Minimal immersions of compact bordered Riemann surfaces with free boundary

Chen, Jingyi; Fraser, Ailana; Pang, Chao

doi:10.48550/arxiv.1209.1165

Cited by 3 publications

(6 citation statements)

References 16 publications

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“…Inequality (1) relates the scalar curvature of M , the mean curvature of ∂M and the topology of the free boundary stable Σ, as in Schoen and Yau's Theorem 1. This connection has also been studied by Chen, Fraser and Pang [4].…”

Section: Introduction and Statements Of The Resultsmentioning

confidence: 74%

Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds

Ambrozio

2013

J Geom Anal

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We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru [12]. We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.

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Section: Introduction and Statements Of The Resultsmentioning

confidence: 74%

Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds

Ambrozio

2013

J Geom Anal

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“…In [28], Shen and Zhu obtained some estimates on the area of compact stable minimal surfaces in three-manifolds with bounds on the scalar curvature. Moreover, Chen, Fraser and Pang [12] obtained the same to the nonempty boundary case and low index. In the same spirit, in Section 3, we obtain some estimates to the volume and area of the boundary of minimal stable free boundary hypersurfaces in terms either of the scalar curvature or the mean convexity of the boundary of the ambient manifold.…”

Section: Introduction and Main Resultsmentioning

confidence: 67%

Free Boundary Hypersurfaces with Non-positive Yamabe Invariant in Mean Convex Manifolds

Barros

Tiarlos²

2019

J Geom Anal

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We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface Σ with nonpositive Yamabe invariant in a Riemannian n-manifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that Σ is locally volume-minimizing in a manifold M n with scalar curvature bounded below by a nonpositive constant and mean convex boundary, we conclude that locally M splits along Σ. In the case that the scalar curvature of M is at least −n(n − 1) and Σ locally minimizes a certain functional inspired by [30], a neighborhood of Σ in M is isometric to ((−ε, ε) × Σ, dt 2 + e 2t g), where g is Ricci flat.2000 Mathematics Subject Classification. Primary 53C42, 53C21; Secondary 58J60.

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“…We will obtain topological and geometrical restrictions for strongly f -stable hypersurfaces under certain curvature and boundary assumptions on the ambient manifold. Our statements and proofs are inspired by previous results for the Riemannian case, see [7], [5], [27], [24], [11], [1], and for hypersurfaces with empty boundary in manifolds with density, see [16], [23] and [15].…”

Section: Topology and Rigidity Of Compact Strongly Stable Hypersurfacesmentioning

confidence: 87%

“…The sharpness of these area estimates comes from the fact that, in case of equality for a locally weighted areaminimizing surface, we get the corresponding rigidity results, see Theorems 4.10 and 4.11 for detailed statements. Previous area estimates in Riemannian 3-manifolds were given by Shen and Zhu [36] when ∂Σ = ∅, and by Chen, Fraser and Pang [11] when ∂Σ = ∅. The associated rigidity results were obtained by Bray, Brendle and Neves [5] under a positive lower bound on the scalar curvature, and by Nunes [27] under a negative one.…”

Section: Introductionmentioning

confidence: 94%

“…These results have been extended for surfaces with empty boundary in manifolds with density by Fan [16] and Espinar [15], respectively. For the case of non-empty boundary, Chen, Fraser and Pang [11], and Ambrozio [1], have recently proved that a compact free boundary stable minimal surface inside a 3-manifold of non-negative scalar curvature and mean convex boundary is either a disk or a totally geodesic flat cylinder. This was previously established in R 3 by Ros [30], who also showed that stable cylinders cannot appear.…”

Section: Introductionmentioning

confidence: 99%

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Free boundary stable hypersurfaces in manifolds with density and rigidity results

Castro

Rosales

2014

Journal of Geometry and Physics

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Abstract. Let M be a weighted manifold with boundary ∂M , i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in ∂M . As a consequence, we obtain variational characterizations of critical points and second order minima of the weighted area with or without a volume constraint. Moreover, in the compact case, we obtain topological estimates and rigidity properties for free boundary stable and area-minimizing hypersurfaces under certain curvature and boundary assumptions on M . Our results and proofs extend previous ones for Riemannian manifolds (constant densities) and for hypersurfaces with empty boundary in weighted manifolds.

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Minimal immersions of compact bordered Riemann surfaces with free boundary

Cited by 3 publications

References 16 publications

Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds

Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds

Free Boundary Hypersurfaces with Non-positive Yamabe Invariant in Mean Convex Manifolds

Free boundary stable hypersurfaces in manifolds with density and rigidity results

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