The Stability of Self-Shrinkers of Mean Curvature Flow in Higher Codimension

Lee, Yng-Ing; Lue, Yang-Kai

doi:10.48550/arxiv.1204.6116

Cited by 2 publications

(3 citation statements)

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“…For any submanifold M n in R n+p , given x 0 ∈ R n+p and t 0 > 0, define the functional F x 0 ,t 0 of M (see [17], [2], [3], [19]) by…”

Section: Self-shrinkers In Euclidean Spacementioning

confidence: 99%

“…On the other hand, in [9], Colding-Minicozzi derived the first and second variational formulas for F functional of self-shrinkers in R n+1 with polynomial volume growth. Later, these variational formulas are generalized to the case of higher codimension ([3], [2], [19]). By the variation formulas, we will obtain that the entropy of a self-shrinker with polynomial volume growth can be achieved by the functional F 0,1 (cf Subsection 7.2 [9]).…”

Section: Self-shrinkers In Euclidean Spacementioning

confidence: 99%

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Eigenvalues of the drifted Laplacian on complete metric measure spaces

Cheng

Zhou

2016

Commun. Contemp. Math.

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In this paper, first we study a complete smooth metric measure space (M n , g, e −f dv) with the (∞)-Bakry-Émery Ricci curvature Ric f ≥ a 2 g for some positive constant a. It is known that the spectrum of the drifted Laplacian ∆ f for M is discrete and the first nonzero eigenvalue of ∆ f has lower bound a 2 . We prove that if the lower bound a 2 is achieved with multiplicity k ≥ 1, then k ≤ n, M is isometric to Σ n−k × R k for some complete (n − k)-dimensional manifold Σ and by passing an isometry, (M n , g, e −f dv) must split off a gradient shrinking Ricci soliton (R k , gcan, a 4 |t| 2 ), t ∈ R k . This result has an application to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian L for properly immersed self-shrinkers in the Euclidean space R n+p , p ≥ 1 and show the discreteness of the spectrum of L and a logarithmic Sobolev inequality.

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“…For any submanifold M n in R n+p , given x 0 ∈ R n+p and t 0 > 0, define the functional F x 0 ,t 0 of M (see [17], [2], [3], [19]) by…”

Section: Self-shrinkers In Euclidean Spacementioning

confidence: 99%

Section: Self-shrinkers In Euclidean Spacementioning

confidence: 99%

Eigenvalues of the drifted Laplacian on complete metric measure spaces

Cheng

Zhou

2016

Commun. Contemp. Math.

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“…This is actually the reflection of the invariance property of the F-functional and the entropy under rescalings and translations. Since Colding-Minicozzi's work [8], classification problem of F-stable self-shrinkers of the mean curvature flow has drawn much attention, see for instance [1,2,16,17].…”

Section: Introductionmentioning

confidence: 99%

Stabilities of homothetically shrinking Yang-Mills solitons

Chen

Zhang

2014

Preprint

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In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of Ffunctional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabilities have connections with the study of Type-I singularities of the Yang-Mills flow. Two byproducts are also included: We show that the Yang-Mills flow in dimension four cannot develop a Type-I singularity; and we obtain a gap theorem for homothetically shrinking solitons.

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The Stability of Self-Shrinkers of Mean Curvature Flow in Higher Codimension

Cited by 2 publications

References 10 publications

Eigenvalues of the drifted Laplacian on complete metric measure spaces

Eigenvalues of the drifted Laplacian on complete metric measure spaces

Stabilities of homothetically shrinking Yang-Mills solitons

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