Topological Uniqueness for Self-expanders of Small Entropy

Bernstein, Jared; Wang, Lu

doi:10.48550/arxiv.1902.02642

Cited by 4 publications

(4 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, it is a very difficult open problem to determine whether or not there exist nontrivial asymptotically conical shrinkers with entropy less than the cylinder. To circumvent this problem, for their recently announced proof of the low entropy Schönflies conjecture in R 4 [BWa], Bernstein-Wang had to write a slew of auxiliary papers [BW17a,BW18b,BW18a,BW19b,BW19a,BWb] to deal with the potential scenario of such asymptotically conical shrinkers.…”

Section: Introductionmentioning

confidence: 99%

Ancient asymptotically cylindrical flows and applications

Choi¹,

Haslhofer²,

Hershkovits³

et al. 2019

Preprint

View full text Add to dashboard Cite

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in R n+1 for all n ≥ 3: we show that if a mean curvature flow {M t } in R n+1 has an S n−1 × R singularity at (x 0 , t 0 ), then there exists an ε = ε(x 0 , t 0 ) > 0 such that M t ∩ B(x 0 , ε) is mean-convex for all t ∈ (t 0 − ε 2 , t 0 + ε 2 ). As in the case n = 2, which was resolved by the first three authors in [CHH18], the existence of such a mean-convex neighborhood follows from classifying a certain class of ancient Brakke flows that arise as potential blowup limits near a neck singularity. Specifically, we prove that any ancient unit-regular integral Brakke flow with a cylindrical blowdown must be either a round shrinking cylinder, a translating bowl soliton, or an ancient oval. In particular, combined with a prior result of the last two authors [HW18], we obtain uniqueness of mean curvature flow through neck singularities.The main difficulty in addressing the higher dimensional case is in promoting the spectral analysis on the cylinder to global geometric properties of the solution. Most crucially, due to the potential wide variety of self-shrinking flows with entropy lower than the cylinder when n ≥ 3, smoothness does not follow from the spectral analysis by soft arguments. This precludes the use of the classical moving plane method to derive symmetry. To overcome this, we introduce a novel variant of the moving plane method, which we call "moving plane method without assuming smoothness" -where smoothness and symmetry are established in tandem.

show abstract

Section: Introductionmentioning

confidence: 99%

Ancient asymptotically cylindrical flows and applications

Choi¹,

Haslhofer²,

Hershkovits³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Compared with the definitions of asymptotically conical given elsewhere, see [BW5] or Chapter 2 in [E] for instance, Definition 3.5 seems more restrictive because of the presence of the second condition. However, this condition turns out to be natural in light of Corollary 3.15, where we show that every time-slice of a selfexpanding MCF coming out of a regular cone does have this property.…”

Section: Asymptotically Conical Mcfmentioning

confidence: 99%

“…Recently a result related to the corollary in Theorem 1.2 has been found in [BW5], where Bernstein and Wang show that any two expanders with the same conical end are isotopic, provided that the entropy of the tangent cone is less than a constant determined by the entropies of cylinders and non-flat minimal cones.…”

Section: Introductionmentioning

confidence: 97%