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arXiv · 2007.08701

Rigidity of low index solutions on $S^3$ via a Frankel theorem for the Allen-Cahn equation

Fritz Hiesmayr
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Abstract

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.

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Fritz Hiesmayr. 2020-07-17. Rigidity of low index solutions on $S^3$ via a Frankel theorem for the Allen-Cahn equation. https://arxiv.org/abs/2007.08701

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