JournalsowrVol. 5 / No. 3DOI 10.4171/owr/2008/34

Geometrie

  • Iskander A. Taimanov

    Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russian Federation
  • Burkhard Wilking

    Universität Münster, Germany
  • John Lott

    University of California, Berkeley, USA
Geometrie cover

You need to subscribe to download the article.

Abstract

The official program consisted of 20 lectures and therefore left plenty of space for fruitful informal collaboration for the 44 participants. One emphasis with 9 talks at this meeting was on geometric flows. Exciting progress could be reported on \begin{itemize} \item existence results for mean curvature as well as for the Ricci flow with singular initial data, \item stability and convergence results for the Ricci flow and new invariant curvature conditions, \item an extension of Perelman's work to open 3-manifolds with positive scalar curvature, and an improved singularity analysis. \end{itemize} Another important theme was metric and Finsler geometry with five talks covering the following topics \begin{itemize} \item a survey on currents in metric spaces, and existence of a curvature tensor in a measured sense on Alexandrov spaces which are noncollapsed limits, \item existence of path isometries to the Euclidean space for a wide range of metric spaces, \item closed geodesics on Finsler manifolds and volume entropy of Hilbert's Finsler metrics on convex sets. \end{itemize} The other six talks covered other aspects of geometry including \begin{itemize} \item dynamics of topological holomorphic maps on 2-sphere and geodesics of the Weil-Petersson metric, \item a cohomogeneity one example of a positively curved manifold and Einstein~/ Ricci-soliton solvmanifolds, \item a discrete analogue of conformal equivalence and isoparametric hypersurfaces. \end{itemize} Several connections between the different areas became apparent during the workshop. For example, the initial value problem for the Ricci flow with singular initial data is closely linked to smoothing problems occuring in Alexandrov geometry.