Inequivalent complexity criteria for free boundary minimal surfaces

Carlotto, Alessandro; Franz, Giada

doi:10.1016/j.aim.2020.107322

Cited by 9 publications

(6 citation statements)

References 31 publications

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“…Li in [Li15] studies the free boundary min-max problem without curvature assumptions on the ambient manifold M . However, as remarked in [CF20, Appendix A] (see also [CF20,pp. 5]), there are problems in defining the Morse index for free boundary minimal surfaces that are not properly embedded in M .…”

Section: Introductionmentioning

confidence: 74%

Equivariant index bound for min-max free boundary minimal surfaces

Franz¹

2021

Preprint

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Given a three-dimensional Riemannian manifold with boundary and a finite group of orientation-preserving isometries of this manifold, we prove that the equivariant index of a free boundary minimal surface obtained via an equivariant min-max procedure à la Simon-Smith with n-parameters is bounded above by n. Contents 1. Introduction 1 2. Basic notation 5 3. Equivariant spectrum 5 4. Free boundary minimal surfaces with bounded equivariant index 8 5. Outline of the proof of regularity and genus bound 10 6. Deformation theorem 15 7. Proof of the main theorem 18 8. Equivariant index of some families of (free boundary) minimal surfaces 20 Appendix A. Spectrum of elliptic operators with Robin boundary conditions 21 Appendix B. Bumpyness is G-generic 22 References 24

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Section: Introductionmentioning

confidence: 74%

Equivariant index bound for min-max free boundary minimal surfaces

Franz¹

2021

Preprint

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“…By arguments of Fischer-Colbrie and Schoen [21, Theorem 1] we only have to show that every bounded open set Ũ ⊂ S accepts a positive Jacobi function. Then it follows from arguments of Fischer-Colbrie and Schoen [21] (see also [16,17]) that S must 1 The definition of the index of a surface touching the boundary of the manifold in its interior can be ambiguous [11]. For this reason we do not state that S is strongly stable.…”

Section: Curvature Bounds For Capillary Surfacesmentioning

confidence: 99%

Capillary surfaces: stability, index and curvature estimates

Hong¹,

Saturnino²

2021

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In this paper we investigate the connection between the index and the geometry and topology of capillary surfaces. We prove an index estimate for compact capillary surfaces immersed in general 3-manifolds with boundary. We also study noncompact capillary surfaces with finite index and show that, under suitable curvature assumptions, such surface is conformally equivalent to a compact Riemann surface with boundary, punctured at finitely many points. We then prove that a strongly stable capillary surface immersed in a half-space of R 3 which is minimal or has a contact angle less than or equal to π/2 must be a half-plane. Using this uniqueness result we obtain curvature estimates for strongly stable capillary surfaces immersed in a 3-manifold with bounded geometry.

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“…In Proposition 6.6, we show that this result generalize straightforwardly to the case free boundary MOTS in an initial data set satisfying dominant energy condition µ − |J| ≥ 0. Moreover, Carlotto and Franz [13], among other things, obtain a diameter estimate for stable free boundary minimal surfaces, from which they then obtain a useful area bound. Their proof combines an argument of Fischer-Colbrie [24], a classical theorem of Hartman [32], and a computation in [44] also employed in [12].…”

Section: Inequalities and The Rigidity Of Stable Free Boundary Mots A...mentioning

confidence: 99%

“…Summing over i and using The proof of the diameter estimate in [13] combines an argument of Fischer-Colbrie [24], a classical theorem of Hartman [32], and a computation in [44] employed in [12]. Those arguments apply in this setting to yield the following.…”

Section: Stable Capillary Marginally Outer Trapped Surfacesmentioning

confidence: 99%

Stable Surfaces and Free Boundary Marginally Outer Trapped Surfaces

Alaee¹,

Lesourd²,

Yau³

2020

Preprint

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We explore various notions of stability for surfaces embedded and immersed in spacetimes and initial data sets. The interest in such surfaces lies in their potential to go beyond the variational techniques which often underlie the study of minimal and CMC surfaces. We prove two versions of Christodoulou-Yau estimate for H-stable surfaces, a Cohn-Vossen type inequality for non-compact stable marginally outer trapped surface (MOTS), and a global theorem on the topology of H-stable surfaces. Moreover, we give a definition of capillary stability for MOTS with boundary. This notion of stability leads to an area inequality and a local splitting theorem for free boundary stable MOTS. Finally, we establish an index estimate and a diameter estimate for free boundary MOTS. These are straightforward generalizations of Chen-Fraser-Pang and Carlotto-Franz results for free boundary minimal surfaces, respectively.

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Inequivalent complexity criteria for free boundary minimal surfaces

Cited by 9 publications

References 31 publications

Equivariant index bound for min-max free boundary minimal surfaces

Equivariant index bound for min-max free boundary minimal surfaces

Capillary surfaces: stability, index and curvature estimates

Stable Surfaces and Free Boundary Marginally Outer Trapped Surfaces

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