# The 5th DIFFERENTIAL GEOMETRY DAY

## at LUND

## Wednesday the 13th of May 2009

### ABSTRACTS

**David Brander** (Lyngby)
**Special submanifolds as maps into loop groups**
It is well know that many special submanifolds are associated to integrable
PDE, such as soliton equations, through their integrability conditions.
On the other hand, the methods for producing solutions to these PDE, and
their representations as the flatness of connections, leads naturally to
maps into loop groups. Finally, this leads to natural and simple
characterizations of a large class of special submanifolds as certain maps
into loop group homogeneous spaces. This talk will discuss this, and
illustrate it with some examples.
**Eric Loubeau** (Brest)
**Biharmonic maps and tensor fields**
Biharmonic and harmonic maps derive from variational problems and ideas from
Physics enable us to construct associated tensor fields. These objects have
interesting properties but can also be a means of studying critical points.
**Andrew Swann** (Odense)
**Abelianised reduction**
In symplectic geometry, moment map reduction is defined at all levels.
However, for a non-Abelian group action, reduction at non-central
values involves use of axillary coadjoint orbits. The process of
implosion gives a way to turn such quotients into quotients by
Abelian groups that do not have this problem. The talk will survey
the symplectic picture and discuss how it might be adapted to complex
symplectic and hyper Kähler reductions.
**Jost-Hinrich Eschenburg** (Augsburg)
**Constant mean curvature surfaces and monodromy of Fuchsian equations**
We will discuss the classical theory (going back to H.A. Schwarz)
of certain Fuchsian equations i.e. the second order linear ODEs
$$ y'' + py' + qy = 0 $$
where $p,q$ are real rational functions with only regular singularities
lying on the real line. In particular we investigate in which cases the
monodromy group is (up to conjugation) contained in the isometry group of
either the 2-sphere or euclidean or hyperbolic plane. As an application we
study punctured spheres of constant mean curvature in euclidean 3-space
where all punctures lie on a common circle.
(Joint work with J. Dorfmeister)