No abstract uploaded!

An inequality relating the Euler characteristic, the signature and the <i>L</i>₂-norm of the curvature of the bundle of densities is proved for a four-dimensional compact Einstein-Weyl manifold. This generalises the Hitchin-Thorpe inequality for Einstein manifolds. The case where equality occurs is discussed and related to Hitchin's classification of Ricci-flat self-dual four-manifolds and to the recent work of Gauduchon on closed non-exact Einstein-Weyl geometries.

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence theorems and applications to isotopy problems in geometry and topology will be presented. The results are based on joint works of the author with his collaborators I. Medo\v{s}, K. Smoczyk, and M.-P. Tsui.

We show an example of a non-archimedean version of the existence part of the Calabi-Yau theorem in complex geometry. Precisely, we study totally degenerate abelian varieties and certain probability measures on their associated analytic spaces in the sense of Berkovich.

We nd a one-parameter family of coordinates f hgh2R which is a deformation of Penner's simplicial coordinate of the decorated Teichmuller space of an ideally triangulated punctured surface (S; T) of negative Euler characteristic. If h > 0, the decorated Teichmuller space in the h coordinate becomes an explicit convex polytope P(T) independent of h, and if h < 0, the decorated Teichmuller space becomes an explicit bounded convex polytope Ph(T) so that Ph(T)  Ph0 (T) if h < h0. As a consequence, Bowditch-Epstein and Penner's cell decomposition of the decorated Teichmuller space is reproduced.