Constant Mean Curvature Research Articles

Theoretical and Numerical Constant Mean Curvature Surface and Liquid Entry Pressure Calculations for a Combined Pillar–Pore Structure

Modern microfabrication techniques have led to a growing interest in micropillars and pillar–pore structures. Therefore, in this paper a study of the liquid entry pressure of a hydrophobic pillar–pore structure and the corresponding liquid–gas interface shape for the pressurized liquid is presented. We theoretically analysed the constant mean curvature problem for the rotationally symmetric case and determined an analytical expression for the liquid entry pressure of a hydrophobic pillar–pore structure. Furthermore, the shape of the liquid–gas interface as well as a formula for the location of the minimum were derived. The results are useful for designing geometries with specific properties, such as preventing or facilitating liquid intrusion into rough structures. We compared these results to multiphase lattice Boltzmann simulations where equilibrium contact angles in the range of 157∘ to 102∘ were tested. In our further analysis, we compared theoretical findings from previous works to our lattice Boltzmann simulations. The presented cases can serve as a benchmark for the development and validation of numerical multiphase models.

Surfaces with Constant Negative Curvature

In this paper, we have considered surfaces with constant negative Gaussian curvature in the simply isotropic 3-Space by defined Sauer and Strubeckerr. Firstly, we have studied the isotropic II-flat, isotropic minimal and isotropic II-minimal, the constant second Gaussian curvature, and the constant mean curvature of surfaces with constant negative curvature (SCNC) in the simply isotropic 3-space. Surfaces with symmetry are obtained when the mean curvatures are equal. Further, we have investigated the constant Casorati, the tangential and the amalgamatic curvatures of SCNC.

Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

Simons type formulas for surfaces in Sol3 and applications

We compute the Laplacian of the squared norm of the second fundamental form of a surface in Sol3 and then use this Simons type formula to obtain some gap results for compact constant mean curvature surfaces of this space.

Gap phenomena for constant mean curvature surfaces

AbstractIn this paper, we prove gap results for constant mean curvature (CMC) surfaces. First, we find a natural inequality for CMC surfaces that imply convexity for distance function. We then show that if is a complete, properly embedded CMC surface in the Euclidean space satisfying this inequality, then is either a sphere or a right circular cylinder. Next, we show that if is a free boundary CMC surface in the Euclidean 3‐ball satisfying the same inequality, then either is a totally umbilical disk or an annulus of revolution. These results complete the picture about gap theorems for CMC surfaces in the Euclidean 3‐space. We also prove similar results in the hyperbolic space and in the upper hemisphere, and in higher dimensions.

Local space time constant mean curvature and constant expansion foliations

Inspired by the small sphere-limit for quasi-local energy we study local foliations of surfaces with prescribed mean curvature. Following the strategy used by Ye in [23] to study local constant mean curvature foliations we use a Lyapunov Schmidt reduction in an n+1 dimensional manifold equipped with a symmetric 2-tensor to construct the foliations around a point, prove their uniqueness and show their nonexistence conditions. To be specific, we study two foliation conditions. First we consider constant space-time mean curvature surfaces. They are used to characterizing the center of mass in general relativity [4]. Second, we study local foliations of constant expansion surfaces [18].

Alexandrov’s Theorem for Anisotropic Capillary Hypersurfaces in the Half-Space

In this paper, we show that any embedded capillary hypersurface in the half-space with anisotropic constant mean curvature is a truncated Wulff shape. This extends Wente’s result (Pac J Math 88:387–397, 1980. https://doi.org/10.2140/pjm.1980.88.387) to the anisotropic case and He–Li–Ma–Ge’s result (Indiana Univ Math J 58(2):853–868, 2009. https://doi.org/10.1512/iumj.2009.58.3515) to the capillary boundary case. The main ingredients in the proof are a new Heintze-Karcher inequality and a new Minkowski formula, which have their own interest.

On the bifurcation of spacelike hypersurfaces with constant mean curvature in spacetimes

In a time-oriented Lorentzian manifold M‾n+1, we use equivariant bifurcation theory in order to establish sufficient conditions that allow us to guarantee the existence of bifurcation instants or the local rigidity of family {Ωτ} of open sets of M‾n+1 whose boundaries ∂Ωτ are closed (compact without boundary) spacelike H-hypersurfaces, this is, closed spacelike hypersurfaces with constant mean curvature H. In particular, we do this study for a certain family of open sets in a spatially closed generalized Robertson-Walker spacetime, namely, Lorentz warped products (−I×fMn,−dt2+f2〈,〉M) with 1-dimensional negative definite base (I,−dt2), closed Riemannian fiber (Mn,〈,〉M) and warping function f:I→(0,+∞). In this case, the results are obtained considering some appropriate hypotheses that depend of f and of the behavior of the eigenvalues of Laplacian operator of Mn. Applications to such families in some spacetimes, like the de Sitter space, are also given.

Hypersurfaces satisfying [formula omitted] in [formula omitted

In this paper, we study hypersurfaces Mr4(r=0,1,2,3,4) satisfying △H→=λH→ (λ a constant) in the pseudo-Euclidean space Es5(s=0,1,2,3,4,5). We obtain that every such hypersurface in Es5 with diagonal shape operator has constant mean curvature, constant norm of second fundamental form and constant scalar curvature. Also, we prove that every biharmonic hypersurface in Es5 with diagonal shape operator must be minimal.

Constant mean curvature hypersurfaces in $\mathbb{H}^n\times\mathbb{R}$ with small planar boundary

We show that constant mean curvature hypersurfaces in {\mathbb H}^n\times\mathbb{R} , with small and pinched boundary contained in a horizontal slice P , are topological disks provided they are contained in one of the two halfspaces determined by P . This is the analogous in {\mathbb H}^n\times\mathbb{R} of a result in \mathbb{R}^3 by A. Ros and H. Rosenberg [J. Differential Geom. 44 (1996), 807–817].

On rotational surfaces with circular boundary and almost constant mean curvature

We analyze compact rotationally symmetric surfaces of {mathbb {R}}^3 with circular boundary and almost constant mean curvature, showing that they must nearly spherical caps.

Harmonic Gauss maps of submanifolds of arbitrary codimension of the Euclidean space and sphere and some applications

AbstractIt is proven results about existence and nonexistence of unit normal sections of submanifolds of the Euclidean space and sphere, which associated Gauss maps, are harmonic. Some applications to constant mean curvature hypersurfaces of the sphere and to isoparametric submanifolds are obtained too.

Long time behaviour of the discrete volume preserving mean curvature flow in the flat torus

We show that the discrete approximate volume preserving mean curvature flow in the flat torus $$\mathbb {T}^N$$ starting near a strictly stable critical set E of the perimeter converges in the long time to a translate of E exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.

Some classifications of 2-dimensional self-shrinkers

It is well-known that self-shrinkers play an important role in the study of mean curvature flows. In this paper, we develop new techniques to study the rigidity of self-shrinkers. We prove that any self-shrinker X:M→R3 with nonzero constant Gauss curvature is the round sphere S2(2). Moreover, we prove that any flat self-shrinker X:M→R3 is a plane R2, a cylinder S1(1)×R, a generalized cylinder Γ×R, where Γ is an Abresch-Langer curve. At last, we show that any self-shrinker X:M→R3 with constant mean curvature is a plane R2, a cylinder S1(1)×R or the round sphere S2(2).

Mean curvature flow for generating discrete surfaces with piecewise constant mean curvatures

Piecewise constant mean curvature (P-CMC) surfaces are generated using the mean curvature flow (MCF). As an extension of the known fact that a CMC surface is the stationary point of an energy functional, a P-CMC surface can be obtained as the stationary point of an energy functional of multiple patch surfaces and auxiliary surfaces between them. A new formulation is presented for the MCF as the negative gradient flow of the energy functional for multiple patch continuous surfaces, which are further discretized so as to determine the change in the vertex positions of triangular meshes on the surface as well as along the internal boundaries between patches. Numerical examples show that multiple patch surfaces approximately reach the specified mean curvatures through the proposed method, which can diversify the options for the shape design using CMC surfaces.

Asymptotically hyperbolic Einstein constraint equations with apparent horizon boundary and the Penrose inequality for perturbations of Schwarzschild-AdS *

We prove the existence of asymptotically hyperbolic solutions to the vacuum Einstein constraint equations with a marginally outer trapped boundary of positive mean curvature, using the constant mean curvature conformal method. As an application of this result, we verify the Penrose inequality for certain perturbations of Schwarzschild Anti-de Sitter black hole initial data.

Stability of surfaces with constant mean curvature bounded by two coaxial circles

Consider two coaxial round circles with the same radius. A mathematical model of a tiny liquid drop trapped between them are constant mean curvature (CMC) surfaces because a CMC surface is a critical point of the area for all variations that preserve the enclosed volume and satisfy given boundary condition. A CMC surface is said to be stable if the second variation of the area for any such variation is nonnegative. In this paper, we judge the stability of one period of the so-called unduloid between two consecutive bulges rigorously for the first time.

CMC and CGC surfaces in Lorentz Heisenberg group H3

The aim of this paper is to study the characterization of some constant mean curvature (CMC) and constant Gauss curvature (CGC) A-net surfaces in the 3-dimensional Lorentz-Heisenberg group H3 endowed with a left invariant Lorentzian metric gi in the three following cases : 1. g1 = -dx2 + dy2 + (xdy + dz)2, 2. g2 = dx2 + dy2 - (xdy + dz)2. 3. g3 = dx2 + (xdy + dz)2 - [(1 - x)dy - dz]2. Using Levi-Civita connection, we will calculate the mean curvature H and the Gaussian curvature K in order to study CMC and CGC A-net surfaces in (H3, g1), (H3, g2) and (H3, g3).

Constant mean curvature $n$-noids in hyperbolic space

Constant mean curvature $n$-noids in hyperbolic space

Symmetry of hypersurfaces and the Hopf Lemma

<abstract><p>A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed.</p></abstract>