Joint Seminar

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.