Complexity and Variational Problems in Riemannian Geometry

Pennsylvania State Univ University Park, University Park PA

Investigators

Abstract

Abstract Award: DMS-0405954 Principal Investigator: Alexander Nabutovsky Firstly, we are going to study the length of a shortest closed geodesic and a shortest closed geodesic net on a compact Riemannian manifold. Secondly, we will also study possible topologies of closed geodesic nets on surfaces. Thirdly, we will try to resolve the following problem posed by M. Gromov: Is it true that all closed geodesic nets on a compact surface always form a dense subset? Fourthly, we will be interested in curvature-free bounds for the smallest area of a non-trivial minimal surface in a closed Riemannian manifold. In another direction we will be studying Morse landscapes of Riemannian functionals and functionals on spaces of submanifolds. Earlier we developed methods based on recursion theory that enable one to prove that Morse landscape of many interesting functionals on these spaces are extremely complicated. Now we will try to develop methods yielding quantitative information about graphs of these functionals and to extend our approach to a wider class of functionals that includes Sobolev norms of various curvatures normalized by a power of volume. When one studies the geometry of the round sphere one naturally notices the equator, meridians and other ``big" circles on it. These curves are examples of closed geodesics. Now let us deform the sphere so that it is not round. Remarkably, there still will be closed geodesics on this surface, and the shortest of them will not be too long. We would like to know how exactly its length is related to the area and the diameter of the deformed surface. We will also study similar questions for Riemannian manifolds that are higher dimensional generalizations of surfaces. On the other hand note that the round sphere has the nicer shape than all deformed spheres. It is less clear what is the nicest shape of the torus (the ``pretzel"). We are going to investigate what happens in higher dimensions where typically there are infinitely many competing ``nice" shapes and a lot depends on what exactly do we mean by ``nice".