# Allen-Cahn min-max on surfaces

## TL;DR

This paper develops a min-max approach using the Allen-Cahn functional to construct geodesics on surfaces, integrating geometric analysis techniques and curvature estimates to analyze singularities.

## Contribution

It introduces a novel method combining Allen-Cahn min-max procedures with curvature estimates to study geodesics and singularities on surfaces.

## Key findings

- Construction of geodesics via Allen-Cahn min-max method
- Analysis of singular points using curvature estimates
- Reduction to understanding specific singularity models

## Abstract

We use a min-max procedure on the Allen-Cahn energy functional to construct geodesics on closed, 2-dimensional Riemannian manifolds, as motivated by the work of Guaraco. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen-Cahn curvature estimates due to Wang-Wei, we manage to study the fine structure of potential singular points at the diffuse level, and show that the problem reduces to that of understanding "entire" singularity models constructed by del Pino-Kowalczyk-Pacard with Morse index 1. The argument is completed by a Morse index estimate on these singularity models.