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

Abstract

The semi-linear, elliptic PDE $A{C}_{\epsilon }(u):=-{\epsilon }^2\Delta u+W^{\prime }(u)=0$ 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 $n+1\ge 3$) is a complete Riemannian manifold of finite volume. Then there exists ${\epsilon }_0>0$, depending on the ambient Riemannian metric, such that for all $0<\epsilon \le {\epsilon }_0$, there exists ${\mathfrak{u}}_{\epsilon }:M\to (-1,1)$ satisfying $A{C}_{\epsilon }({\mathfrak{u}}_{\epsilon })=0$ with the energy $E_{\epsilon }({\mathfrak{u}}_{\epsilon })<\infty$ and the Morse index $\text{Ind}({\mathfrak{u}}_{\epsilon })\le 1$. Moreover, $0<{\mathrm{liminf}}_{\epsilon \to 0}{E}_{\epsilon }({\mathfrak{u}}_{\epsilon })\le {\mathrm{limsup}}_{\epsilon \to 0}{E}_{\epsilon }({\mathfrak{u}}_{\epsilon })<\infty .$ 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.

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