Finiteness of Index and Total Scalar Curvature for Minimal Hypersurfaces

Tysk, Johan

doi:10.2307/2046961

Cited by 12 publications

(29 citation statements)

References 4 publications

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“…But sinceM 1 k is stable on every ball of radius less than 1 2 , we must have that Y = ∅ and therefore the convergence is locally smooth and of multiplicity one on every such V . Moreoverg k → g 0 smoothly and uniformly on every V , so that A(Σ 1 ) < ∞ by a result of Tysk [23].…”

Section: Thus By Settingmentioning

confidence: 87%

“…For a contradiction we suppose the contrary; this guarantees that on some such set V we have λ p (M ∩ V ) = α < 0 and following the proof in [1, p. 8] we find eventually that λ p (M k ∩ V ) ≤ α 2 contradicting the assumption that lim inf λ p (M k ) ≥ 0. If g = g 0 then M has finite total curvature by the main result in Tysk [23]. If also Y = ∅ then it is stable on every compact set (and therefore stable), it has finite Euclidean volume growth and therefore by Proposition 2.1 it is a plane.…”

Section: Localised Compactness Theorymentioning

confidence: 88%

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

Buzano

Sharp

2018

Trans. Amer. Math. Soc.

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We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index, the total curvature along the sequence is quantised in terms of the total curvature of some limit hypersurface, plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in Euclidean space -all of which are finite. Thus, we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary.

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Section: Thus By Settingmentioning

confidence: 87%

Section: Localised Compactness Theorymentioning

confidence: 88%

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

Buzano

Sharp

2018

Trans. Amer. Math. Soc.

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“…Here, we shall denote as B b,m t the geodesic t-ball in the real space form IK m (b), and as S b,m−1 t the geodesic t-sphere in IK m (b). On the other hand, we can find also in Lemma 4 in [35] an extrinsic approach to the use of the monotonicity formula of the extrinsic volume growth Vol(Dt) Vol(B b,m t ) to prove that a minimal hypersurface ϕ : P n−1 → R n is proper when the extrinsic volume growth is finite, (a more restrictive hypothesis than our condition (1.1)).…”

Section: Introductionmentioning

confidence: 85%

Mean curvature, volume and properness of isometric immersions

Gimeno

Palmer

2017

Trans. Amer. Math. Soc.

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ABSTRACT. We explore the relation among volume, curvature and properness of a mdimensional isometric immersion in a Riemannian manifold. We show that, when the L p -norm of the mean curvature vector is bounded for some m ≤ p ≤ ∞, and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact 2-Riemannian manifold M with non-positive Gaussian curvature and finite topology, using the study of the focal points of the transverse Jacobi fields to a geodesic ray in M . In particular, we have explored the relation between the minimal focal distance of a geodesic ray and the total curvature of an end containing that geodesic ray.

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“…Now, going back to the definitions it is clear that by restriction index(Σ) ≥ Ind E (Q |Ǎ| 2 ), while the converse inequality is easily obtained by considering a basis for the maximal subspace where Q |A| 2 is negative definite, extending each function by even symmetry and then smoothing along the edge that may be created on ∂Σ. In higher dimensions, one can follow the same argument modulo concluding by invoking the main theorem of J. Tysk in [45].…”

Section: Lemma 18 a Two-dimensional Half-bubble Has Finite Morse Indmentioning

confidence: 95%

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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Finiteness of Index and Total Scalar Curvature for Minimal Hypersurfaces

Cited by 12 publications

References 4 publications

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

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

Mean curvature, volume and properness of isometric immersions

Bubbling analysis and geometric convergence results for free boundary minimal surfaces

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