Rigidity and existence of Busemann profiles for the Allen--Cahn equation on Cartan--Hadamard manifolds

Abstract

We study entire solutions of the Allen--Cahn equation $Δ_g u = W&#39;(u)$ on Cartan--Hadamard manifolds of the form $u = U\circ b$, where $b$ is a Busemann function and $U\in C^2(\mathbb{R})$; we call these Busemann profiles. Our main result is a rigidity principle requiring no homogeneity of the manifold: if $\inf Δ_g b>0$ and $U$ has finite limits $\ell_\pm$, then $W(\ell_-)\le W(\ell_+)$, with equality exactly when $U$ is constant. In particular no nonconstant Busemann layer joins the two wells of a balanced double-well potential, in any dimension $n\ge 2$; under $K\le-κ_2<0$ the hypothesis is automatic by Hessian comparison, and if $W&#39;$ is locally Lipschitz it holds under the mere nonvanishing of $Δ_g b$. The mechanism is a dissipation identity for the Hamiltonian $\tfrac12(U&#39;)^2-W(U)$ along the flow lines of $\nabla b$, in which the mean curvature of the horospheres acts as a wave speed. Conversely, when $Δ_g b\equiv c$ and $W$ is bistable and unbalanced, strictly increasing Busemann fronts joining the wells exist precisely when $c$ is the Fife--McLeod speed; their interfaces are then horospheres of constant mean curvature. On warped-line and rotational surfaces we test sharpness: monotone warpings admit no equal-level profile and rotational surfaces no radial two-well profile, while an even convex warping carries a nonconstant monotone layer whose nodal set is the central geodesic leaf.