Index estimates for free boundary minimal hypersurfaces

Ambrozio, Lucas; Carlotto, Alessandro; Sharp, Benjamin

doi:10.1007/s00208-017-1549-8

Cited by 45 publications

(62 citation statements)

References 28 publications

(51 reference statements)

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“…Proof. The proof is close to Savo [25], Ambrozio-Carlotto-Sharp [3], small difference arises from the boundary computation. We prove it here for reader's convenience.…”

Section: Morse Index Estimatesupporting

confidence: 66%

“…We use Ind(M ) to denote the Morse index for a type-II stationary hypersurface M . Following the argument of Savo [25] and Ambrozio-Carlotto-Sharp [2,3], by using the coordinates of harmonic one-forms, we are able to prove the following lower bound for the index. where Rm and Ric denote the Riemannian curvature tensor and Ricci curvature tensor ofM respectively, H ∂B denotes the mean curvature of ∂B ⊂M , and II denotes the second fundamental form for the embeddingM ⊂ R d .…”

Section: Introductionmentioning

confidence: 92%

“…Since Thanks to above proposition, to estimate the Morse index of M , one only needs to estimate the number of negative eigenvalues of (5.1). Next we use the method of Savo [25], Ambrozio-Carlotto-Sharp [2,3] to find an estimate of number of negative eigenvalues of (5.1) in terms of topological invariant.…”

Section: Morse Index Estimatementioning

confidence: 99%

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Stability for a Second Type Partitioning Problem

Guo

Xia

2020

J Geom Anal

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In this paper, we study stability and instability problem for type-II partitioning problem. First, we make a complete classification of stable type-II stationary hypersurfaces in a ball in a space form as totally geodesic n-balls. Second, for general ambient spaces and convex domains, we give some topological restriction for type-II stable stationary immersed surfaces in two dimension. Third, we give a lower bound for the Morse index for type-II stationary hypersurfaces in terms of their topology.MSC 2010: 53A10, 53C24, 53C40.

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“…Proof. The proof is close to Savo [25], Ambrozio-Carlotto-Sharp [3], small difference arises from the boundary computation. We prove it here for reader's convenience.…”

Section: Morse Index Estimatesupporting

confidence: 66%

Section: Introductionmentioning

confidence: 92%

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Stability for a Second Type Partitioning Problem

Guo

Xia

2020

J Geom Anal

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“…It follows that both our main theorems can be rephrased with those assumptions, instead. Moreover, we note that when n = 2 the theorem by V. Lima can be regarded as a partial converse to the results in [4], where the Morse index is proven to be bounded from below by an affine function of the genus and the number of boundary components of the surface in question, under suitable curvature conditions on the ambient manifold.…”

Section: Introductionmentioning

confidence: 77%

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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“…Remark 1.1. The f -minimal equation (2), together with the rules of conformal change, tells us that Σ m is f -minimal in (M m+1 , g) if and only if it is minimal (in the usual sense) in the manifold (M m+1 , e − 2f m g). Moreover, the f -index coincides with the usual index of the minimal immersion Σ m → (M m+1 , e − 2f m g).…”

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confidence: 99%

Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers

2019

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We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of f -minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the first Betti number is large, this improves index estimates known in literature. In the complete non-compact case, the lower bound is in terms of the dimension of the space of weighted square summable f -harmonic 1-forms; in particular, in dimension 2, the procedure gives an index estimate in terms of the genus of the surface.2010 Mathematics Subject Classification. 53C42, 53C21.

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Index estimates for free boundary minimal hypersurfaces

Cited by 45 publications

References 28 publications

Stability for a Second Type Partitioning Problem

Stability for a Second Type Partitioning Problem

Bubbling analysis and geometric convergence results for free boundary minimal surfaces

Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers

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