Weil-Petersson Geometry, Renormalized Volume and Higher Teichmuller Theory

拓扑表面是允许通过拉伸或弯曲改变其形状但不撕裂或执行任何不连续动作的空间。人们可以通过考虑曲面可能具有的各种形状的空间来研究曲面的性质。这个形状空间称为曲面的模空间。一个例子是，如果一个圆可以改变它的形状，但仍然是椭圆形，那么模空间将由两个数字来描述，短轴的长度和长轴的长度，因此是二维的。拓扑曲面（和高维物体）可以通过考虑其模空间的形状或几何来研究。其中一种几何是威尔-彼得森几何，它在数学和物理学中起着重要的作用。该NSF奖支持一个专注于模空间的Weil-Petersson几何的项目。在之前的工作中，PI和合作者在表面的模空间上引入了一种流动，称为Weil-Petersson重整体积梯度流。这种流动揭示了三维空间的模空间的许多结构。对于一大类三维空间来说，这是一种均匀化流，流动任何形状，使其尽可能对称：在圆的类比中，使椭圆变成圆。其中一个主要的方向是表明这种流动对于某种类型的所有空间都是均匀化的。这项工作处于数学和物理的交叉点，有望在这两个领域之间建立新的联系。该项目将支持一名研究生，并允许PI通过会议和研讨会传播其工作。该项目侧重于两个主要研究领域，1)重整体积及其Weil-Petersson梯度流和2)更高Teichmuller空间的Weil-Petersson几何。这两个领域相对较新，是在过去15年里发展起来的。在1)PI计划使用重整体积来研究双曲型三流形的结构。双曲流形的重整体积与其凸核体积密切相关，但具有较好的解析性质，如在模空间上是光滑函数。在之前的工作中，PI及其合作者引入了重归一化体积的Weil-Petersson梯度流来研究三维流形上凸紧双曲结构的变形空间几何。特别指出，当空间为非圆柱形时，流线是Weil-Petersson拟测地线，并且在具有凸核边界的独特结构处，重归一化体积最小。此外，激流是均匀流，每个点都流向具有凸核边界的独特结构，完全是测地线。一个主要的项目是证明在边界不可压缩情况下，流动限制沿其窗口和非圆柱形碎片的推测分解。在高等Teichmuller理论中，PI和合作者考虑将经典Teichmuller理论的解析和度量结构扩展到高阶李群的几何表示。在早期的工作中，PI和合作者引入了Weil-Petersson度规的自然扩展到更高的Teichmuller理论。最近，他们推广了基于相关李代数的单根来定义扩展的构造。PI将调查这些Weil-Petersson扩展并研究它们的几何结构。这已经导致了一些刚性结果，与简单的光谱长度和希钦表示的刘维尔体积有关。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

A topological surface is a space which is allowed to change its shape by stretching or bending but without tearing or performing any discontinuous actions. One can study the properties of the surface by considering the space of all shapes it can have. This space of shapes is called the moduli space of the surface. One example would be if a circle is allowed to change its shape but remain an ellipse, then the moduli space would be described by two numbers, the length of the short axis and the length of the long axis and therefore be two dimensional. Topological surfaces (and higher dimensional objects) can be studied by considering the shape or geometry of its moduli space. One such geometry is the Weil-Petersson geometry which plays an important role in mathematics and physics. This NSF award supports a project with a focus on the Weil-Petersson geometry of a moduli space. In prior work, the PI and collaborators introduced a flow on the moduli space of a surface, called the Weil-Petersson renormalized volume gradient flow. This flow reveals much of the structure of the moduli space of three-dimensional spaces. For a large class of three-dimensional spaces this is a uniformizing flow, flowing any shape to make it as symmetric as possible: in the circle analogy, making the ellipse become a round circle. One of the major directions is to show that this flow is uniformizing for all spaces of a certain type. This work is at the intersection of mathematics and physics and is expected to lead to new connections between the two fields. The project will support a graduate student and allow the PI to disseminate the work through conferences and seminars. The project focuses on two main areas of research, 1) renormalized volume and its Weil-Petersson gradient flow and 2) the Weil-Petersson geometry of higher Teichmuller spaces. These two areas are relatively new, having developed over the last fifteen years. In 1) the PI plans to use renormalized volume to study the structure of hyperbolic three-manifolds. The renormalized volume of a hyperbolic manifold is closely related to its convex core volume but has nicer analytic properties such as being a smooth function on moduli space. In prior work, the PI and collaborators introduced the Weil-Petersson gradient flow of renormalized volume to study the geometry of the deformation space of convex cocompact hyperbolic structures on a three dimensional manifold. In particular this work showed that when the space is acylindrical then the flowlines are Weil-Petersson quasigeodesics and that the renormalized volume is minimized at the unique structure which has convex core boundary totally geodesic. Furthermore, a surgered version of the flow is a uniformizing flow, flowing every point to the unique structure which has convex core boundary totally geodesic. A major project is to show that in the boundary incompressible case, the flow limits to the conjectured decomposition along its windows and acylindrical pieces. In higher Teichmuller theory the PI and collaborators consider extending the analytic and metric structure of classical Teichmuller theory to geometric representations into higher rank Lie groups. In earlier work, the PI and collaborators introduced a natural extension of the Weil-Petersson metric to higher Teichmuller theory. More recently they have generalized the construction to define extensions based on the simple roots of the associated Lie algebra. The PI will investigate these Weil-Petersson extensions and study their geometric structure. This has already led to a number of rigidity results, related to simple spectral length and the Liouville volume for Hitchin representations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.