GEOMETRIC RIGIDITY FOR MAPS, FOLIATIONS, AND BOUNDARY STRUCTURES OF NONPOSITIVELY CURVED SPACES
The PI plans to establish rigidity results for manifolds, foliations, and quasiconformal structures associated with nonpositively curved spaces. There are two main components of this project. The first is to broaden the scope of Mostow rigidity by giving new characterizations of locally symmetric spaces of noncompact type. Our proposed methods focus on expanding the sharp relationship between volume, entropy and the degree of maps initiated in its current form by work of Besson, Courtois and Gallot. We then wish to extend weaker forms of this relationship to spaces admitting nontrivial maps into nonpositively curved targets. The second component involves understanding the connection between quasiconformal structures and special geometric measures which live on the boundary of most Hadamard spaces. The initial steps toward this goal also play an important role in the first component. In the effort to establish these results, we aim to significantly enhance our understanding of the interaction between the geometry, topology and geodesic dynamics of such spaces.
Based on a long standing principle, researchers have come to expect that the most efficient solutions to analytic problems are often achieved by those objects which have the most symmetry. For instance, the modern version of a conjecture by Pappus of Alexandria asserts that the circle is "rigid:" any other curve enclosing a region of the plane with the same area must have longer length than the circle. This was finally proved in 1841. In higher dimensions, the analogous result turns out to be true for the spheres of constant positive curvature. We can ask related questions about nonpositively curved spaces; these have the property that the sum of the angles of any small triangle does not exceed 180 degrees. We propose to show that most nonpositively curved spaces with a sufficient amount of symmetry exhibit similar rigid behavior, but of a more intrinsic nature. Moreover, we expect many nonpositively curved spaces which are not symmetric to also exhibit a sort of weak rigidity. From such results, we can partly determine the rough shape of many asymmetrical spaces, regardless of curvature, and even draw some purely algebraic conclusions related to their underlying structure. This naturally leads to dynamical information arising, for instance, from the physical interpretation of these objects as phase spaces.
