Remarks on the self-shrinking Clifford torus

TL;DR

This paper investigates the stability and rigidity properties of the Clifford torus in complex Euclidean space under mean curvature flow, revealing its instability under Hamiltonian perturbations and its local uniqueness as a self-shrinker.

Contribution

It provides new insights into the instability and rigidity of the Clifford torus, showing it is unstable under Hamiltonian perturbations but uniquely rigid as a self-shrinker.

Findings

Clifford torus is unstable for Lagrangian mean curvature flow under small Hamiltonian perturbations.

It is Hamiltonian F-stable and locally area minimizing under Hamiltonian variations.

The torus is locally unique as a self-shrinker despite infinitesimal non-rigid deformations.

Abstract

On the one hand, we prove that the Clifford torus in $\mathbb{C}^2$ is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian $F$-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is rigid: it is locally unique as a self-shrinker for mean curvature flow, despite having infinitesimal deformations which do not arise from rigid motions. The proofs rely on analysing higher order phenomena: specifically, showing that the Clifford torus is not a local entropy minimiser even under Hamiltonian variations, and demonstrating that infinitesimal deformations which do not generate rigid motions are genuinely obstructed.