Joint Seminar

24.05.2018, 16:15  –  University of Potsdam, Campus Golm, Building 9, Room 2.22
Seminar "Topics in Geometric Analysis"

Tobias Marxen and Volker Branding

Tobias Marxen
Ricci Flow on Warped Product Manifolds    

The Ricci flow has become famous via the solution of Thurston's
geometrization conjecture and the Poincare conjecture by Perelman in
2002-2003. We consider the Ricci flow on warped product manifolds R times T^n with flat fibres (T^n denotes the n-dimensional torus).
Assuming that the initial manifold is spatially asymptotic to a cylinder
(R times T^n, product metric) (and if it is complete with bounded
curvature) we show that the Ricci flow exists for all positive times
(longtime existence) and converges, as t goes to infinity, smoothly uniformly
to the cylinder (after pullback by a family of diffeomorphisms). During
the proof we derive and apply a new convergence result for the heat
equation with time-dependent metric.


Volker Branding  
On semi-biharmonic maps between Riemannian manifolds

We introduce an action functional for maps between Riemannian manifolds that interpolates between the actions for harmonic and biharmonic maps. Action functionals of this type are used in both elasticity theory and string theory. Critical points of this functional will be called semi biharmonic maps. We will report on the basic properties of semi-biharmonic maps and present several existence and non-existence results.
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