealing it only approximates the area functional to first order, and relates Morse indices of critical points to minimal hypersurfaces.

Contribution

It establishes the relationship between second inner variations of the Allen-Cahn functional and its Gamma-limit, and links Morse indices of critical points to minimal hypersurfaces.

Findings

Allen-Cahn functional approximates the area functional only to first order.

Morse indices of Allen-Cahn critical points are bounded below by those of the limiting minimal hypersurface.

Results depend on the single-multiplicity property of the limiting energy.

Abstract

In this paper, we study the relation between the second inner variations of the Allen-Cahn functional and its Gamma-limit, the area functional. Our result implies that the Allen-Cahn functional only approximates well the area functional up to the first order. However, as an application of our result, we prove, assuming the single-multiplicity property of the limiting energy, that the Morse indices of critical points of the Allen-Cahn functional are bounded from below by the Morse index of the limiting minimal hypersurface.