Oberwolfach Reports

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Volume 4, Issue 3, 2007, pp. 1865–1912
DOI: 10.4171/OWR/2007/32

Published online: 2008-06-30

Differentialgeometrie im Grossen

Olivier Biquard[1], Bruce Kleiner[2], Bernhard Leeb[3] and Gang Tian[4]

(1) Université Paris VI, France
(2) Courant Institute of Mathematical Sciences, New York, United States
(3) Universität München, Germany
(4) Princeton University, USA

The meeting continued the biannual conference series {\em Differentialgeometrie im Gro{\ss}en} at the MFO which was established in the 60's by Klingenberg and Chern. Traditionally, the conference series covers a wide scope of different aspects of global differential geometry and its connections with topology, geometric group theory and geometric analysis. The Riemannian aspect is emphasized, but the interactions with the developments in complex geometry, symplectic/contact geometry/topology and mathematical physics play also an important role. Within this spectrum each particular conference gives special attention to two or three topics of particular current relevance. The scientific program consisted of 22 (almost) one hour talks leaving ample time for informal discussions. This time, a main focus of the workshop were {\em geometric evolution equations}. 6 talks discussed the Ricci flow and its applications to the geometrization of 3-manifolds, the Ricci flow in arbitrary dimension and applications to Riemannian manifolds of positive curvature, the K\"ahler-Ricci flow and the (inverse) mean curvature flow. A second focus was the {\em geometry of singular spaces} (5 talks), that is, metric spaces with sectional curvature bounds (in the sense of Aleksandrov), Gromov-hyperbolic spaces and Carnot spaces with connections to geometric group theory. One of the talks discussed the theory of collapse with lower curvature bound which is another ingredient (independent of Ricci flow) in the argument for geometrization in dimension $3$. Other talks covered results about {\em geometric structures on manifolds} (hyperbolic geometry and representation varieties), from {\em geometric analysis} (Dirac operators, metrics of positive scalar curvature), {\em symplectic and contact geometry} (open book decompositions, confoliations, construction of special Lagrangian submanifolds with isolated conical singularities), and {\em complex geometry} (extremal metrics on K\"ahler manifolds, K\"ahler-Ricci flow). There were 52 participants from 8 countries, more specifically, 23 participants from Germany, 12 from France, 6 from other European countries, and 11 from North-America. 6\ were young researchers (less than 10 years after diploma or B.A.), both on doctoral and postdoctoral level. The organizers would like to thank the institute staff for their great hospitality and support before and during the conference. The financial support of the European Union (in particular for young participants) is gratefully acknowledged.

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Biquard Olivier, Kleiner Bruce, Leeb Bernhard, Tian Gang: Differentialgeometrie im Grossen. Oberwolfach Rep. 4 (2007), 1865-1912. doi: 10.4171/OWR/2007/32