Mathematisches Institut

Georg-August-Universität Göttingen

[zurück zum Institut] [Fakultät], [Universität]

Updates: letztes: 05.09.97 jp   nächstes: nach Bedarf      [verantwortlich]
Andrejewski Vorlesung

Degeneration of Riemannian Spaces under Curvature Bounds

Jeff Cheeger, Courant Institute
von Montag, 20.10.97 bis Freitag, 24.10.97
Die Vorträge finden dienstags, mittwochs und donnerstags jeweils um 17,15 Uhr
im Auditorium Maximum des Mathematischen Institutes statt.
  1. Vortrag (Di) : Collapse of manifolds of bounded sectional curvature

  2. Vortrag (Mi) : Generalized torus actions and applications

  3. Vortrag (Do) : The small scale structure of spaces with Ricci curvature bounded below
Abstract
On an infinitesimal scale, the geometry of any riemannian manifold is Euclidean. At a given point, the curvature of the manifold measures the asymptotic rate of approach to Euclidean geometry as the scale parameter goes to zero. The following (closely related) questions are of basic importance:
``What are truly bad examples of Riemannian manifolds whose curvature satisfies definite bounds?''
``What are kinds of spaces arise as possible singular limits of families of such examples?''
Our lecture will provide answers to these questions for the case of bounded sectional curvature and for the case in which the Ricci curvature is bounded below.
As explained in our first lecture, the bad examples of Riemannian manifolds of bounded sectional curvature, say |K|\le 1, are precisely those which are very collapsed. Roughly, this means that the apparant dimension is less than the true dimension. For example, the surface of a very thin wire has apparant dimension 1, although the true dimension is 2. Remarkably, it turns out that all (sufficiently collapsed) such spaces must exhibit a high degree of (generalized) circular symmetry; in the present instance, this symmetry is given by rotation about the axis of the wire.
In our second lecture, we will give various examples of collapsed manifolds of bounded curvature, as well as applications of the generalized circular symmetry structure.
In our third lecture, we will discuss the more general case in which the Ricci curvature satisfies a definite lower bound. Although there are many new possibilities for how degenerations can occur, a meaningful theory is still possible. In particular we will make precise the notion of limiting generate case and describe the infinitesimal structure of such limits.