[Excerpt] We have constructed two minimal surfaces of theoretical interest. The first is a complete, embedded, singly-periodic minimal surface (SPEMS) that is asymptotic to the helicoid, has infinite genus, and whose quotient by translations has genus one. The quotient of the helicoid by translations has genus zero and the helicoid itself is simply-connected.

[From the introduction] We will survey what is known about minimal surfaces $S\subset\R^3$, which are complete, embedded, and have finite total curvature: \[\]. The only classically known examples of such surfaces were the plane and the catenoid. The discovery by Costa [14, 15], early in the last decade, of a new example that proved to be embedded sparked a great deal of research in this area. Many new examples have been found, even families of them, as will be described below. The central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples.

[From the introduction] In this paper we shall demonstrate a surprising relationship between the topology of a properly embedded periodic minimal surface in R³ and its global geometry. We shall call a minimal surface periodic if it is connected and invariant under a group $G$ of isometries that acts freely on R³. We will analyze these surfaces by studying their quotient in $R^3/G$. We have already carried out this study for doubly-periodic minimal surfaces [16].

[From the introduction] Recently Freedman, He and Wang [FHW], following work of O'Hara [O], introduced an energy $E(\Gamma)$ for a simple closed curve $\Gamma\subset R^3$. The functional $E$ is continuous on each isotopy class of curves, and tends to infinity as $\Gamma$ nears self-intersection. Moreover, $E$ is "proper" on the set of all isotopy classes, in the sense that there are only finitely many knot types below a given energy level.

[From the introduction] The only connected minimal surfaces foliated by circles and lines are domains on one of the following surfaces: the helicoid, the catenoid, the plane, and the examples of Riemann ([Ri] p329-33, [En] p403-6, [Ni] p85-6). All these surfaces are complete and embedded. Topologically they are planar domains: the helicoid is simply-connected, the catenoid is an annulus (conformally a twice-punctured sphere), and each Riemann example (see Figure 1) is conformal to the plane minus the points $\{(n,0),(\frac{1}{n},0)|n\in Z\} [HKR]. In this section we will show that the plane, helicoid, and catenoid arise naturally as limits of well-chosen and properly normalized sequences of Riemann examples. The local behavior of domains on Riemann examples accounts for the existence of these limits, thus allowing a change of topology in the limit surfaces.

[From the introduction] Minimal immersions or, more generally, harmonic maps of a Riemann surface $S$ into $S^n$; $CP^n$ and other Riemannian symmetric spaces have been intensively studied over the past twenty five years. The subject has been given considerable impetus by the interest of mathematical physicists in non-linear $\sigma$-models, which are harmonic maps of S² into $CP^n$, and in related problems which may be handled using twistor theory and methods of complex geometry.

[From the introduction] Over the past few years significant progress has been made in the understanding of various completely integrable nonlinear PDE (soliton equations) and their relationship to classical problems in differential geometry. It has been shown in a series of recent papers [41, 31, 20, 24, 14, 7, 11] that constant mean and Gauss curvature surfaces, Willmore surfaces, minimal surfaces in spheres and projective spaces and generally harmonic maps from a Riemann surface M into various homogeneous spaces may be described as solutions to various soliton equations. Moreover, these solutions are algebraic in the sense that they are obtained by integrating ordinary differential equations of Lax type which linearise on the Jacobian of an appropriate algebraic curve.

[From the introduction] Around 1780, soon after Lagrange derived the minimal surface equation, Meusnier found the first nonplanar solutions: Euler's catenoid and the helicoid. More than 200 years later, in 1980, the catenoid was still the only known finite total curvature embedded minimal surface and the helicoid was the only known infinite total curvature embedded minimal surface of finite topology. First, the finite total curvature situation changed. Chen-Gackstatter [CG] found an immersed torus with one end and - soon after that - Costa [C] found an embedded torus with three ends. We now know that for every genus $\ge$ 1, there exists a one-parameter family of embedded finite total curvature minimal surfaces with three ends ([HM]), and we have some more examples - less well understood - with four and five ends. ([CHM,BW,W1,W2,HK]). In this paper we describe the construction of a minimally embedded torus with one end and infinite total curvature ([HKW]). The (elliptic) Gauss map of this surface has an essential singularity at the puncture and this makes the surface fundamentally different from all previous embedded examples. The other infinite total curvature and finite topology embedded minimal surface, the helicoid, can be described with the exponential map as Gauss map, i.e. with an essential singularity. The new surface is substantially more complicated than the helicoid because it does not cover a finite total curvature surface in a flat spaceform (namely in R²\times S^1$ in the case of the helicoid). Therefore it cannot be described on a quotient surface in terms of a meromorphic Gauss map. (In the case of the helicoid, $g(z)=z$ on $C\setminus\{0\}$.)

A $K$-surface is a surface whose Gauss curvature $K$ is equal to a positive constant. In this paper, we will consider $K$-surfaces that are defined by a nonlinear boundary value problem. In this setting, existence follows from some recent results on nonlinear second-order elliptic partial differential equations. The analytical techniques used to establish these results motivate effective numerical methods for computing $K$-surfaces. In theory, the solvability of the boundary value problem reduces to the existence of a subsolution. In an analogous way, if an approximate numerical subsolution can be determined, then the corresponding $K$-surface can be computed. We will consider two boundary value problems. In the first problem, the $K$-surface is a graph over a plane. In the second problem, the $K$-surface is a radial graph over a sphere. From certain geometrical considerations, it follows that there is a maximum allowable Gauss curvature $K_{max} for these problems. The principal results in this paper are numerical estimates of $K_{max} for a variety of geometries and boundary data. Using a continuation method, we determine numerically the unique one-parameter family of $K$-surfaces that exist for $K$ \in (0, $K_{max} ). We can compare our numerical estimates for $K_{max} to the true value when the $K$-surface is a subset of a hyperbolic spherical surface of revolution. In this case, we find that our numerical estimates for $K_{max} are in close agreement with the expected values.

[From the introduction] In [2], two of the authors and W. Meeks found examples of translation-invariant, embedded minimal surfaces with an infinite number of topological ends. For each $k>0$, a surface $M_k$ was constructed, which was invariant with respect to a translation parallel to the x₃-axis, and under a rotation group of order $k+1$ around the x₃-axis. The method of construction was generalized in [3] to obtain the first known examples of embedded, singly-periodic minimal surfaces with an infinite number of topological ends, invariant under screw-motions (with a nontrivial rotational component). For each integer $k>0$ and angle $\theta$, with $|\theta| < \frac{\pi}{k+1}$, there exists an embedded surface $M_{k,\theta} whose orientation-preserving symmetry group contains a rotation of order $k+1$ around the x₃ -axis and a screw motion - a unit translation in the x₃-direction, followed by a $2\theta$ rotation around it. See Figure 0. Although the surfaces $M_{k,\theta}$ were conceived1 as smooth deformations of the singly-periodic examples $M_k$, the proof in [3] did not construct these deformations. In fact, it left open the following questions: 1. Is the family $M_{k,\theta} smooth in $\theta$? If so, are they deformations of $M_k$? 2. Are the surfaces $M_{k\theta} unique? 3. Is there a surface $M_{k,\pi/(k+1)$? To be more precise: the symmetry groups of the $M_{k,\theta}$ have a single limit as $\theta\leftarrow\pi/(k+1)$ and $\theta\leftarrow -\pi/(k+1)$. This group contains a translation but is different from the symmetries of $M_{k,\theta}. The question becomes, is there a surface with this symmetry group and obvious properties generalizing the $M_{k,\theta}$?

There has been recent interest in knot energies, especially those which are invariant under Möbius transformations of space. We describe computer experiments with such energies, and discuss ways of extending these to energies for higher-dimensional submanifolds. The appendix gives a table of computed energy-minimizing knots and links through eight crossings.

[From the introduction] In recent years mathematical physicists have been studying discrete (in space and time) analogs of integrable non-linear field models motivated by questions arising in statistical mechanics (spin models) and quantum field theory [10]. Perhaps it is necessary to explain what we mean by integrable . We call a non-linear field equation integrable if it arises as the flatness (zero-curvature) condition of a connection with values in a loop Lie algebra. A standard example for this is the non-linear \033-model, i.e., the harmonic map equation for maps of a surface into a symmetric space [14, 19, 18, 7].

[From the introduction] Over the past five years substantial progress has been made in the understanding of harmonic maps $f:M\rightarrow G/K$ of a compact Riemann surface M into a compact symmetric space $G=K$ [29, 30, 12, 14, 17, 22, 5, 7, 10, 31]. When $M$ is the Riemann sphere all such maps may b e obtained from holomorphic curves into some associated twistor space [11, 29, 14]. In contrast, if $M$ is a 2-torus, then every harmonic map $\phi:M\rightarrow G/K$ (satisfying specific nondegeneracy assumptions) may be obtained from a solution to a certain family of completely integrable finite dimensional systems of Hamiltonian ODE in Lax form on a loop algebra [5, 7, 17]. Such harmonic maps are called harmonic maps of finite type and, e.g., all harmonic 2-tori in $S^n$, $CP^n$ are accounted for in this way [4]. So far there is no comparable systematic theory when $M$ has higher genus. Nevertheless there are examples: the harmonic maps $f:M\rightarrow S^2$ arising as Gauss maps of Kapouleas' [18, 19] constant mean curvature surfaces of genus $g\ge 2$ and the minimal (and hence harmonic) maps $f:M \rightarrow S^3$ given by Lawson's [20] minimal surfaces of genus $g\ge 2$.

[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.