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

DMS 0306594- PI：Christopher Connell非正曲空间映射、叶理和边界结构的几何刚性PI计划建立与非正曲空间相关的流形、叶理和拟共形结构的刚性结果。 第一，通过给出非紧型局部对称空间的新刻画，拓宽了Mostow刚性的范围。我们所提出的方法集中于扩展体积，熵和度的映射之间的尖锐的关系，在其目前的形式开始由贝松，Courtois和Gallot的工作。然后，我们希望将这种关系的较弱形式扩展到允许非平凡映射到非正弯曲目标的空间。第二部分涉及到理解准共形结构和生活在大多数Hadamard空间边界上的特殊几何测度之间的联系。实现这一目标的最初步骤在第一个组成部分中也发挥着重要作用。在建立这些结果的过程中，我们的目标是显著提高我们对这些空间的几何、拓扑和测地线动力学之间相互作用的理解。基于一个长期存在的原则，研究人员已经开始期望，分析问题的最有效的解决方案往往是由那些具有最大对称性的对象实现的。 例如，亚历山大的帕普斯（Pappus of Alexandria）的一个猜想的现代版本断言，圆是“刚性的“：任何其他包围平面上具有相同面积的区域的曲线都必须比圆长。这一点最终在1841年得到了证明。在更高的维度上，类似的结果对常正曲率的球面也是成立的。我们可以问一些关于非正曲空间的相关问题，这些空间具有任何小三角形的角之和不超过180度的性质。我们证明了大多数具有足够对称性的非正曲空间都表现出类似的刚性行为，但具有更内在的性质，而且，我们期望许多非对称的非正曲空间也表现出某种弱刚性。从这些结果中，我们可以部分地确定许多非对称空间的粗略形状，而不管曲率如何，甚至可以得出一些与其基本结构有关的纯代数结论。这自然会导致动态信息的产生，例如，从物理解释这些对象作为相空间。

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.