On the entropy of closed hypersurfaces and singular self-shrinkers

Abstract

Self-shrinkers are the special solutions of mean curvature flow in R n+1 that evolve by shrinking homothetically; they serve as singularity models for the flow. The entropy of a hypersurface introduced by Colding-Minicozzi is a Lyapunov functional for the mean curvature flow, and is fundamental to their theory of generic mean curvature flow.In this paper we prove that a conjecture of Colding-Ilmanen-Minicozzi-White, namely that any closed hypersurface in R n+1 has entropy at least that of the round sphere, holds in any dimension n. This result had previously been established for the cases n ≤ 6 by Bernstein-Wang using a carefully constructed weak flow.The main technical result of this paper is an extension of Colding-Minicozzi's classification of entropy-stable self-shrinkers to the singular setting. In particular, we show that any entropy-stable self-shrinker whose singular set satisfies Wickramasekera's α-structural hypothesis must be a round cylinder S k ( √ 2k) × R n−k .

Round spheres are Hausdorff stable under small perturbation of entropy

“…In [3], Bernstein-Wang showed that the round spheres S n uniquely minimize the entropy (modulo dilations and rigid motions) among closed hypersurfaces in R n+1 for 2 ≤ n ≤ 6, giving an affirmative answer to a conjecture made by Colding-Ilmanen-Minicozzi-White in [8]. Later, Zhu [25] extended the result to all higher dimensions. In [5], Bernstein-Wang further showed that, for n = 2, the round sphere is Hausdorff stable under small perturbations of entropy.…”

Hausdorff stability of the round two-sphere under small perturbations of the entropy

“…This verifies a conjecture of Colding-Ilmanen-Minicozzi-White [6, Conjecture 0.9] (cf. [16,24] for related interesting results). We further show, in [3,Corollary 1.3], that surfaces in R 3 of small entropy are topologically rigid.…”

Entropy and generic mean curvature flow in curved ambient spaces

“…As Colding and Minicozzi point out in [6], the entropy λ(Σ s ) does not necessarily depend smoothly on s; we are only interested though in when we can say it is continuous. We start by proving what one can interpret as a very weak version of Bernstein and Wang's results [3] (see also [21] for the extension to higher dimensions), where as a consequence of their work it was shown the entropy of a closed hypersurface is bounded below by the entropy of the round sphere: Proof: Suppose not. Then there exists a sequence Σ i ∈ Π C,D so that λ(Σ i ) < 1+ 1 i .…”