Title: On stable solutions to the Allen-Cahn equation with bounded energy density in $\mathbb{R}^4$ Show Chinese title

Title: 关于在$\mathbb{R}^4$中具有有界能量密度的 Allen-Cahn 方程的稳定解

Abstract: We show that stable solutions $u:\mathbb{R}^4\to (-1,1)$ to the Allen-Cahn equation with bounded energy density (or equivalently, with cubic energy growth) are one-dimensional. This is known to entail important geometric consequences, such as robust curvature estimates for stable phase transitions, and the multiplicity one and Morse index conjectures of Marques-Neves for Allen-Cahn approximations of minimal hypersurfaces in closed 4-manifolds. Show Chinese abstract

Abstract: 我们证明了具有有界能量密度（或等价地，具有三次能量增长）的Allen-Cahn方程的稳定解$u:\mathbb{R}^4\to (-1,1)$是一维的。 这已知会带来重要的几何结果，例如稳定相变的鲁棒曲率估计，以及Marques-Neves关于闭四流形中极小超曲面的Allen-Cahn近似中的多重性一和Morse指标猜想。