论文标题
$ s^3 $的低索引解决方案的刚度通过弗兰克尔定理的allen-cahn方程
Rigidity of low index solutions on $S^3$ via a Frankel theorem for the Allen-Cahn equation
论文作者
论文摘要
我们证明了三个球体上的艾伦·卡恩方程式的Urbano风格的刚性定理：莫尔斯索引五的关键点是在克利福德圆环上消失的对称函数。此外，他们意识到Allen-Cahn功能的最低最大光谱的第五宽度。我们通过分析这些关键点的无效和对称性来解决这个问题。然后，我们证明了适合其淋巴结集的Frankel型定理，通常在带正曲率曲率的歧管中有效。这在建立结论中起着关键作用，并进一步使我们能够在较大维度的球体中得出辅助刚度。
We prove a rigidity theorem in the style of Urbano for the Allen-Cahn equation on the three-sphere: the critical points with Morse index five are symmetric functions that vanish on a Clifford torus. Moreover they realise the fifth width of the min-max spectrum for the Allen-Cahn functional. We approach this problem by analysing the nullity and symmetries of these critical points. We then prove a suitable Frankel-type theorem for their nodal sets, generally valid in manifolds with positive Ricci curvature. This plays a key role in establishing the conclusion, and further allows us to derive ancillary rigidity results in spheres with larger dimension.
