AI 中文总结
该研究在Cartan-Hadamard流形上探讨Allen–Cahn方程的Busemann剖面，证明了相关刚性原理，明确了双稳不平衡势下Busemann前沿存在的条件，并在特定曲面上验证了结果的尖锐性。
AI 中文摘要
我们在形如$u = U\circ b$的Cartan-Hadamard流形上研究Allen–Cahn方程$\Delta_g u = W'(u)$的整体解，其中$b$是Busemann函数，$U\in C^2(\mathbb{R})$；我们将这些解称为Busemann剖面。我们的主要结果是一个刚性原理，不要求流形具有齐次性：若$\inf \Delta_g b>0$且$U$具有有限极限$\ell_\pm$，则$W(\ell_-)\le W(\ell_+)$，当且仅当$U$为常数时等号成立。特别地，在任意$n\ge 2$维空间中，不存在非恒定的Busemann层连接平衡双势阱的两个阱；在$K\le-\kappa_2<0$的条件下，该假设由 Hessian 比较自动成立，且若$W'$局部Lipschitz，则仅需$\Delta_g b$非零即可满足。其机制是哈密顿量$\tfrac12(U')^2-W(U)$沿$\nabla b$流线的耗散恒等式，其中horosphere的平均曲率充当波速。反之，当$\Delta_g b\equiv c$且$W$为双稳且不平衡时，仅当$c$为Fife–McLeod速度时，才存在连接两阱的严格递增Busemann前沿；此时它们的界面是常平均曲率的horosphere。我们在扭曲线曲面和旋转曲面上检验尖锐性：单调扭曲不允许存在等水平剖面，旋转曲面不允许存在径向双阱剖面，而偶凸扭曲则承载着非恒定单调层，其节点集为中心测地线叶。
英文摘要
We study entire solutions of the Allen--Cahn equation $Δ_g u = W'(u)$ 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 $W'$ is locally Lipschitz it holds under the mere nonvanishing of $Δ_g b$. The mechanism is a dissipation identity for the Hamiltonian $\tfrac12(U')^2-W(U)$ 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.
