Index estimates for free boundary constant mean curvature surfaces

Cavalcante, Marcos P.; Oliveira, Darlan F. de

doi:10.2140/pjm.2020.305.153

Cited by 7 publications

(5 citation statements)

References 24 publications

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“…In particular, as a byproduct, they showed that the index of free boundary CMC surfaces in a mean convex domain of R 3 is bounded below by (2g − r − 4)/6 where g is the genus of the surface and r is the number of boundary components of the surface. This particular result was also obtained by Cavalcante-de Oliveira in [12]. Here we generalize these results to compact capillary surfaces.…”

Section: Introductionsupporting

confidence: 85%

Capillary surfaces: stability, index and curvature estimates

Hong¹,

Saturnino²

2021

Preprint

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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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Section: Introductionsupporting

confidence: 85%

Capillary surfaces: stability, index and curvature estimates

Hong¹,

Saturnino²

2021

Preprint

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“…There are plenty of works on the index estimate for closed minimal hypersurfaces or minimal hypersurfaces with free boundary, see for example, Ros [20], Savo [25] and Ambrozio-Carlotto-Sharp [2,3]. See also [19,11,12] for index estimate for CMC surfaces with free boundary, which is related to type-I partitioning problem. The technique in [19,11,12] for non-minimal CMC case only applies for two dimension.…”

Section: Introductionmentioning

confidence: 99%

“…See also [19,11,12] for index estimate for CMC surfaces with free boundary, which is related to type-I partitioning problem. The technique in [19,11,12] for non-minimal CMC case only applies for two dimension.…”

Section: Introductionmentioning

confidence: 99%

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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“…In the free boundary hypersurface case, Ambrozio, Carlotto and Sharp [3] and Sargent [11] explored the same strategy and also obtain interesting comparison results. In codimension one case, we also point out the paper [1, 4], where the authors were able to adapt the main ideas above cited to the free boundary constant mean curvature setting.…”

Section: Introductionmentioning

confidence: 95%