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)