# Gradient estimates for the Allen-Cahn equation on Riemannian manifolds

## TL;DR

This paper derives gradient estimates for positive solutions to the Allen-Cahn equation on Riemannian manifolds and applies these results to establish a Liouville theorem under nonnegative Ricci curvature.

## Contribution

It provides new gradient estimates for the Allen-Cahn equation on Riemannian manifolds and proves a Liouville theorem for nonnegative Ricci curvature cases.

## Key findings

- Gradient estimates for solutions on manifolds
- Liouville theorem for nonnegative Ricci curvature
- Extension of classical results to Riemannian setting

## Abstract

In this paper, we consider bounded positive solutions to the Allen-Cahn equation on complete noncompact Riemannian manifolds without boundary. We derive gradient estimates for those solutions. As an application, we get a Liouville type theorem on manifolds with nonnegative Ricci curvature.