A stable self-shrinking M\"{o}bius bundle in $\mathbb R^4$

Abstract

We prove the linear stability of an embedded mean curvature flow shrinker in $\mathbb R^4$ with the topology of the M\"{o}bius bundle. This provides a non-flat, non-spherical stable shrinker in higher codimension, in sharp contrast with the classification of stable hypersurface shrinkers. Topologically, the M\"obius shrinker models the reversal of a real blow-up, suggesting that stable higher-codimension singularities may encode non-trivial topological operations under mean curvature flow.