We examine the space of 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 space $M_{k}$ of all such surfaces with $k$ ends (where surfaces are identified if they differ by an isometry of R³)is locally a finite dimensional real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has no L²-nullspace we prove that $M_{k}$ is locally the quotient of a real analytic manifold of dimension 3$k$-6 by a finite group (i.e., a real analytic orbifold), for $k>2$. 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 of $M_{k}$ is independent of the topology of the underlying punctured Riemann surface to which SIGMA is conformally equivalent. These results also apply to hypersurfaces of H^{n+1} with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.

The simplest negatively curved (schwartzene) surface allotrope of carbon, $C_{32}$-trousers is a stable unit and also demonstrate the viability of several related schwartzenes. In contrast to many schwartzene structures proposed earlier, $C_{32}$ is a reasonable target for organic synthesis.

[From the introduction] At the heart of the modern theory of harmonic maps from a Riemann surface to a Riemannian symmetric space is the observation that, in this setting, the harmonic map equations have a zero curvature representation [19, 24, 28] and so correspond to loops of flat connections. This fact was first exploited in the mathematical literature by Uhlenbeck in her study [24] of harmonic maps $R\leftarrow G$ into a compact Lie group $G$. Uhlenbeck discovered that harmonic maps correspond to certain holomorphic maps, the extended solutions, into the based loop group $\Omega G$ and used this to define an action of a certain loop group on the space of harmonic maps. However, the main focus of [24] was on harmonic maps of a 2-sphere and for these maps the action reduces to an action of a finite-dimensional quotient group (see also [1, 9]).

We study Darboux and Christoffel transforms of isothermic surfaces in Euclidean space. The transformations play a significant role in relation to integrable system theory. Surfaces of constant mean curvature turn out to be special among all isothermic surfaces: Their parallel constant mean curvature surfaces are Christoffel and Darboux transforms at the same time. We prove - as a generalization of Bianchi's theorem on minimal Darboux transforms of minimal surfaces - that constant mean curvature surfaces in Euclidean space allow infinity ^3 Darboux transforms into surfaces of constant mean curvature. We indicate the connection of these Darboux transforms to Bäcklund transforms of spherical surfaces.

[From the introduction] In this paper we investigate the interplay between spin structures on a Riemann surface $M$ and immersions of $M$ into three-space. Here, a spin structure is a complex line bundle S over $M$ such that $S\otimes S$ is the holomorphic (co)tangent bundle $T(M)$ of $M$. Thus we may view a section of a S as a "square root" of a holomorphic 1-form on M . Using this notion of spin structure, in the first part of this paper we develop the notion of the spinor representation of a surface in space , based on an observation of Dennis Sullivan [27]. The classical Weierstrass representation is \[\] where $g$ and $\eta$ are respectively a meromorphic function and one-form on the underlying compact Riemann surface. The spinor representation (Theorem 5) is \[\] where s₁ and s₂ are meromorphic sections of a spin structure S . Either representation gives a (weakly) conformal harmonic map $M\rightarrow R^3$ , which therefore parametrizes a (branched) minimal surface.

We study Ising models with general spin flip dynamics obeying the detailed balance law. After passing to suitable macroscopic limits, we obtain interfaces moving with normal velocity depending anisotropically on their principal curvatures and direction. In addition we deduce (direction-dependent) Kubo- Green-type formulae for the mobility and the hessian of the surface tension, thus obtaining an explicit description of anisotropy in terms of microscopic quantities. The choid of dynamics affects only the mobility, a scalar function of the direction.

We give sharp, necessary conditions on complete embedded CMC surfaces with three of four ends subject to reflection symmetries. The respective submoduli spaces are two-dimensional varieties in the moduli spaces of general CMC surfaces. Fundamental domains of our CMC surfaces are characterized by associated great circle polygons in the three-sphere.

[From the introduction] A classical question in surface theory is which data are sufficient to describe a surface in space up to rigid motions. Bonnet suggested that mean curvature and metric should suffice to determine the surface generically. The local theory (without umbilic points) was developed by Bonnet [6], Cartan [4] and Chern [5] who showed the existence of various 1-parameter families of Bonnet surfaces, i.e., surfaces with the same induced metric and mean curvature. A comprehensive study of this problem and its relationship to the Painleve equations has recently been completed by Bobenko and Eitner [2]. On the other hand, Lawson and Tribuzy [9] have shown that for embedded compact surfaces there are at most two surfaces to a given metric and mean curvature. Moreover, uniqueness can be established under various global assumptions [8]. Up to date it is unknown whether such compact Bonnet pairs exist and if, how to construct them.

[From the introduction] What length of rope (of given diameter) is required to tie a particular knot? Or, to turn the problem around, given an embedded curve, how thick a regular neighborhood of the curve also is embedded? Intuitively, the diameter of the possible rope is bounded by the distance between strands at the closest crossing in the knot. But of course the distance between two points along a curve goes to zero as the points approach each other, so to make the notion precise, we need to exclude some neighborhood of the diagonal.

We introduce a relaxation model for front propagation problems. Our proposed relaxation approximation is a semilinear hyperbolic system without singularities. It yields a direction-dependent normal velocity at the leading term and captures, in the Chapman-Enskog expansion, the higher order curvature dependent corrections, including possible anisotropies.

For each embedded constant mean curvature surface in R³ with three ends and genus zero, we construct a conjugate cousin boundary contour in S³. The moduli space of such contours is parametrized by the space of triangles on S². This imposes necessary conditions on these CMC surfaces, and we expect this space of triangles (up to rotation of S²) exactly parametrizes the moduli space of these surfaces (up to rigid motion of R³). Remarks on the implications for CMC surfaces with more ends and higher genus are also included.

We study discrete and semidiscrete relaxation schemes for multidimensional scalar conservation laws. We show convergence of the relaxation schemes to the entropy solution of the conservation law and derive error estimates that exhibit the precise interaction between the relaxation time and the space/time discretization parameters of these schemes.

R. Schoen has asked whether the sphere and the cylinder are the only complete (almost) embedded constant mean curvature surfaces with finite absolute total curvature. We propose an infinite family of such surfaces. The existence of examples of this kind is supported by results of computer experiments carried out using an algorithm developed by Oberknapp and Polthier.

Classically, isothermic surfaces are characterized as those surfaces which "are divisible into infinitesimal squares by their curvature lines". This characterization is the direct analogue to the definition of discrete isothermic nets. In order to understand the relations between the discrete and smooth theory better, it is described how to give the classical characterizations a rigorous meaning in the sense of modern differential geometry.

A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. This calculus is then used to discuss the relation between curved flats in the symmetric space of pair points and Darboux and Christoffel pairs of isothermic surfaces/ A new viewpoint on relations between surfaces of constant mean curvature in certain space forms is presented - in particular, a new form of Bryant's Weierstrass-type representation for surfaces of constant mean curvature 1 in hyperbolic 3-space is given.