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.

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关于克利福德自缩环面的评论

一方面，我们证明了克利福德环 ${ℂ}^2$对于拉格朗日平均曲率流，在任意小的哈密顿量扰动下，它是不稳定的，即使它是哈密顿量F稳定且在哈密顿量变化下局部面积最小。另一方面，我们表明Clifford环面是刚性的：尽管局部变形不是由刚性运动引起的，但它作为局部平均曲率流的自收缩器是局部唯一的。证明依赖于分析高阶现象：具体地说，表明即使在哈密顿量变化的情况下，克利福德环面也不是局部熵极小值，并且证明了不会产生刚性运动的无穷小变形确实受到了阻碍。