Cylindrically bounded constant mean curvature surfaces in $\mathbb {H} ^2\times \mathbb {R}$

Mazet, Laurent

doi:10.1090/s0002-9947-2015-06171-6

Cited by 9 publications

(5 citation statements)

References 18 publications

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“…The proof of this result is similar to the one of the gradient estimate proved by Spruck [25, Theorem 1.1] and Mazet [13,Proposition 16]. Before beginning the proof, let us make some preliminary computation.…”

Section: Gradient Estimatementioning

confidence: 52%

The Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$, with possible infinite boundary data

Nguyen¹

2014

Illinois J. Math.

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In this paper, we study the Dirichlet problem for the minimal surface equation in Sol3 with possible infinite boundary data, where Sol3 is the non-Abelian solvable 3-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometries. Our main result is a Jenkins-Serrin type theorem which establishes necessary and sufficient conditions for the existence and uniqueness of certain minimal Killing graphs with a non-unitary Killing vector field in Sol3.

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Section: Gradient Estimatementioning

confidence: 52%

The Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$, with possible infinite boundary data

Nguyen¹

2014

Illinois J. Math.

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“…This number is usually called the CMC flux of M along [α] associated to K. Observe that the mean curvature of M is computed with respect to the unit normal vector that, along α, points to the same closed component of M ∪ β as N , and that N, η ≤ 0 along α. The above argument shows the homological invariance of the CMC flux; we refer the reader to the papers [13,17,18,21] for applications of this CMC flux.…”

Section: Casementioning

confidence: 75%

Constant mean curvature spheres in homogeneous three-manifolds

Meeks¹,

Mira²,

Pérez³

et al. 2017

Preprint

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“…For a vertical axis γ the periodicity implies that Σ is contained in a vertical cylinder. Mazet has shown in [Maz15] that such surfaces in H 2 × R are rotationally invariant.…”

Section: Sketch Of Proof (I)mentioning

confidence: 99%

On the existence problem for tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$

Vržina¹

2017

Preprint

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We study the existence problem for tilted unduloids in H 2 × R. These are singly periodic annuli with constant mean curvature H > 1/2 in H 2 × R, and the periodicity of these surfaces is with respect to a discrete group of translations along a geodesic that is neither vertical nor horizontal in the Riemannian product H 2 × R. Via the Daniel correspondence we are able to reduce this existence problem to a uniqueness problem in the Berger spheres: if a pair of linked horizontal geodesics bounds exactly two embedded minimal annuli (for a fixed orientation of the boundary curves) then tilted unduloids in H 2 × R exist.

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Cylindrically bounded constant mean curvature surfaces in $\mathbb {H} ^2\times \mathbb {R}$

Abstract: In this paper it is proved that a properly embedded constant mean curvature surface in H 2 × R which has finite topology and stays at a finite distance from a vertical geodesic line is invariant by rotation around a vertical geodesic line.

Cited by 9 publications

References 18 publications

The Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$, with possible infinite boundary data

The Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$, with possible infinite boundary data

Constant mean curvature spheres in homogeneous three-manifolds

On the existence problem for tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$

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