418 research outputs found
Pseudospherical surfaces with singularities
We study a generalization of constant Gauss curvature -1 surfaces in
Euclidean 3-space, based on Lorentzian harmonic maps, that we call
pseudospherical frontals. We analyze the singularities of these surfaces,
dividing them into those of characteristic and non-characteristic type. We give
methods for constructing all non-degenerate singularities of both types, as
well as many degenerate singularities. We also give a method for solving the
singular geometric Cauchy problem: construct a pseudospherical frontal
containing a given regular space curve as a non-degenerate singular curve. The
solution is unique for most curves, but for some curves there are infinitely
many solutions, and this is encoded in the curvature and torsion of the curve.Comment: 26 pages, 11 figures. Version 3: examples added (new Section 6).
Introduction section revise
Spherical Surfaces
We study surfaces of constant positive Gauss curvature in Euclidean 3-space
via the harmonicity of the Gauss map. Using the loop group representation, we
solve the regular and the singular geometric Cauchy problems for these
surfaces, and use these solutions to compute several new examples. We give the
criteria on the geometric Cauchy data for the generic singularities, as well as
for the cuspidal beaks and cuspidal butterfly singularities. We consider the
bifurcations of generic one parameter families of spherical fronts and provide
evidence that suggests that these are the cuspidal beaks, cuspidal butterfly
and one other singularity. We also give the loop group potentials for spherical
surfaces with finite order rotational symmetries and for surfaces with embedded
isolated singularities.Comment: 23 pages, 18 figures. Version 3: Typos correcte
Remarks on the boundary curve of a constant mean curvature topological disc
We discuss some consequences of the existence of the holomorphic quadratic
Hopf differential on a conformally immersed constant mean curvature topological
disc with analytic boundary. In particular, we derive a formula for the mean
curvature as a weighted average of the normal curvature of the boundary curve,
and a condition for the surface to be totally umbilic in terms of the normal
curvature.Comment: 6 pages, 1 figure. Version 2: comments and references adde
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 A-determined map-germs from the plane to
the plane can be represented by harmonic maps.Comment: 30 pages, 7 figures. Version 2: substantial revision compared with
version 1. The results are essentially the same, but some of the arguments
are improved or correcte
Singularities of spacelike constant mean curvature surfaces in Lorentz-Minkowski space
We study singularities of spacelike, constant (non-zero) mean curvature (CMC)
surfaces in the Lorentz-Minkowski 3-space . We show how to solve the
singular Bj\"orling problem for such surfaces, which is stated as follows:
given a real analytic null-curve , and a real analytic null vector
field parallel to the tangent field of , find a conformally
parameterized (generalized) CMC surface in which contains this curve
as a singular set and such that the partial derivatives and are
given by \frac{\dd f_0}{\dd x} and along the curve. Within the class of
generalized surfaces considered, the solution is unique and we give a formula
for the generalized Weierstrass data for this surface. This gives a framework
for studying the singularities of non-maximal CMC surfaces in . We use
this to find the Bj\"orling data -- and holomorphic potentials -- which
characterize cuspidal edge, swallowtail and cross cap singularities.Comment: 28 pages, 2 figures. Version 2: Figure 2 adde
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