| 講演タイトル・アブストラクトおよびプログラム： |
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| Mason Pember (University of Bath, UK & Kobe University)
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| 10:00-11:00 "Lie Sphere Geometry" |
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We will give a basic introduction to Lie sphere geometry, the study of oriented spheres
and their oriented contact in 3-dimensional Riemannian and Lorentzian space forms.
We will see how this can be used to study the differential geometry of surfaces, and
more generally fronts, in these space forms. For simplicity's sake, we will mainly focus
on S^3.
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| David Brander (Technical University of Denmark & Kobe University)
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| "DPW-type Methods in Surface Theory" |
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| 11:30-12:30 Lecture (I) |
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| 14:30-15:30 Lecture (II) |
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| 16:00-17:00 Lecture (III)
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Over the last 20 years loop group methods have been used
to study many "classical" classes of surfaces in space forms; for
example, constant mean or Gauss curvature surfaces. These types of
surfaces are sometimes called "integrable surfaces", and the
relevant property is that the surfaces are characterized by the
existence of a special type of map (which is easy to describe)
into a subgroup of the group of loops in a complex semisimple
Lie group. There are at least three different basic ideas which
can be used to generate solutions: dressing, the AKS-type theory
for finite type solutions, and the "DPW"-type method.
In these talks I will describe the last mentioned of these. This
method has the property that it produces ALL solutions for the
problem at hand from simpler data, in analogue to the classical
Weierstrass representation for minimal surfaces. The challenge,
however, is that it is difficult to read geometric information
about the surface in the generalized Weierstrass data, due to a
loop group splitting involved in passing between the two
equivalent pieces of information. A basic idea which can be
exploited is that the Weierstrass data associated to a given
surface is not unique, and therefore one may try to choose this in
such a way as to make this issue easier. I will illustrate this
with two examples: one is the solution of the generalization of
Bjorling's problem to other surface classes; the other is the use
of special potentials to analyze singularities of surfaces.
These ideas can be implemented both for surfaces associated to
Riemannian harmonic maps, which have a generalized Weierstrass
representation in terms of holomorphic functions, and for those
associated to Lorentzian harmonic maps, which have a generalized
d'Alembert type solution.
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