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

Abstract

The semi-linear, elliptic PDE AC ε (u) := −ε 2 ∆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 ≥ 3) is a complete Riemannian manifold of finite volume. Then there exists ε 0 > 0, depending on the ambient Riemannian metric, such that for all 0 < ε ≤ ε 0 , there existsOur result is motivated by the theorem of Chambers-Liokumovich [CL20] and Song [Son19], which says that M contains a complete minimal hypersurface Σ with 0 < H n (Σ) < ∞. This theorem can be recovered from our result.