CMC Surfaces Research Articles

Isotropic ppmc immersions

Isometric immersions with parallel pluri-mean curvature (“ppmc”) in euclidean n-space generalize constant mean curvature (“cmc”) surfaces to higher dimensional Kähler submanifolds. Like cmc surfaces they allow a one-parameter family of isometric deformations rotating the second fundamental form at each point. If these deformations are trivial the ppmc immersions are called isotropic. Our main result drastically restricts the intrinsic geometry of such a submanifold: Locally, it must be a symmetric space or a Riemannian product unless the immersion is holomorphic or a superminimal surface in a sphere. We can give a precise classification if the codimension is less than 7. The main idea of the proof is to show that the tangent holonomy is restricted and to apply the Berger–Simons holonomy theorem.

Monodromy of Constant Mean Curvature Surface in Hyperbolic Space

In this paper we give a global version of the Bryant representation of surfaces of constant mean curvature one (cmc-1) in hyperbolic space. This allows to set the associated non-abelian period problem in the framework of flat unitary vector bundles on Riemann surfaces. We use this machinery to prove the existence of certain cmc-1 surfaces having prescribed global monodromy.

Ribaucour Transformations for Hypersurfaces in Space Forms

The theory of Ribaucour transformations for hypersurfaces in space forms is established. For any such hypersurface M, that admits orthonormal principal vector fields, it was shown the existence of a totally umbilic hypersurface locally associated to M by a Ribaucour transformation. A method of obtaining linear Weingarten surfaces in a three-dimensional space form is provided. By applying the theory, a new one-parameter family of complete constant mean curvature (cmc) surfaces in the unit sphere, locally associated to the flat torus, is obtained. The family contains a class of complete cmc cylinders in the sphere. In particular, one gets a family of complete minimal surfaces and minimal cylinders, locally associated to the Clifford torus.

Constant mean curvature surfaces with circular boundary in R³

In this work we deal with surfaces immersed in R3 with constant mean curvature and circular boundary. We improve some global estimates for area and volume of such immersions obtained by other authors. We still establish the uniqueness of the spherical cap in some classes of cmc surfaces.

Assemblies of Concanavalin A onto Carboxymethylcellulose

The immobilization of Concanavalin A, (Con A), onto flat surfaces formed by ultrathin films of carboxymethylcellulose, CMC, silicon wafers or spin-coated poly(methyl methacrylate), (PMMA), was studied by ellipsometry, contact angle measurements and atomic force microscopy (AFM). The formation of Con A monolayer was only observed onto CMC films. The adsorption constant of Con A onto CMC films was determined as being (2.1+/-0.2) x 10(6) L mol(-1). After assembling Con A onto CMC surfaces, these became more hydrophobic, indicating a molecular orientation of Con A hydrophilic residues to the polysaccharide and Con A hydrophobic residues to the air. The affinity of Con A for hydroxyl rich silicon surfaces or for more hydrophobic PMMA films was very weak, evidencing that nonspecific interactions play a marginal role. For comparison, the immobilization of Con A onto hybrid particles of PMMA/CMC was investigated by means of UV-spectrophotometry. Such particles carry CMC chains attached to the surface, as evidenced by mean zeta potential value of -40 mV. The adsorption constant determined for Con A onto PMMA/CMC particles was one order of magnitude smaller than that found for Con A onto CMC films. This finding indicates that the substrate geometry might influence the molecular arrangement of sugar residues on the surface, consequently affecting the sugar-Con A interaction (cluster effect).

WETTING PHENOMENA AND CONSTANT MEAN CURVATURE SURFACES WITH BOUNDARY

In a microscopic scale or microgravity environment, interfaces in wetting phenomena are usually modeled by surfaces with constant mean curvature (CMC surfaces). Usually, the condition regarding the constancy of the contact angle along the line of separation between different phases is assumed. Although the classical capillary boundary condition is the angle made at the contact line, configurations also occur in which a Dirichlet condition is appropriate. In this article, we discuss those with vanishing boundary conditions, such as those that occur on a thin flat portion of a plate of general shape covered with water. In this paper, we review recent works on the existence of CMC surfaces with non-empty boundary, with a special focus on the Dirichlet problem for the constant mean curvature equation.

Curvature estimates for CMC surfaces in three dimensional manifolds

In this paper, we give curvature estimates for strongly stable CMC surfaces in complete three dimensional manifolds.

Ripening of porous media

We address the surface-tension-driven dynamics of porous media in nearly saturated pore-space solutions. We linearize this dynamics in the reaction-limited regime near its fixed points--surfaces of constant mean curvature (CMC surfaces). We prove that the only stable interface for this dynamics is the plane and estimate the time scale for a CMC surface to become unstable. We also discuss the differences between dynamics in open and closed environments, pointing out the unlikelihood that CMC surfaces are ever realized in such environments on any time scale.

Asymptotics of CMC surfaces with polynomial Hopf differential

Asymptotics of CMC surfaces with polynomial Hopf differential

A Hopf differential for constant mean curvature surfaces in S2×R and H2×R

A basic tool in the theory of constant mean curvature (cmc) surfaces Σ in space forms is the holomorphic quadratic differential discovered by H. Hopf. In this paper we generalize this differential to immersed cmc surfaces Σ in the product spaces S × R and H × R and prove a corresponding result about the geometry of immersed cmc spheres S in these target spaces.

Hyperbolic constant mean curvature one surfaces: Spinor representation and trinoids in hypergeometric functions

We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive explicit formulas in hypergeometric functions for CMC-1 surfaces of genus 0 with three regular ends which are asymptotic to catenoid cousins (CMC-1 trinoids).

Construction of non-simply connected CMC surfaces via dressing

Construction of non-simply connected CMC surfaces via dressing

Constant mean curvature one surfaces in hyperbolic 3-space using the Bianchi-Calò method

In this note we present a method for constructing constant mean curvature on surfaces in hyperbolic 3-space in terms of holomorphic data first introduced in Bianchi's Lezioni di Geometria Differenziale of 1927, therefore predating by many years the modern approaches due to Bryant, Small and others. Besides its obvious historical interest, this note aims to complement Bianchi's analysis by deriving explicit formulae for CMC-1 surfaces and comparing the various approaches encountered in the literature.

Triply Periodic Bicontinuous Cubic Microdomain Morphologies by Symmetries

In response to thermodynamic driving forces, the domains in microphase-separated block copolymers have distinct intermaterial dividing surfaces (IMDS). Of particular interest are bicontinuous and tricontinuous, triply periodic morphologies and their mathematical representations. Level surfaces are represented by functions F: R3 → R of points (x, y, z) ∈ R3, which satisfy the equation F(x, y, z) = t, where t is a constant. Level surfaces make attractive approximations of certain recently computed triply periodic constant mean curvature (cmc) surfaces and they are good starting surfaces to obtain cmc surfaces by mean curvature flow. The functions F(x, y, z) arise from the nonzero structure factors F(hkl) of a particular space group, such that the resulting surfaces are triply periodic and maintain the given symmetries. This approach applies to any space group and can, therefore, yield desired candidate morphologies for novel material structures defined by the IMDS. We present a technique for generating such level surfaces, give new examples, and discuss certain bicontinuous cubic IMDS in detail.

Monodromy of isometric deformations of CMC surfaces

We investgate relationship between the monodromy of isometric deformation of CMC-surfaces in the space forms and that of the canonical deformation of CMC surfaces into the other space forms. As an application, we show that the number of isometric closed CMC-surfaces is finite.

Some characterization, uniqueness and existence results for Euclidean graphs of constant mean curvature with planar boundary

We establish the existence and uniqueness of solutions to the Dirichlet problem for the cmc surface equation, including the minimal one, for zero boundary data, in certain domains of the plane. We obtain results that characterize the sphere and cmc graphs among compact embedded cmc surfaces with planar boundary satisfying certain geometric conditions. We also find conditions that imply that a compact embedded cmc surface which is a graph near the boundary is indeed a global graph.

Möbius Geometry of Surfaces of Constant Mean Curvature 1 in Hyperbolic Space

An overview of the various transformations of isothermic surfaces and their interrelations is given using aquaternionic formalism. Applications to the theory of cmc-1 surfaces inhyperbolic space are given and relations between the two theories are discussed. Within this context, we give Mobius geometric characterizations for cmc-1 surfaces in hyperbolic space and theirminimal cousins.

Transformations of discrete isothermic nets and discrete cmc-1 surfaces in hyperbolic space

Using a quaternionic calculus, the Christoffel, Darboux, Goursat, and spectral transformations for discrete isothermic nets are described, with their interrelations. The Darboux and spectral transformations are used to define discrete analogs for cmc-1 surfaces in hyperbolic space and to obtain a discrete version of Bryant's Weierstrass type representation.

Investigation and Application of the Dressing Action on Surfaces of Constant Mean Curvature

We investigate the dressing action on surfaces of constant mean curvature (CMC surfaces) in Euclidean space. In particluar, we show that for CMC surfaces with umbilics the isotropy group under dressing is always trivial. This result is applied to the investigation of CMC surfaces whose metric is invariant under a group of automorphisms of the parameter domain.

Embedded constant mean curvature surfaces with special symmetry

We give sharp, necessary conditions on complete embedded CMC surfaces with three ends and an extra reflection symmetry. The respective submoduli space is a two-dimensional variety in the moduli space of general CMC surfaces. Fundamental domains of our CMC surfaces are characterized by associated great circle polygons in the three-sphere.