The meaning of curvature: A distance geometric approach

This talk is essentially concerned with the shape molding forces of curvature and with the old question of how to detect the presence of curvature in manifolds and length spaces at various scales ranging from global to local to microlocal. As the old saying goes: Positive curvatures produce fat (geodesic) triangles and negative curvatures slim triangles. Some basic ideas from metric (length-)space geometry and from the geometric analysis of the Laplacian will be surveyed and applied. Via the Laplacian the curvature tensor is in control of a variety of very natural phenomena ranging from heat diffusion to volume growth. We are e.g. interested in obtaining precise bounds for mean exit times for Brownian motions and for isoperimetric inequalities. One way to obtain such bounds are via curvature controlled comparison with corresponding values in constant curvature spaces and in other tailor-made so-called warped product spaces.