Title:

Ricci solitons and geometric analysis

This thesis studies Ricci solitons of cohomogeneity one and uniform Poincaré inequalities for differentials on Riemann surfaces. In the two summands case, which assumes that the isotropy representation of the principal orbit consists of two inequivalent Adinvariant irreducible summands, complete steady and expanding Ricci solitons have been detected numerically by BuzanoDancerGallaugherWang. This work provides a rigorous construction thereof. A Lyapunov function is introduced to prove that the Ricci soliton metrics lie in a bounded region of an associated phase space. This also gives an alternative construction of noncompact Einstein metrics of nonpositive scalar curvature due to Böhm. It is explained how the asymptotics of the Ricci flat trajectories induce Böhm's Einstein metrics on spheres and other low dimensional spaces. A numerical study suggests that all other Einstein metrics of positive scalar curvature which are induced by the generalised Hopf fibrations occur in an entirely nonlinear regime of the Einstein equations. Extending the theory of cohomogeneity one steady and expanding Ricci solitons, an estimate which allows to prescribe the growth rate of the soliton potential at any given time is shown. As an application, continuous families of Ricci solitons on complex line bundles over products of Fano Kähler Einstein manifolds are constructed. This generalises work of Appleton and Stolarski. The method also applies to the LüPagePope setup and allows to cover an optimal parameter range in the two summands case. The Ricci soliton equation on manifolds foliated by torus bundles over products of Fano Kähler Einstein manifolds is discussed. A rigidity theorem is obtained and a preserved curvature condition is discovered. The cohomogeneity one initial value problem is solved for mquasiEinstein metrics and complete metrics are described. L^{p}Poincaré inequalities for kdifferentials on closed Riemann surfaces are shown. The estimates are uniform in the sense that the Poincaré constant only depends on p &GE;1, k ≥ 2 and the genus γ ≥ 2 of the surface but not on its complex structure. Examples show that the analogous estimate for 1differentials cannot be uniform. This part is based on joint work with Melanie Rupflin.
