Geometric rigidity for maps, foliations, and boundary structures of nonpositively curved spaces

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DMS 0306594- PI: Christopher ConnellABSTRACTGEOMETRIC RIGIDITY FOR MAPS, FOLIATIONS, AND BOUNDARY STRUCTURES OFNONPOSITIVELY CURVED SPACESThe PI plans to establish rigidity results for manifolds, foliations, andquasiconformal structures associated with nonpositively curved spaces.There are two main components of this project. The first is to broadenthe scope of Mostow rigidity by giving new characterizations of locallysymmetric spaces of noncompact type. Our proposed methods focus onexpanding the sharp relationship between volume, entropy and the degreeof maps initiated in its current form by work of Besson, Courtois andGallot. We then wish to extend weaker forms of this relationship to spacesadmitting nontrivial maps into nonpositively curved targets. The secondcomponent involves understanding the connection between quasiconformalstructures and special geometric measures which live on the boundary ofmost Hadamard spaces. The initial steps toward this goal also play animportant role in the first component. In the effort to establish theseresults, we aim to significantly enhance our understanding of theinteraction between the geometry, topology and geodesic dynamics of suchspaces.Based on a long standing principle, researchers have come to expect thatthe most efficient solutions to analytic problems are often achieved bythose objects which have the most symmetry. For instance, the modernversion 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 areamust have longer length than the circle. This was finally proved in 1841.In higher dimensions, the analogous result turns out to be true for thespheres of constant positive curvature. We can ask related questions aboutnonpositively curved spaces; these have the property that the sum of theangles of any small triangle does not exceed 180 degrees. We propose toshow that most nonpositively curved spaces with a sufficient amount ofsymmetry exhibit similar rigid behavior, but of a more intrinsic nature.Moreover, we expect many nonpositively curved spaces which are notsymmetric to also exhibit a sort of weak rigidity. From such results, wecan partly determine the rough shape of many asymmetrical spaces,regardless of curvature, and even draw some purely algebraic conclusionsrelated to their underlying structure. This naturally leads to dynamicalinformation arising, for instance, from the physical interpretation ofthese objects as phase spaces.