Remarks on topology of stable translating solitons

Kunikawa, Keita; Saito, Shunsuke

doi:10.1007/s10711-018-0399-1

Cited by 8 publications

(11 citation statements)

References 16 publications

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“…As a consequence of Theorem D we can conclude that any f -stable translator with |A| ∈ L ∞ has at most genus 1. We remark that this last fact was indeed independently improved in the very recent preprint [10] (which appeared on the Arxiv preprint server while we were reviewing a final version of this work). In that paper it is actually obtained that every f -stable translator has genus 0.…”

Section: Introduction and Main Resultsmentioning

confidence: 75%

“…In that paper it is actually obtained that every f -stable translator has genus 0. The proof in [10] relies on a general proposition due to M. Gaffney, [8], as well as on an adaptation of a computation in [17]. Note however that the result in [10] only concerns f -stable translators, while the main focus in our Theorem D is to relate quantitatively the f -index and the genus.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Quantitative index bounds for translators via topology

Impera

Rimoldi

2019

Math. Z.

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We obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form. The estimate involves the dimension of the space of weighted square integrable f -harmonic 1-forms. By the adaptation to the weighted setting of Li-Tam theory developed in previous works, this yields estimates in terms of the number of ends of the hypersurface when this is contained in a upper halfspace with respect to the translating direction. When there exists a point where all principal curvatures are distinct we estimate the nullity of the stability operator. This permits to obtain quantitative estimates on the stability index via the topology of translators with bounded norm of the second fundamental form which are either two-dimensional or (in higher dimension) have finite topological type and are contained in a upper halfspace.2010 Mathematics Subject Classification. 53C42, 53C21.

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Section: Introduction and Main Resultsmentioning

confidence: 75%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Quantitative index bounds for translators via topology

Impera

Rimoldi

2019

Math. Z.

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“…Let us assume ∂H ⊆ π − (∂ Conv (π (Σ))). From Theorem 10 and the -periodicity assumption (30), it follows that Σ ∩ ∂H ≠ ∅. The conclusion follows from the separating tangency principle.…”

Section: Sup σ δmentioning

confidence: 70%

“…Remark 5. Simply connected translating solitons are particularly interesting because it is known (see [28], [29], [30]) that complete -dimensional stable translaters in R must be simply connected. By stable translaters we mean translaters which are linearly stable as minimal surfaces w.r.t.…”

Section: Conjecturementioning

confidence: 99%

Simply connected translating solitons contained in slabs

Chini¹

2020

Geometric Flows

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In this work we show that 2-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of ℝ3 with entropy strictly less than 3 must be mean convex and thus, thanks to a result by Spruck and Xiao are convex. Recently, such 2-dimensional convex translating solitons have been completely classified, up to an ambient isometry, as vertical planes, (tilted) grim reaper cylinders, Δ-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples by Ho man, Martín and White show that the bound on the entropy is necessary.

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“…Remark 5. Simply connected translating solitons are particularly interesting because it is known (see [IR17], [IR19], [KS18]) that complete 2-dimensional stable translaters in R 3 must be simply connected. By stable translaters we mean translaters which are linearly stable as minimal surfaces w.r.t.…”

Section: Introductionmentioning

confidence: 99%

Simply connected translating solitons contained in slabs

Chini¹

2019

Preprint

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In this work we show that 2-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of R 3 with entropy strictly less than 3 must be mean convex and thus, thanks to a result by J. Spruck and L. Xiao [SX17], are convex. Recently, such 2-dimensional convex translating solitons have been completely classified in [HIMW19a], up to an ambient isometry, as vertical plane, (tilted) grim reaper cylinders, ∆-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples by [HMW19a] show that the bound on the entropy is necessary.

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Remarks on topology of stable translating solitons

Abstract: We show that any complete f -stable translating soliton M admits no codimension one cycle which does not disconnect M . As a corollary, it follows that any two dimensional complete f -stable translating soliton has genus zero.

Cited by 8 publications

References 16 publications

Quantitative index bounds for translators via topology

Quantitative index bounds for translators via topology

Simply connected translating solitons contained in slabs

Simply connected translating solitons contained in slabs

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