Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area

Buzano, Reto; Sharp, Benjamin

doi:10.1090/tran/7168

Cited by 18 publications

(39 citation statements)

References 29 publications

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“…See also the subsequent work of Buzano-Sharp[BS18].5 The same bound with a worse constant was originally proven by Tysk[Tys87]. See also[Nay93,GY03].…”

mentioning

confidence: 69%

On the topology and index of minimal surfaces

Chodosh¹,

Máximo

2016

J. Differential Geom.

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For an immersed minimal surface in R 3 , we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index.Our new estimate resolves several conjectures made by J. Choe and D. Hoffman concerning the classification of low-index minimal surfaces: we show that there is no complete two-sided immersed minimal surface in R 3 of index two, complete embedded minimal surface with index three, or complete one-sided minimal immersion with index one.

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“…See also the subsequent work of Buzano-Sharp[BS18].5 The same bound with a worse constant was originally proven by Tysk[Tys87]. See also[Nay93,GY03].…”

mentioning

confidence: 69%

On the topology and index of minimal surfaces

Chodosh¹,

Máximo

2016

J. Differential Geom.

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“…This result is based on a localized version of the compactness result for free boundary minimal hypersurfaces from [3]. Once we have proved Theorem 33, we can then follow a scheme related to the bubbling analysis for closed minimal hypersurfaces developed in Section 3 of [5] to construct various point-scale sequences which will detect all the non-trivial bubbles and half-bubbles that develop at points of curvature concentration on ∂N, yielding Theorem 5.…”

Section: Bubbling Analysismentioning

confidence: 95%

“…These results apply, as a special case, to all classes of the form M(Λ, I) (since this is clearly the same as M I+1 (Λ, 0)). A significant part of the present article is aimed at an accurate description of the local picture around those points of bad convergence, in analogy with the results obtained, for the closed case, by Buzano and Sharp in [5]. We then specify these results to the three dimensional scenario, and derive various sorts of new geometric results based on these tools.…”

Section: Introductionmentioning

confidence: 92%

“…Proof. Case I: We can follow the arguments exactly as in [5,Corollary 2.6] to conclude all the statements of the theorem.…”

Section: Bubbling Analysismentioning

confidence: 99%

“…As a first step in this program, we quantify the lack of smooth compactness (via a blow-up argument) by a finite number of non-trivial, complete, properly embedded minimal hypersurfaces of finite total curvature Σ n ֒→ R n+1 as in [5], except this time it is possible that only 'half' of Σ is captured at a boundary point of bad convergence. In order to formalize this picture, let us introduce some terminology: we employ the word bubble to denote a complete, connected and properly embedded minimal hypersurface of finite total curvature in R n+1 ; we employ the word half-bubble to denote a complete, connected, properly embedded minimal hypersurface that is contained in a half-space and has (non-empty) free boundary with respect to the boundary of this half-space, and has finite total curvature.…”

Section: Introductionmentioning

confidence: 99%

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Bubbling analysis and geometric convergence results for free boundary minimal surfaces

Ambrozio¹,

Buzano²,

Carlotto³

et al. 2019

Journal De L’École Polytechnique — Mathématiques

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We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient dimension less than eight and applicable to possibly improper limit hypersurfaces. In dimension three, this identity can be combined with the Gauss-Bonnet theorem to provide a constraint relating the topology of the free boundary minimal surfaces in a converging sequence, of their limit, and of the bubbles or half-bubbles that occur as blow-up models. We present various geometric applications of these tools, including a description of the behaviour of index one free boundary minimal surfaces inside a 3-manifold of non-negative scalar curvature and strictly mean convex boundary. In particular, in the case of compact, simply connected, strictly mean convex domains in R 3 unconditional convergence occurs for all topological types except the disk and the annulus, and in those cases the possible degenerations are classified.Mathematics Subject Classification (MSC 2010): Primary 53A10; Secondary 53C42, 49Q05.which holds true, with the setup as in Theorem 1, for all k sufficiently large. Actually, the second summand on the right-hand side can also be expressed only in terms of topological data (see Subsection 2.3 for a detailed discussion), so that we can derive the formula we will employ in all of our applications:Corollary 7. In the setting of Theorem 1, specified to n = 2, we have for all k sufficiently largewhere χ(Σ j ) denotes the Euler characteristic of Σ j and b j denotes the number of its ends. This is the starting point for our primary geometric applications. We present three instances, which are meant to illustrate the method, and leave other possible extensions in the form of remarks.Here is the first application we wish to discuss: since we can fully classify bubbles and half-bubbles of Morse index less than two (see Corollary 22 and Corollary 24) we can then get novel, unconditional, geometric convergence results for sequences of free boundary minimal

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Degeneration of 7-Dimensional Minimal Hypersurfaces Which are Stable or Have a Bounded Index

Edelen

2024

Arch Rational Mech Anal

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A 7-dimensional area-minimizing embedded hypersurface $$M^7$$ M 7 will in general have a discrete singular set, and the same is true if M is locally stable provided $${\mathcal {H}}^6(\textrm{sing}M) = 0$$ H 6 ( sing M ) = 0 . We show that if $$M_i^7$$ M i 7 is a sequence of 7D minimal hypersurfaces which are minimizing, stable, or have bounded index, then $$M_i \rightarrow M$$ M i → M can limit to a singular $$M^7$$ M 7 with only very controlled geometry, topology, and singular set. We show that one can always “parameterize” a subsequence $$i'$$ i ′ with controlled bi-Lipschitz maps $$\phi _{i'}$$ ϕ i ′ taking $$\phi _{i'}(M_{1'}) = M_{i'}$$ ϕ i ′ ( M 1 ′ ) = M i ′ . As a consequence, we prove the space of smooth, closed, embedded minimal hypersurfaces M in a closed Riemannian 8-manifold $$(N^8, g)$$ ( N 8 , g ) with a priori bounds $${\mathcal {H}}^7(M) \leqq \Lambda $$ H 7 ( M ) ≦ Λ and $$\textrm{index}(M) \leqq I$$ index ( M ) ≦ I divides into finitely-many diffeomorphism types, and this finiteness continues to hold if one allows the metric g to vary, or M to be singular.

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Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area

Cited by 18 publications

References 29 publications

On the topology and index of minimal surfaces

On the topology and index of minimal surfaces

Bubbling analysis and geometric convergence results for free boundary minimal surfaces

Degeneration of 7-Dimensional Minimal Hypersurfaces Which are Stable or Have a Bounded Index

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