Arithmetic groups and variational problems for Riemannian functionals
We propose a new approach to variational problems for Riemannian functionals.
This approach involves cohomology of arithmetic groups, surgery, hyperbolic
geometry and computability theory. We prove that for any compact manifold
M of dimension greater than four, diameter, regarded as a functional on
the space of isometry classes of Riemannian metrics on M such that the
absolute value of sectional curvature does not exceed one, has infinitely
many "very deep" local minima. (Joint work with Shmuel Weinberger.)
Spring 2001 meeting of the PNGS