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article

Modified mean curvature flow of entire locally Lipschitz radial graphs in hyperbolic space

Patrick Allmann, Longzhi Lin, Jingyong Zhu

Mathematische Nachrichten > 293 > 5 > 861 - 878

The Asymptotic Plateau Problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space

H^{n + 1}

. The modified mean curvature flow (MMCF) was firstly introduced by Xiao and the second author a few years back in [15], and it provides a tool using geometric flow to find such hypersurfaces with constant mean...

article

Mesh Parametrization Driven by Unit Normal Flow

Hui Zhao, Kehua Su, Chenchen Li, Boyu Zhang, more

Computer Graphics Forum > 39 > 1 > 34 - 49

Based on mesh deformation, we present a unified mesh parametrization algorithm for both planar and spherical domains. Our approach can produce intermediate frames from the original meshes to the targets. We derive and define a novel geometric flow: ‘unit normal flow (UNF)’ and prove that if UNF converges, it will deform a surface to a constant mean curvature (CMC) surface, such as planes and spheres...

article

Translation surfaces with non-zero constant mean curvature in Euclidean space

Thomas Hasanis

Journal of Geometry > 2019 > 110 > 2 > 1-8

We prove that the only surface in 3-dimensional Euclidean space $${\mathbb {R}}^3$$ R3 with constant and non-zero mean curvature H, constructed by the sum of a planar curve and a space curve, is the circular cylinder of radius $$\frac{1}{2|H|}$$ 12|H| .

article

Stability of constant mean curvature surfaces in three-dimensional warped product manifolds

Gregório Silva Neto

Annals of Global Analysis and Geometry > 2019 > 56 > 1 > 57-86

In this paper, we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary, oriented nonzero constant mean curvature surfaces in a large class of three-dimensional...

article

Capillary Surfaces in Circular Cylinders

Thomas I. Vogel

Journal of Mathematical Fluid Mechanics > 2019 > 21 > 1 > 1-14

In the absence of gravity, a capillary surface within a circular cylinder will have constant mean curvature and will make a constant contact angle with the cylinder. Two types of surfaces with these properties are Delaunay surfaces and cylinders (i.e., the free surface of the liquid is another circular cylinder). Stability and energy minimality of these particular capillary surfaces are investigated.

article

Algebraic CMC hypersurfaces of order 3 in Euclidean spaces

Oscar Perdomo, Vladimir G. Tkachev

Journal of Geometry > 2019 > 110 > 1 > 1-7

Understanding and finding of general algebraic constant mean curvature surfaces in the Euclidean spaces is a hard open problem. The basic examples are the standard spheres and the round cylinders, all defined by a polynomial of degree 2. In this paper, we prove that there are no algebraic hypersurfaces of degree 3 in $$\mathbb {R}^n$$ Rn , $$n\ge 3$$ n≥3 , with nonzero constant mean curvature.

article

Constant Mean Curvature Spacelike Hypersurfaces in Spatially Open GRW Spacetimes

José A. S. Pelegrín, Marco Rigoli

The Journal of Geometric Analysis > 2019 > 29 > 4 > 3293-3307

In this paper, we provide under certain geometric and physical assumptions new uniqueness and non-existence results for complete spacelike hypersurfaces of constant mean curvature in spatially open Generalized Robertson–Walker spacetimes. Some of our results are then applied to relevant spacetimes as the steady-state spacetime, Einstein–de Sitter spacetime, and certain radiation models.

article

Stable Big Bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems

Igor Rodnianski, Jared Speck

Selecta Mathematica > 2018 > 24 > 5 > 4293-4459

We prove a stable blowup result for solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Our result applies to small perturbations of the spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) solution with topology $$(0,\infty ) \times \mathbb {T}^3$$ (0,∞)×T3 . The FLRW solution models a spatially uniform scalar-field/stiff fluid evolving in a spacetime that expands towards...

article

Triply periodic constant mean curvature surfaces

William H. Meeks, Giuseppe Tinaglia

Advances in Mathematics > 2018 > 335 > C > 809-837

Given a closed flat 3-torus N, for each H>0 and each non-negative integer g, we obtain area estimates for closed surfaces with genus g and constant mean curvature H embedded in N. This result contrasts with the theorem of Traizet [31], who proved that every flat 3-torus admits for every positive integer g with g≠2, connected closed embedded minimal surfaces of genus g with arbitrarily large area.

article

Stability of Delaunay surface solutions to capillary problems

Thomas I. Vogel

Lithuanian Mathematical Journal > 2018 > 58 > 3 > 329-340

The stability of rotationally symmetric solutions to capillary problems is examined, with applications to three specific problems.

article

Existence of CMC Cauchy surfaces from a spacetime curvature condition

Gregory J. Galloway, Eric Ling

General Relativity and Gravitation > 2018 > 50 > 9 > 1-7

In this note we present a result establishing the existence of a compact CMC Cauchy surface from a curvature condition related to the strong energy condition.

article

Dual quadratic differentials and entire minimal graphs in Heisenberg space

José M. Manzano

Annals of Global Analysis and Geometry > 2019 > 55 > 2 > 197-213

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $${\mathbb {L}}(\kappa ,\tau )$$ L ( κ , τ ) with isometry group of dimension 4, which are dual to the Abresch–Rosenberg differentials in the Riemannian counterparts $${\mathbb {E}}(\kappa ,\tau )$$ E ( κ , τ ) , and obtain some consequences. On the...

article

The homogeneous ruled real hypersurface in a complex hyperbolic space

Makoto Kimura, Sadahiro Maeda, Hiromasa Tanabe

Journal of Geometry > 2018 > 109 > 1 > 1-8

We characterize the homogeneous ruled real hyperurface of a complex hyperbolic space in the class of ruled real hypersurfaces having constant mean curvature.

article

Characterizations of complete CMC spacelike hypersurfaces satisfying an Okumura type inequality

A. Gervasio Colares, Eudes L. de Lima, Henrique F. de Lima

Differential Geometry and its Applications > 2018 > 56 > C > 295-307

We deal with complete constant mean curvature spacelike hypersurfaces immersed in a Lorentzian space form and satisfying a suitable Okumura type inequality, which is a weaker hypothesis than to assume that the spacelike hypersurface has two distinct principal curvatures. In this setting, we obtain estimates for the norm of the traceless part of the second fundamental form and we show that these estimates...

article

A new optimal estimate for the first stability eigenvalue of closed hypersurfaces in Riemannian space forms

Eudes L. Lima, Henrique F. Lima

Rendiconti del Circolo Matematico di Palermo Series 00002 > 2018 > 67 > 3 > 533-537

In this paper, we obtain a new upper bound for the first eigenvalue $$\lambda _1^J$$ λ1J of the stability operator J of a closed constant mean curvature hypersurface in a Riemannian space form, in terms of the mean curvature and the length of the total umbilicity operator of $$\Sigma ^n$$ Σn . When the ambient space is the Euclidean sphere, through the calculus of $$\lambda _1^J$$ λ1J of the Clifford...

article

Compact CMC Graphs in $$M\times \mathbb {R}$$ M × R with Boundary in Two Horizontal Slices

A. Aiolfi, G. Nunes, L. Sauer, R. Soares

Bulletin of the Brazilian Mathematical Society, New Series > 2018 > 49 > 3 > 659-672

We establish existence and uniqueness of compact graphs of cons tant mean curvature in $$M\times \mathbb {R}$$ M × R , over bounded domains contained in $$M\times \left\{ 0\right\} $$ M × 0 , with boundary lying in two horizontal slices of $$M\times \mathbb {R}$$ M × R .

article

Height Estimates for Killing Graphs

Debora Impera, Jorge H. Lira, Stefano Pigola, Alberto G. Setti

The Journal of Geometric Analysis > 2018 > 28 > 3 > 2857-2885

The paper aims at proving global height estimates for Killing graphs defined over a complete manifold with non-empty boundary. To this end, we first point out how the geometric analysis on a Killing graph is naturally related to a weighted manifold structure, where the weight is defined in terms of the length of the Killing vector field. According to this viewpoint, we introduce some potential theory...

article

A New Calabi–Bernstein Type Result in Spatially Closed Generalized Robertson–Walker Spacetimes

C. Aquino, H. Baltazar, H.F. Lima

Milan Journal of Mathematics > 2017 > 85 > 2 > 235-245

The aim of this article is to study the uniqueness of a complete spacelike hypersurface $${\sum^{n}}$$ ∑ n immersed with constant mean curvature H in a spatially closed generalized Robertson–Walker spacetime $${\overline{M}^{n+1} = -I {\times_{f}} {M^{n}}}$$ M ¯ n + 1 = - I × f M n , whose Riemannian fiber $${M^n}$$ M n has positive curvature. Supposing that...

article

Embeddedness of the solutions to the H-Plateau problem

Baris Coskunuzer

Advances in Mathematics > 2017 > 317 > C > 553-574

We generalize Meeks and Yau's embeddedness result on the solutions of the Plateau problem to constant mean curvature disks. We show that any minimizing H-disk in an H0-convex domain is embedded for any H∈[0,H0). In particular, for the unit ball B in R3, this implies that for any H∈[0,1], any Jordan curve in ∂B bounds an embedded H-disk in B.

article

Strongly 2-Hopf hypersurfaces in complex projective and hyperbolic planes

José Carlos Díaz-Ramos, Miguel Domínguez-Vázquez, Cristina Vidal-Castiñeira

Annali di Matematica Pura ed Applicata ( 01923 -) > 2018 > 197 > 2 > 469-486

We give a geometric characterization of certain hypersurfaces of cohomogeneity one in the complex projective and hyperbolic planes. We also obtain some partial classifications of austere hypersurfaces and of Levi-flat hypersurfaces with constant mean curvature in these spaces.

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Search results

Modified mean curvature flow of entire locally Lipschitz radial graphs in hyperbolic space

Mesh Parametrization Driven by Unit Normal Flow

Translation surfaces with non-zero constant mean curvature in Euclidean space

Stability of constant mean curvature surfaces in three-dimensional warped product manifolds

Capillary Surfaces in Circular Cylinders

Algebraic CMC hypersurfaces of order 3 in Euclidean spaces

Constant Mean Curvature Spacelike Hypersurfaces in Spatially Open GRW Spacetimes

Stable Big Bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems

Triply periodic constant mean curvature surfaces

Stability of Delaunay surface solutions to capillary problems

Existence of CMC Cauchy surfaces from a spacetime curvature condition

Dual quadratic differentials and entire minimal graphs in Heisenberg space

The homogeneous ruled real hypersurface in a complex hyperbolic space

Characterizations of complete CMC spacelike hypersurfaces satisfying an Okumura type inequality

A new optimal estimate for the first stability eigenvalue of closed hypersurfaces in Riemannian space forms

Compact CMC Graphs in $$M\times \mathbb {R}$$ M × R with Boundary in Two Horizontal Slices

Height Estimates for Killing Graphs

A New Calabi–Bernstein Type Result in Spatially Closed Generalized Robertson–Walker Spacetimes

Embeddedness of the solutions to the H-Plateau problem

Strongly 2-Hopf hypersurfaces in complex projective and hyperbolic planes

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