The Allen-Cahn equation on the complete Riemannian manifolds of finite volume

TL;DR

This paper proves the existence of finite energy solutions to the Allen-Cahn equation on certain complete, finite volume Riemannian manifolds, linking solutions to minimal hypersurfaces and extending previous geometric results.

Contribution

It establishes the existence of finite energy solutions with Morse index at most one for the Allen-Cahn equation on non-compact manifolds of finite volume, connecting PDE solutions to minimal hypersurface theory.

Findings

Existence of solutions with finite energy and Morse index ≤ 1.

Solutions' energies are bounded and non-vanishing as epsilon approaches zero.

The results recover and extend theorems about minimal hypersurfaces in manifolds.

Abstract

The semi-linear, elliptic PDE $AC_{\varepsilon}(u):=-\varepsilon^2\Delta u+W'(u)=0$ is called the Allen-Cahn equation. In this article we will prove the existence of finite energy solution to the Allen-Cahn equation on certain complete, non-compact manifolds. More precisely, suppose $M^{n+1}$ (with $n+1\geq 3$) is a complete Riemannian manifold of finite volume. Then there exists $\varepsilon_0>0$, depending on the ambient Riemannian metric, such that for all $0<\varepsilon\leq\varepsilon_0$, there exists $\mathfrak{u}_{\varepsilon}:M\rightarrow (-1,1)$ satisfying $AC_{\varepsilon}(\mathfrak{u}_{\varepsilon})=0$ with the energy $E_{\varepsilon}(\mathfrak{u}_{\varepsilon})<\infty$ and the Morse index $\text{Ind}(\mathfrak{u}_{\varepsilon})\leq 1$. Moreover, $0<\liminf_{\varepsilon\rightarrow 0}E_{\varepsilon}(\mathfrak{u}_{\varepsilon})\leq\limsup_{\varepsilon\rightarrow 0}E_{\varepsilon}(\mathfrak{u}_{\varepsilon})<\infty.$ Our result is motivated by the theorem of Chambers-Liokumovich and Song, which says that $M$ contains a complete minimal hypersurface $\Sigma$ with $0<\mathcal{H}^n(\Sigma)<\infty.$ This theorem can be recovered from our result.