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Remarks on the self-shrinking Clifford torus

Evans, CG; Lotay, JD; Schulze, F; (2020) Remarks on the self-shrinking Clifford torus. Journal für die reine und angewandte Mathematik , 2020 (765) pp. 139-170. 10.1515/crelle-2019-0015. Green open access

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Abstract

On the one hand, we prove that the Clifford torus in 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.

Type: Article
Title: Remarks on the self-shrinking Clifford torus
Open access status: An open access version is available from UCL Discovery
DOI: 10.1515/crelle-2019-0015
Publisher version: https://doi.org/10.1515/crelle-2019-0015
Language: English
Additional information: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10073301
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