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

非正弯曲空间的地图、叶状结构和边界结构的几何刚性

• 批准号: 0420432
• 负责人: Christopher Connell
• 金额: --
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-09-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0420432&HistoricalAwards=false
• 关键词: Geometric; rigidity; maps; foliations; boundary

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.

PI计划建立与非正弯曲空间相关的流形、叶状和拟共形结构的刚性结果。这个项目有两个主要组成部分。首先，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。我们提出的方法侧重于扩展体积，熵和地图程度之间的尖锐关系，这些关系是由贝松，库尔图瓦和加洛的工作以其当前形式发起的。然后，我们希望将这种关系的较弱形式扩展到空间，允许非平凡映射到非正弯曲目标。第二个部分涉及到理解拟正形结构和存在于大多数哈达玛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起着重要作用。在努力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那些具有最大对称性的对象。例如，亚历山德里亚的帕普斯的一个猜想的现代版本断言圆是“刚性的”：任何其他以相同面积包围平面区域的曲线都必须比圆长。这最终在1841年得到了证明。在更高的维度中，类似的结果对于常正曲率的球体是正确的。我们可以问一些关于非正弯曲空间的相关问题；它们的性质是任何小三角形的内角之和都不超过180度。我们提出证明，大多数具有足够对称性的非正弯曲空间表现出类似的刚性行为，但具有更固有的性质。此外，我们期望许多非对称的非正弯曲空间也表现出一种弱刚性。从这些结果中，我们可以部分地确定许多不对称空间的大致形状，而不考虑曲率，甚至可以得出一些与它们的底层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，从这些对象作为相空间的物理解释。