On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature

Li, Haozhao; Wang, Bing

doi:10.48550/arxiv.1811.08654

Cited by 2 publications

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References 59 publications

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“…In [16] Li-Wang studied the flow with type-I mean curvature and confirmed the multiplicity-one conjecture in this case. The present paper follows the method of [18] and can be seen as an attempt to understand more about the asymptotic behaviour of self-shrinkers in terms of HA.…”

Section: Introductionmentioning

confidence: 58%

Self-shrinkers with bounded HA

Wang¹

2021

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We study integral and pointwise bounds on the second fundamental form of properly immersed self-shrinkers with bounded HA. As applications, we discuss gap and compactness results for self-shrinkers. Contents 1 Introduction 2 Preliminaries 3 L p estimate and growth rate 4 Gap and compactness theorems From [3] one sees the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space. Lemma 2.3. (Theorem 1.1 of [3]) Let Σ n be a complete noncompact properly immersed self-shrinker in Eucildean space R n+1 . Then Σ has finite weighted volume Volwhere C is a positive constant depending only on Σ e −f dv.Provided bounded mean curvature we can also get volume ratio lower bound. See Lemma 3.5 in Li-Wang [15].Lemma 2.4. (Lemma 3.5 of [15]) Let Σ n ֒→ R n+1 be a properly immersed hypersurface in B(x 0 , r 0 ) with x 0 ∈ Σ and sup Σ |H| ≤ Λ. Then for any s ∈ (0, r 0 ) we have Vol Σ (B(x 0 , s) ∩ Σ)ω n s n ≤ e Λr 0 Vol Σ (B(x 0 , r 0 ) ∩ Σ) ω n r 0 n .In particular, Vol(B(x 0 , r) ∩ Σ) ≥ e −Λr ω n r n , ∀ r ∈ (0, r 0 ].

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Section: Introductionmentioning

confidence: 58%

Self-shrinkers with bounded HA

Wang¹

2021

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“…Multiplicity-one first appeared as an hypothesis in Brakke's work [Bra78], and then has been upgraded to a conjecture by Ilmanen. Some recent progress under additional assumptions has been made in [Sun18,LW18], though the general case remains widely open. Wang proved that the ends of shrinkers are always either conical or cylindrical [Wan16].…”

Section: Another Well-known Related Open Problem Ismentioning

confidence: 99%

Lectures on mean curvature flow of surfaces

Haslhofer

2021

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Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton's Ricci flow from intrinsic geometry. In this lecture series, I will provide an introduction to the mean curvature flow of surfaces, with a focus on the analysis of singularities. We will see that the surfaces evolve uniquely through neck singularities and nonuniquely through conical singularities. Studying these questions, we will also learn many general concepts and methods, such as monotonicity formulas, epsilon-regularity, weak solutions, and blowup analysis that are of great importance in the analysis of a wide range of partial differential equations. These lecture notes are from the 2021 summer school in PDE at UT Austin, and also contain a detailed discussion of open problems and conjectures.

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On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature

Cited by 2 publications

References 59 publications

Self-shrinkers with bounded HA

Self-shrinkers with bounded HA

Lectures on mean curvature flow of surfaces

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