A rigidity theorem for the Allen-Cahn equation in $S^3$

Abstract: We present a recent rigidity theorem for the Allen-Cahn equation in the three-sphere: critical points with Morse index are symmetric and vanish on a Clifford torus. One key ingredient is a novel Frankel-type property we establish for the nodal sets of any two distinct solutions: they intersect if they are connected. This in fact holds in all manifolds with positive Ricci curvature. Time permitting we will discuss additional rigidity results in higher-dimensional spheres.

analysis of PDEsdifferential geometry

Audience: researchers in the topic

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Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

Registration and links to videos available at blatt.sbg.ac.at/onlineseminar.php

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