arXiv subjects
Publications and source records attributed to Wayne Rossman.
We give a Weierstrass type representation for semi-discrete minimal surfaces in Euclidean 3-space. We then give explicit parametrizations of various smooth, semi-discrete and fully-discrete catenoids, determined from either variational or integrable systems principles. Finally, we state the shared properties that those various catenoids have.
We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in hyperbolic 3-space, and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. We also examine discrete linear Weingarten surfaces of Bryant type in hyperbolic 3-space, and consider an example of a discrete flat surface related to the Airy equation that exhibits swallowtail singularities and a Stokes phenomenon.
CMC-1 trinoids (i.e. constant mean curvature one immersed surface with three regular embedded ends) in hyperbolic 3-space H^3 are irreducible generically, and the irreducible ones have been classified. However, the reducible case has not yet been fully treated, so in this paper we give an explicit description of CMC-1 trinoids in H^3 that includes the reducible case.
The quantum cohomology of CP^1 provides a distinguished solution of the third Painleve equation. S. Cecotti and C. Vafa discovered this from a physical viewpoint. We derive it from a differential geometric viewpoint, using the theory of harmonic maps and in particular the generalized Weierstrass representation (DPW representation) for spacelike surfaces of constant mean curvature in Minkowski space. The nontrivial aspects are the characterization of the solution, and its global behaviour. From our point of view, the latter property says that the extended harmonic map remains entirely within a single Iwasawa orbit of the appropriate loop group.
This survey article is about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure of the smooth surfaces using conserved quantities, which can be converted into a discrete theory in a natural way.
The quantum cohomology of CP^1 is generated by some potential (Frobenius manifold) that also has an interpretation as a potential of some harmonic map. Actually, the potential induces harmonic maps into three different symmetric spaces and each of these harmonic maps induces an immersion of an integrable surface. The final goal of this paper is to investigate if any of the associated immersions closes around the singularity at z=0 (none does). The crucial tool for deciding this question is the main technical result of the paper, which states that the dressing isotropy of the CP^1 potential is trivial.
We propose a Lie geometric point of view on flat fronts in hyperbolic space as special omega-surfaces and discuss the Lie geometric deformation of flat fronts.
We make observations about constant mean curvature surfaces in Euclidean 3-space and their dual surfaces, and the resulting pairs of surfaces in hyperbolic 3-space under the Lawson correspondence.
We show existence of constant mean curvature 1 surfaces in both hyperbolic 3-space and de Sitter 3-space with two complete embedded ends and any positive genus up to genus twenty. We also find another such family of surfaces in de Sitter 3-space, but with a different non-embedded end behavior.
We consider the question of existence of embedded doubly periodic minimal surfaces in Euclidean 3-space with Scherk-type ends, surfaces that topologically are Scherk's doubly periodic surface with handles added in various ways. We extend the existence results of H. Karcher and F. Wei to more cases, and we find other cases where existence does not hold.
This is an elementary introduction to a method for studying harmonic maps into symmetric spaces, and in particular for studying constant mean curvature (CMC) surfaces, that was developed by J. Dorfmeister, F. Pedit and H. Wu. There already exist a number of other introductions to this method, but all of them require a higher degree of mathematical sophistication from the reader than is needed here. The authors' goal was to create an exposition that would be readily accessible to a beginning graduate student, and even to a highly motivated undergraduate student. Constant mean curvature surfaces in Euclidean 3-space, and also spherical 3-space and hyperbolic 3-space, are described, along with the Lax pair equations that determine their frames. The simplest examples, including Delaunay surfaces and Smyth surfaces, are described in detail.
Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean 3-space given by the third and fourth authors in a previous paper. In this paper, we discuss singularities and several global properties of maximal surfaces, and give explicit examples of such surfaces of arbitrary genus. When the genus is one, our examples are embedded outside a compact set. Moreover, we deform such examples to CMC-1 faces (mean curvature one surfaces with admissible singularities in de Sitter 3-space) and obtain "cousins" of those maximal surfaces. Cone-like singular points on maximal surfaces are very important, although they are not stable under perturbations of maximal surfaces. It is interesting to ask if cone-like singular points can appear on a maximal surface having other kinds of singularities. Until now, no such examples were known. We also construct a family of complete maximal surfaces with two complete ends and with both cone-like singular points and cuspidal edges.
We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space $\real^{2,1}$. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group $SU_2$ with $SU_{1,1}$. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form $SU_{1,1}$, and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in $\real^{2,1}$. In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant.
The first author studied spacelike constant mean curvature one (CMC-1) surfaces in de Sitter 3-space when the surfaces have no singularities except within some compact subset and are of finite total curvature on the complement of this compact subset. However, there are many CMC-1 surfaces whose singular sets are not compact. In fact, such examples have already appeared in the construction of trinoids given by Lee and the last author via hypergeometric functions. In this paper, we improve the Osserman-type inequality given by the first author. Moreover, we shall develop a fundamental framework that allows the singular set to be non-compact, and then will use it to investigate the global behavior of CMC-1 surfaces.
In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space H^3. Galvez, Martinez and Milan showed that when the singular set does not accumulate at an end, the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that the asymptotic order (called "pitch" p) of the end determines the limiting shape, even when the singular set does accumulate at the end. If the singular set is bounded away from the end, we have -1<p<=0. If the singular set accumulates at the end, the pitch p is a positive rational number not equal to 1. Choosing appropriate positive integers n and m so that p=n/m, suitable slices of the end by horospheres are asymptotic to d-coverings (d-times wrapped coverings) of epicycloids or d-coverings of hypocycloids with 2n_0 cusps and whose normal directions have winding number m_0, where n=n_0d, m=m_0d (n_0, m_0 are integers or half-integers) and d is the greatest common divisor of m-n and m+n. Furthermore, it is known that the caustics of flat surfaces are also flat. So, as an application, we give a useful explicit formula for the pitch of ends of caustics of complete flat fronts.
We show that the index of a constant mean curvature 1 surface in hyperbolic 3-space is completely determined by the compact Riemann surface and secondary Gauss map that represent it in Bryant's Weierstrass representation. We give three applications of this observation. Firstly, it allows us to explicitly compute the index of the catenoid cousins and some other examples. Secondly, it allows us to be able to apply a method similar to that of Choe (using Killing vector fields on minimal surfaces in Euclidean 3-space) to our case as well, resulting in lower bounds of index for other examples. And thirdly, it allows us to give a more direct proof of the result by do Carmo and Silveira that if a constant mean curvature 1 surface in hyperbolic 3-space has finite total curvature, then it has finite index. Finally, we show that for any constant mean curvature 1 surface in hyperbolic 3-space that has been constructed via a correspondence to a minimal surface in Euclidean 3-space, we can take advantage of this correspondence to find a lower bound for its index.
Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete m-dimensional manifolds with singularities on sets of codimension at least 2. With certain restrictions on the metric, we establish that the spectrum is discrete and satisfies Weyl's asymptotic formula.
We show numerically that any of the constant mean curvature tori first found by Wente must have index at least eight.