16:15 | Tobias Marxen (Oldenburg) | 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. |