Bouncing Jacobi fields and the Allen-Cahn equation on surfaces

TL;DR

This paper studies the Allen-Cahn equation on surfaces, showing under certain conditions the existence of critical points whose nodal sets approximate geodesics, with bounded energy and Morse index as the parameter tends to zero.

Contribution

It establishes the existence of special critical points of the Allen-Cahn functional on surfaces with nodal sets converging to geodesics, a phenomenon not present in higher dimensions.

Findings

Critical points with nodal sets converging to geodesics

Bounded energy and Morse index of solutions

Specific to two-dimensional surfaces

Abstract

The Allen-Cahn functional is a well studied variational problem which appears in the modeling of phase transition phenomenon. This functional depends on a parameter $\varepsilon >0$ and is intimately related to the area functional as the parameter $\varepsilon$ tends to $0$. In the case where the ambient manifold is a compact surface, we give sufficient assumptions which guarantee the existence of countable families of critical points of the Allen-Cahn functional whose nodal sets converge with multiplicity $2$ to a given embedded geodesic, while their energies and Morse indices stays uniformly bounded, as the parameter $\varepsilon$ tends to $0$. This result is specific to two dimensional surfaces and, for generic metric, it does not occur in higher dimension.