# Remarks on the self-shrinking Clifford torus

Evans, C
Lotay, J
Schulze, F

16 July 2019

## Journal:

Journal für die reine und angewandte Mathematik

## Last Updated:

2021-08-23T07:46:44.01+01:00

765

2020

## DOI:

10.1515/crelle-2019-0015

139-170

## 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.

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Submitted

Journal Article