具有有界指数的相变：与De Giorgi猜想的类比
Phase transitions with bounded index: Parallels to De Giorgi's conjecture
AI总结：
本文研究了具有有界指数的相变问题，证明了在高维空间中相变解的刚性行为，并与De Giorgi猜想进行了类比。
摘要：
De Giorgi的一个著名猜想——受最小曲面Bernstein问题的启发——断言在R^{d+1}中Allen-Cahn方程的单调解的刚性，其中d≤7。我们为具有有界Morse指数的一般解建立了与De Giorgi猜想类似的结论，远比最小曲面类比所暗示的更强：即任何具有有界能量密度的Allen-Cahn方程有限指数解在R^4中是一维的，并且在稳定解分类的条件下，对于所有4≤n≤7的情况也是如此。作为几何应用，闭合四流形中具有有界能量和指数的相变具有光滑过渡层，其行为类似于最小超曲面。因此，相变在高维中表现出异常刚性的行为。这与三维情况形成鲜明对比，在三维情况下，已知存在大量非平凡的整个解，这些解具有有限指数（和能量密度），由Del Pino-Kowalczyk-Wei的工作所揭示。作者推测任何这样的解必须具有平行的端点，这些端点要么是平面的，要么是猫鼻形的，这在这一框架中被视为De Giorgi猜想的类比。我们在有界能量密度假设下确认了这一图景。
英文摘要：
A well-known conjecture of De Giorgi -- motivated by analogy with the Bernstein problem for minimal surfaces -- asserts the rigidity of monotone solutions to the Allen--Cahn equation in $\mathbb{R}^{d+1}$, with $d\leq 7$. We establish close parallels to De Giorgi's conjecture for general solutions of bounded Morse index, far stronger than the minimal surface analogy would suggest: Namely, any finite index solution to the Allen--Cahn equation with bounded energy density in $\mathbb{R}^4$ is one-dimensional, and -- conditionally on the classification of stable solutions -- the same holds for all $4\leq n \leq 7$. As a geometric application, phase transitions with bounded energy and index in closed four-manifolds have smooth transition layers which behave like minimal hypersurfaces. Consequently, phase transitions exhibit a remarkably rigid behaviour in higher dimensions. This is in stark contrast with the 3D case, in which a wealth of nontrivial entire solutions with finite index (and energy density) is conversely known to exist, by work pioneered by Del Pino--Kowalczyk--Wei. The authors conjectured that any such solution must have parallel ends which are either planar or catenoidal, suggesting it as a parallel to De Giorgi's conjecture in this framework. We confirm this picture under the bounded energy density assumption.
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