Nonzero Constant Mean Curvature Research Articles

Spacelike hypersurfaces with constant higher order mean curvature in Minkowski space–time

In this paper, we develop a series of general integral formulae for compact spacelike hypersurfaces with hyperplanar boundary in the ( n+1)-dimensional Minkowski space–time L n+1 . As an application of them, we prove that the only compact spacelike hypersurfaces in L n+1 having constant higher order mean curvature and spherical boundary are the hyperplanar balls (with zero higher order mean curvature) and the hyperbolic caps (with nonzero constant higher order mean curvature). This extends previous results obtained by the first author, jointly with Pastor, for the case of constant mean curvature [J. Geom. Phys. 28 (1998) 85] and the case of constant scalar curvature [Ann. Global Anal. Geom. 18 (2000) 75].

Constant mean curvature surfaces with Delaunay ends

In this paper we shall present a construction of complete surfaces M in R3 with finitely many ends and finite topology, and with nonzero constant mean curvature (CMC). This construction is parallel to the well-known original construction by Kapouleas [5], but we feel that ours is somewhat simpler analytically, and controls the resulting geometry more closely. On the other hand, the surfaces we construct have a rather different, and usually simpler, geometry than those of Kapouleas; in particular, all of the surfaces constructed here are noncompact, so we do not obtain any of his immersed compact examples. The method we use here closely parallels the one we developed recently [10] to study the very closely related problem of constructing Yamabe metrics on the sphere with k isolated singular points, just as Kapouleas’ construction parallels the earlier construction of singular Yamabe metrics by Schoen [18]. The original examples of noncompact CMC surfaces were those in the one-parameter family of rotationally invariant surfaces discovered by Delaunay in 1841 [2]. One extreme element of this family is the cylinder; the ‘Delaunay surfaces’ are periodic, and the embedded members of this family (which are called unduloids) interpolate between the cylinder and an infinite string of spheres arranged along a common axis. The family continues beyond this, but the elements now are immersed (and are called nodoids). The role of Delaunay surfaces in the theory of complete CMC surfaces is analogous to the role of catenoids (and planes) in the study of complete minimal surfaces of finite total curvature. For example, just as any complete minimal surface with two ends must be a catenoid [19], it was proved by Meeks [14] and Korevaar, Kusner and Solomon [8] that any Alexandrov embedded constant mean curvature surface with at most two ends is necessarily a Delaunay surface. A rather more remarkable theorem, paralleling the fact that any end of a complete minimal surface of finite total curvature must be asymptotic to a catenoid or a plane, is the fact that any embedded end of a CMC surface must be asymptotic to one of these rotationally symmetric Delaunay surfaces (and in particular, must be cylindrically bounded).

A Maximum Principle at Infinity for Surfaces with Constant Mean Curvature in Euclidean Space

Maximum principles at infinity generalize Hopf's maximum principle for hypersurfaces with constant mean curvature in R n . We establish such a maximum principle for parabolic surfaces in R3 with nonzero constant mean curvature and bounded Gaussian curvature.

The ABCs of Self-Assembly

Block copolymers consisting of two or three different polymer chains linked by covalent bonds form intricate domain structures that can be exploited in applications ranging from gas separation to optical communication. Recently, previously predicted gyroid structures, which consist of three or five separate three-dimensionally continuous regions, have been synthesized. Their characterization is challenging and was aided by mathematicians9 studies of minimal and nonzero constant mean curvature surfaces. The author is the Morris Cohen Professor of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. E-mail elt@mit.edu

A Bernstein‐type theorem in hyperbolic spaces

The following Bernstein-type theorem in hyperbolic spaces is proved. Let ∑ be a non-zero constant mean curvature complete hypersurface in the hyperbolic space ℍn. Suppose that there exists a one-to-one orthogonal projection from ∑ into a horosphere. (1) If the projection is surjective, then ∑ is a horosphere. (2) If the projection is not surjective and its image is simply connected, then ∑ is a hypersphere.

Some remarks on compact constant mean curvature hypersurfaces in a halfspace of ? n+1

We consider embedded compact hypersurfacesM in a halfspace of hyperbolic space with boundary∂M in the boundary geodesic hyperplaneP of the halfspace and with non-zero constant mean curvature. We prove the following. Let {M n } be a sequence of such hypersurfaces with∂M n contained in a disk of radiusr n centered at a pointσ e P such thatr n → 0 and that eachM n is a large. H-hypersurface,H > 1. Then there exists a subsequence of {M n } converging to the sphere of mean curvatureH tangent toP atσ. In the case of smallH-hypersurfaces orH ≤ 1, if we add a condition on the curvature of the boundary, there exists a subsequence of {M n } which are graphs. The convergence is smooth on compact subset of ℍ3 σ.

Spacelike surfaces of constant mean curvature with free boundary in the Minkowski space

In this paper we introduce a variational problem for spacelike surfaces in the Minkowski space whose critical points turn out to be constant mean curvature spacelike surfaces which intersect a given support surface at a constant hyperbolic angle. In the case where the support surface is either a spacelike plane or a hyperbolic plane we prove that the only such surfaces are the planar discs (with zero mean curvature) and the hyperbolic caps (with non-zero constant mean curvature). We also include some additional discussions on this variational problem for the general n-dimensional case.

Constant Mean Curvature Surfaces Foliated by Circles in Lorentz-Minkowski Space

We prove that a spacelike surface in L 3 with nonzero constant mean curvature and foliated by pieces of circles in spacelike planes is a surface of revolution. When the planes containing the circles are timelike or null, examples of nonrotational constant mean curvature surfaces constructed by circles are presented. Finally, we prove that a nonzero constant mean curvature spacelike surface foliated by pieces of circles in parallel planes is a surface of revolution.

Kenmotsu-Bryant type representation formula for constant mean curvature spacelike surfaces in [formula omitted

Our primary object of this paper is to give a representation formula for constant mean curvature spacelike surfaces (including maximal surfaces) in the anti-de Sitter 3-space H 3 1(c 2) of constant negative curvature  c 2, similar to the Kenmotsu type representation formula for nonzero constant mean curvature spacelike surfaces in the Minkowski 3-space L 3. This formula implies that every simply connected spacelike surface M with constant mean curvature in H 3 1(c 2) can be represented by a harmonic map from M to the hyperbolic 2-plane H 2.

Constant mean curvature spacelike hypersurfaces with spherical boundary in the Lorentz-Minkowski space

We study compact spacelike hypersurfaces (necessarily with non-empty boundary) with constant mean curvature in the ( n + 1)-dimensional Lorentz-Minkowski space. In particular, when the boundary is a round sphere we prove that the only such hypersurfaces are the hyperplanar round balls (with zero mean curvature) and the hyperbolic caps (with non-zero constant mean curvature).

Willmore tori and Willmore-Chen submanifolds in pseudo-Riemannian spaces

We exhibit a new method to find Willmore tori and Willmore-Chen submanifolds in spaces endowed with pseudo-Riemannian warped product metrics, whose fibres are homogeneous spaces. The chief points are the invariance of the involved variational problems with respect to the conformal changes of the metrics on the ambient spaces and the principle of symmetric criticality. They allow us to relate the variational problems with that of generalized elastic curves in the conformal structure of the base space. Among other applications we get a rational one-parameter family of Willmore tori in the standard anti De Sitter 3-space shaped on an associated family of closed free elastic curves in the once punctured standard 2-sphere. We also obtain rational one-parameter families of Willmore-Chen submanifolds in standard pseudo-hyperbolic spaces. As an application of a general approach to our method, we give nice examples of pseudo-Riemannian 3-spaces which are foliated with leaves being non-trivial Willmore tori. More precisely, the leaves of this foliation are Willmore tori which are conformal to non-zero constant mean curvature flat tori.

On slant surfaces with constant mean curvature in C2

In this paper, we completely classify proper slant surfaces with constant Gaussian curvature and nonzero constant mean curvature in C2.

The moduli space of complete embedded constant mean curvature surfaces

We examine the space of finite topology surfaces in ℝ3 which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the spaceMk of all such surfaces withk ends (where surfaces are identified if they differ by an isometry of ℝ3) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has noL2-nullspace, we prove thatMk is locally the quotient of a real analytic manifold of dimension 3k−6 by a finite group (i.e. a real analytic orbifold), fork ≥ 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension ofMk is independent of the genus of the underlying punctured Riemann surface to which Σ is conformally equivalent. These results also apply to hypersurfaces of Hn+1 with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.

Constant mean curvature surfaces constructed by fusing Wente tori

Constant mean curvature surfaces constructed by fusing Wente tori

Hypersurfaces with constant scalar curvature and constant mean curvature

Non-spherical hypersurfaces in E 4 with non-zero constant mean curvature and constant scalar curvature m'e the only hypersurfaces possessing the following property: Its position vector can be written as a sum of two non-constant maps, which are eigenmaps of the Laplacian operator with corresponding eigenvalues the zero and a non-zero constant.

Constant mean curvature discs with bounded area

It has been long conjectured that the two spherical caps are then only discs in the Euclidean three-space R 3 {\mathbb {R}^3} with non-zero constant mean curvature spanning a round circle. In this work, we prove that it is true when the area of such a disc is less than or equal to that of the big spherical cap.

The global structure of constant mean curvature surfaces

In a recent ground-breaking paper Kapouleas constructed this century's first examples of complete, finite topology, properly embedded surfaces with (non-zero) constant mean curvature [Ka l ] . Geometrically these surfaces are quite easy to visualize, as we explain below.The purpose of the present paper is to characterize the extent to which Kapouleas ' examples are typical of all embedded constant mean curvature surfaces. First, and throughout this paper, we scale space so that the (constant) mean curvature (the trace of the second fundamental form) of our surface L" is identically one, and we call such a surface an MC1 surface. The only complete embedded MC1 surfaces of finite topology known before Kapouleas ' work arise from the 1parameter family of axially-symmetric (and periodic) examples found by Delaunay in 1841: Using the maximum axial radius R of such a surface as a parameter, this family begins with the cylinder (R = 1), and evolves through embedded, periodic surfaces until the family degenerates to a linear chain of touching spheres (R = 2); the Delaunay family continues for all R > 2 as periodic immersions. Kapouleas ' construction of embedded surfaces begins with a piecewise linear graph in R 3, having the property that edges are either finite segments (with length a multiple of 4) or half-infinite rays, and with the further property that at each vertex some linear combination of their direction vectors, with positive coefficients, adds to zero. (See Fig. 1.) He constructs an approximate solution to the problem by centering radius R = 2 spheres at all vertices, and connecting these smoothly by Delaunay segments having R < 2 (but near 2) with axes along the graph-edges. (Since the periods of the Delaunay surfaces are slightly longer than 4, he must assume also a "flexibility" condition on the graph, to guarantee that such a gluing

On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space

Let RV1 be the (n+l)-dimensional Minkowski space, that is, Rn+l with the Lorentz metric < ,)―(dx1)2+ ・・・+(dxn)2―(dxn+1)z. It has been known that in R +1 hyperplanes are the only complete space-like hypersurfaces whose mean curvatures are zero. This Bernstein type theorem was proposed by Calabi, and solved by him [3] (for n^4) and by Cheng and Yau [5] (for all n) (see alsoIshihara [10] or Nishikawa [14]). On the other hand, for complete space-like hypersurfaces of nonzero constant mean curvature in Rnx+ there are many nonlinear examples constructed by Treibergs [18], Hano and Nomizu [7], Isbihara and Hara [11] and others. In his recent paper, Palmer [17] discussed the Gauss map of a complete space-like hypersurface of constant mean curvature in 1??+1 and showed a condition for the hypersurface to be a hyperplane. This is a result analogous to the one obtained by Hoffman, Osserman and Schoen [9], who proved that the normals to a complete surface of constant mean curvature in the 3-dimensional Euclidean space Es cannot lie in a closed hemisphere of S*, unless the surface is a plane or a right circular cylinder. Note that a right circular cylinder is the simplest example of a complete non-umbilical surface of constant mean curvature in Es. In R+1 the simplest example of a complete non-umbilical space-like hypersurface of constant mean curvature is given by the following:

Algorithms for the computer simulation of two-dimensional projections from structures determined by dividing surfaces

Transmission electron microscopy (TEM) data come in the form of two-dimensional projections of specimens with finite width. For visual interpretation of such data it is valuable, and for quantitative comparison with models essential, to have available a method for generating simulated projections from model structures. We present three computer algorithms that simulate TEM micrographs, from structures determined by contrast between two materials separated by complicated dividing surfaces. Three very general forms used to represent dividing surfaces are treated: traditional coordinate systems, finite element representations, and tesselations adapted to complex line integration schemes. Because surface tension drives dividing surfaces to minimize their area, particular attention is paid to minimal surfaces, surfaces of nonzero constant mean curvature, and parallel surfaces that may form in specimens constrained to small dimensions. Besides the model structure itself, the other inputs are the orientation of the model structure, the location of (parallel) upper and lower truncating planes representing the finite specimen thickness or the form of the bounding surface for microdroplet samples, and the direction and magnitude of a linear deformation representing distortion due to the microtoming process. The values of these registration parameters providing the optimal fit with digitized TEM data are found by a relaxation method. Remarkable matches are obtained between micrographs of block copolymer morphologies and model structures determined by surfaces of constant mean curvature.

The asymptotic blow-up of a surface in Euclidean 3-space

Given a smoothly immersed surface in Euclidean (or affine) 3-space, the asymptotic directions define a subset in the Grassmann bundle of unoriented one-dimensional subspaces over the surface. This links the Euler characteristic of the region where the Gauss curvature is nonpositive with the index of singularities in a natural line field defined on this subset. To apply this we need only identify mechanisms which restrict the index of the singularities. In Section 2.1 we show that specific configurations of nonpositive Gauss curvature cannot be realized by an immersed surface and that specific configurations in 2-sphere cannot be realized as Gauss images of surfaces. In Section 2.2 we prove an existence theorem for surfaces which satisfy regularity conditions and a Symplectic Monge Ampere PDE. In general, a PDE of this type will not restrict the indices of the singularities over a solution. However, we show that over a surface of nonzero constant mean curvature the indices are restricted and, hence, that specific configurations of nonpositive Gauss curvature cannot be realized by a constant mean curvature surface.