David Brander

Holomorphic representation of constant mean curvature surfaces in Minkowski space: Consequences of non-compactness in loop group methods

Abstract We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space R 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 R 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.

Loop group decompositions in almost split real forms and applications to soliton theory and geometry

Abstract We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action–by the whole subgroup of loops which extend holomorphically to the exterior disc–on the U -hierarchy of the ZS-AKNS systems, on curved flats and on various other integrable systems, is global for compact cases. It also implies a global infinite dimensional Weierstrass-type representation for Lorentzian harmonic maps ( 1 + 1 wave maps) from surfaces into compact symmetric spaces. An “Iwasawa-type” decomposition of the same type of real form, with respect to a fixed point subgroup of an involution of the second kind, is also proved, and an application given.

Robotic Hot-Blade Cutting

This paper presents a novel method for cost-effective, robotic production of double curved formwork in Expanded Polystyrene (EPS) for in situ and prefabricated concrete construction. A rationalization and segmentation procedure is developed, which allows for the transliteration of double curved NURBS surfaces to Euler elastica surface segments, while respecting various constraints of production. An 18 axis, tri-robot system approximates double curved NURBS surfaces by means of an elastically deformed and heated blade, mounted on the flanges of two manipulators. Re-orienting or translating either end of the blade dynamically deforms the blade’s curvature. The blade follows the contours of the rationalized surface by continuous change in position and orientation of the end-effectors. The concept’s potential is studied by a pilot production of a full-scale demonstrator panel assembly.

Approximation by planar elastic curves

We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good initial guess for the approximating curve. A gradient-driven optimization is then used to find the approximating elastic curve.

CONSTANT GAUSSIAN CURVATURE SURFACES IN THE 3-SPHERE VIA LOOP GROUPS

In this paper we study constant positive Gauss curvature K surfaces in the 3-sphere S 3 with 0 < K < 1, as well as constant negative curvature surfaces. We show that the so-called normal Gauss map for a surface in S 3 with Gauss curvature K < 1 is Lorentz harmonic with respect to the metric induced by the second fundamental form if and only if K is constant. We give a uniform loop group formulation for all such surfaces with K⁄ 0, and use the generalized d’Alembert method to construct examples. This representation gives a natural correspondence between such surfaces with K <0 and those with 0 < K < 1.

Timelike Constant Mean Curvature Surfaces with Singularities

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behavior of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized surfaces are cuspidal edges, swallowtails, and cuspidal cross caps.

Monge surfaces and planar geodesic foliations

A Monge surface is a surface obtained by sweeping a generating plane curve along a trajectory that is orthogonal to the moving plane containing the curve. Locally, they are characterized as being foliated by a family of planar geodesic lines of curvature. We call surfaces with the latter property PGF surfaces, and investigate the global properties of these two naturally defined objects. The only compact orientable PGF surfaces are tori; these are globally Monge surfaces, and they have a simple characterization in terms of the directrix. We show how to produce many examples of Monge tori and Klein bottles, as well as tori that do not have a closed directrix.

Families of spherical surfaces and harmonic maps

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relationship between the two types of maps and their singularities. Finally, we determine which finitely

Designing interactively with elastic splines

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