[From the introduction] In the last twenty years, triply-periodic embedded minimal surfaces have been of great interest and utility to researchers in chemistry, cystallography and material science. The classical examples of Schwarz, Neovious et al., dating from the 19th century were known, but seemed to rest on the periphery of modern mathematical interest in minimal surfaces. The monograph of Schoen, which may fairly be said to be the source of modern interest in the sub ject, presented many new examples, but this work was much better known to scientists than to mathematicians. Except for the gyroid, few of Schoen's examples have had very much impact, until quite recently, in the field of mathematics proper. This has changed recently due in part to the work of Karcher, Fischer-Koch, Wohlgemuth, Nitsche, and Ross.

[From the introduction] The subject of this paper is embedded minimal annuli bounded by two straight lines. The only known examples of such surfaces are given by subdomains of the singly periodic Riemann examples, $R$. There is a 1-parameter family of these surfaces. A fundamental domain of a Riemann example consists of a minimal annulus bounded by two straight lines, and a copy of that surface produced by Schwarz reflection about one of the boundary lines. (See Figure 1. and the analytic description of these surfaces in Section 2.)

[From the introduction] The compact smooth surfaces in R³ with constant positive Gauss Curvature ($K$-surfaces) form a natural class. A $K$-surface without boundary is itself the boundary of a convex body, so it must be embedded. The surfaces of interest to us have non-empty boundary and so are not necessarily embedded. A fundamental question is this: given a collection $\gamma = \{C_1,\dots,C_n\}$ of Jordan curves in R³, what are the $K$-surfaces with boundary $\gamma$? For example, if $\gamma$ is a single Jordan curve with no inflection points, does $\gamma$ bound a $K$-surface?

[From the introduction] In Part I of this paper [11], we have proven the existence of Abrikosov like periodic vortices for the bosonic sector models proposed by Ambjorn and Olesen [4] of the full Glashow-Salam-Weinberg electroweak theory, where the gauge group is $SU(2)\times U(1)$. These solutions were found from a Bogomol'nyi system of first order equations which take on a more complicated form than in the classical abelian case due to the anti- screening of the magnetic field. As a result, this system further reduces to a semilinear elliptic system of nonstandard type and we showed in Part I that the number of such vortices is bounded above in terms of the relevant physical parameters (although the locations may be prescribed arbitrarily). The goal of the present paper is to study this Bogomol'nyi system for the self- dual electroweak interactions in the full space R². These solutions are necessarily of infinite energy and thus the method of Part I cannot be directly applied. Our main strategy then, is to combine the method of weighted Sobolev spaces used by McOwen [8] in his study of conformal deformation equations, with the crucial change of variables introduced in Part I to reduce our elliptic system to a lower diagonal form. As a result, we are able to show (Theorem 3.3) that for any distribution of vortex locations there is a two parameter family of gauge-distinct solutions.

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 to 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.

[From the introduction] In this short presentation we will discuss two ways to compute minimal surfaces. The first way takes advantage of the classical conformal representation of a minimal surface using analytic data on the surface. The second way involves solving the Plateau Problem for a specific boundary, usually polygonal. Our interest is in complete and properly embedded minimal surfaces. Since we will be discussing ways to compute, there will have to be some finiteness imposed on the problem. We will assume compactness, not of the surfaces themselves, of course, but of the underlying conformal structure. According to Osserman's Theorem, a complete minimal surface of finite total curvature is conformally a compact Riemann surface that has been punctured in a finite number of points, each point corresponding to an end of the surface. In particular, such a surface has finite topology. The converse is not true as the example of the helicoid shows; the helicoid is singly-periodic and not flat, so it has infinite total curvature. Whether there are more examples is an open question. See [?] for more details.

[From the introduction] In these notes we will sketch the known necessary conditions for properly embedded surfaces $\Sigma\subset R^3$ which have non-zero constant mean curvature and finite topology. Many examples of constant mean curvature immersions and embeddings have been constructed in recent years through the work of H. Wente [13], N. Kapouleas [4, 5], H. Karcher [6], U. Pinkall and I. Sterling [12]. Immersions of tori and cylinders can already have extremely complicated behavior [12, 13], so general immersed constant mean curvature surfaces must be quite varied in structure. If \006 is required to be properly embedded, it turns out that its structure can be characterized fairly well. Not only is this structure interesting in its own right, but it could also aid in the analysis of physical systems involving several materials or phases, such as those described by the Cahn-Hilliard equation [2, 3], where it is believed that slowly evolving interfaces may have time-varying, approximately constant mean curvature.

[From the introduction] A basic problem in differential geometry is the global structure and classification of complete properly embedded constant mean curvature surfaces in three-manifolds of constant sectional curvature. The models for these manifolds are three-dimensional Euclidean space R³, the three sphere S³, and hyperbolic three-space H³.

[From the introduction] In ([19, 20]), Tryggvason and Aref use a boundary integral method and the vortex-in-cell method to evolve the interface between two fluids in a Hele-Shaw cell. The method gives excellent results for intermediate values of the nondimensional surface tension parameter. The results are different from the predicted results of McLean and Saffman for small surface tension. For large surface tension, there are some numerical problems. In this paper, we implement the method of Tryggvason and Aref but use the point vortex method instead of the vortex-in-cell method. A parametric spline is used to represent the interface. The finger widths obtained agree well with those predicted by McLean and Saffman. We conclude that the method of Tryggvason and Aref can provide excellent results but that the vortex-in-cell method may not be the method of choice for extreme values of the surface tension parameter. In a second method, we represent the interface with a Fourier representation. In addition, an alternative way of discretizing the boundary integral is used. Our results are compared to the linearized theory and the results of McLean and Saffman and are shown to be highly accurate.

[From the introduction] A circle $C$ in R³ is the boundary of two spherical caps of constant mean curvature H for any positive number $H$, which is at most the radius of $C$. It is natural to ask whether spherical caps are the only possible examples. Some examples of constant mean curvature immersed tori by Wente [7] indicate that there are compact genus-one immersed constant mean curvature surfaces with boundary C that are approximated by compact domains in Wente tori; however, this has not been proved. Still one has the conjecture: Conjecture 1. A compact constant mean curvature surface bounded by a circle is a spherical cap if either of the following conditions hold: 1. The surface has genus 0 and is immersed; 2. The surface is embedded.

[From the introduction] In 1956 M. Shiffman [17] proved several beautiful theorems concerning the geometry of a minimal annulus $A$ whose boundary consists of two closed convex curves in parallel planes $P_1, P_2$. The first theorem stated that the intersection of $A$ with any plane $P$, between P₁ and P₂, is a convex Jordan curve. In particular it follows that $A$ is embedded. He then used this convexity theorem to prove that every symmetry of the boundary of $A$ extended to a symmetry of $A$. In the case that @ $A$ consists of two circles Shiffman proved that $A$ was foliated by circles in parallel planes. Earlier B. Riemann [15] described, in terms of elliptic functions, all minimal annuli in R³ that can be expressed as the union of circles in parallel planes (also see [3] for a description of these surfaces as well as a computer graphics image of one of them). Together these results yield a classification of all minimal annuli with boundary consisting of circles in parallel planes.

[From the introduction] In this paper we shall prove two fundamental theorems on the topological uniqueness of certain surfaces in R³. The first of these theorems, which will depend on the second theorem, shows that a properly embedded minimal surface in R³ with one end is unknotted. More precisely, Theorem 1.1 Two properly embedded one-ended minimal surfaces in R³ of the same genus are ambiently isotopic.

[From the introduction] In 1956 Shiffman [14] proved that every minimally immersed annulus in R³ bounded by convex curves in parallel planes is embedded. He proved this theorem by showing that the minimal annulus was foliated by convex curves in parallel planes. We are able to prove a related embeddedness theorem for extremal convex planar curves. Recall that a subset of R³ is extremal if it is contained on the boundary of its convex hull. We will call a pair of convex curves extremal if their union is extremal.

[From the introduction] A fundamental problem in the classical theory of minimal surfaces is to describe the asymptotic geometry of properly embedded minimal surfaces in R³. In the special case that the surface has finite total curvature 1 its asymptotic behavior is well understood. For, in this case, the surface is conformally diffeomorphic to a finitely punctured closed Riemann surface and each end of the surface, one for each puncture point, is asymptotic to a plane or an end of a catenoid (see [19]). Thus the plane and the catenoid are the models for describing the asymptotic behavior of these minimal surfaces. When the properly embedded minimal surface has infinite total curvature, but still finite topology, the question has been asked whether the surface must be asymptotic to a helicoid.

[From the introduction] In this paper we study the conformal structure and the asymptotic behavior of properly embedded minimal surfaces of finite topology in R³. One consequence of our study is that when such a surface has at least two ends, then it has finite conformal type, i.e., it is conformally diffeomorphic to a compact Riemann surface punctured in a finite number of points.

[From the introduction] In this paper we shall discuss hypersurfaces M of space forms of constant curvature; where curvature means one of the symmetric functions of curvature associated to the second fundamental form. The values of the constant will be chosen so that the linearized equation will be an elliptic equation on M . For example, for surfaces in R³ the two possible curvatures are the mean curvature H and the Gaussian curvature $K$. The linearized equation for $H$ is always elliptic and for $K$ it is elliptic when the constant $K$ is positive. In hyperbolic 3-space, the constant $K > - 1$ yields an elliptic equation. Hypersurfaces of constant scalar curvature S₂ will be elliptic when $S_2 > 0$.

We consider the problem of constructing the middle surface of a deformed elastic shell from its first and second fundamental forms, $\hat{a}_{\alpha\beta}$ and $\hat{b}_{\alpha\beta}. The undeformed shell is a spherical cap of radius $R$ and thickness $h$ with an angular width $2\theta_0$ where $0 < \theta_0 < \pi/2$. The cap is subjected to a constant uniform load $\lambda$ and is simply supported at its edge. We seek to compute the one-parameter families of buckled states which branch from the unbuckled state of the shell. This is accomplished in two steps. First, a finite element method is used to solve the governing shell equations, a pair of fourth{order nonlinear partial differential equations. A solution of this system is a curvature potential $w$, a stress potential $f$, and the load $\lambda$. Using Liapunov-Schmidt reduction, it can be shown that solutions possessing a variety of symmetries bifurcate from the unbuckled state of the shell. In the work that is presented here, we will numerically continue these local branches. We parametrize solution branches in terms of a pseudo-arc-length parameter $\rho (i.e., $(\lambda,f,w)=(\lambda(\rho),f_\rho,w_\rho)$), enabling us to track them around turning points. The second step in our solution process is to solve numerically for the parametrization $\hat{X}_\rho$ corresponding to the middle surface of the buckled shell $\hat{X}_\rho$. We do so by integrating the partial differential equations of $\hat{S}_\rho$. The coefficients in these differential equations involve the first and second fundamental forms of the deformed shell $\hat{S}_\rho$ which can be computed from $(\lambda(\rho),f_\rho,w_\rho)$. A number of bifurcation diagrams corresponding to the first three branch points of a spherical cap of size $\theta_0=12.85\degree$ are presented. For this example, a secondary bifurcation point was found connecting two distinct nonaxisymmetric solution branches. Computer graphics are used to display images of various buckled surfaces which branch from the unbuckled state of the shell.

We describe first the analytic structure of Riemann's examples of singly-periodic minimal surfaces; we also characterize them as extensions of minimal annuli bounded by parallel straight lines between parallel planes. We then prove their uniqueness as solutions of the perturbed problem of a punctured annulus, and we present standard methods for determining finite total curvature periodic minimal surfaces and solving the period problems.

[From the introduction] A compound bubble is a partition of a Riemannian 3-manifold M into domains whose boundaries are smooth constant mean curvature surfaces, meeting 3 to a smooth edge or 6 to an isolated vertex, at equal angles. Observe that the equal angle condition means that at any point on the support $\Sigma$ (that is, the union of the boundary surfaces) of the compound bubble the tangent cone is either a plane, the product of a line with an equiangular "Y" or the central cone over the 1-skeleton of a regular tetrahedron.