发表机构
Beijing Technology and Business University; University of Science and Technology of China; Wuhan University; Chinese University of Hong Kong; Yunnan Normal University（北京工商大学; 中国科学技术大学; 武汉大学; 香港中文大学; 云南师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了 Allen--Cahn 方程稳定解在维度 2 至 10 的片状定理，并由此推出低维稳定整体解的一维性及重数一刚性。
AI 中文摘要
我们证明了在环境维度 $2$ 到 $10$ 中 Allen--Cahn 方程稳定解的片状定理。我们表明，测度 $\varepsilon|Du_\varepsilon|^2\\,dX$ 弱收敛到具有整数重数的超平面意味着零集是不相交的光滑图。作为推论，在 $\mathbb R^N$（$N\le7$）中具有 $O(R^{N-1})$ 能量增长的稳定整体解是一维的。当 $W''(-1)=W''(1)$ 时，对于 $4\le N\le10$，具有有限 Morse 指标和该能量增长的整体解的每一个 blowdown 都具有重数一。对于偶势，我们还在维度 $3$ 到 $7$ 的闭流形上，在 bumpiness 或正 Ricci 曲率条件下，获得了具有有界能量和 Morse 指标的极限的重数一。
英文摘要
We prove a sheeting theorem for stable solutions to the Allen--Cahn equation in ambient dimensions $2$ through $10$. We show that weak convergence of the measures $\varepsilon|Du_\varepsilon|^2\,dX$ to a hyperplane with integer multiplicity implies that the zero sets are disjoint smooth graphs. As a consequence, stable entire solutions in $\mathbb R^N$, $N\le7$, with $O(R^{N-1})$ energy growth are one-dimensional. When $W''(-1)=W''(1)$, every blowdown of an entire solution with finite Morse index and this energy growth has multiplicity one for $4\le N\le10$. For even potentials, we also obtain multiplicity one for limits with bounded energy and Morse index on closed manifolds of dimensions $3$ through $7$, under bumpiness or positive Ricci curvature.
