Weil-Petersson Geometry, Renormalized Volume and Higher Teichmuller Theory

拓扑曲面是一个空间，它可以通过拉伸或弯曲来改变其形状，但不会撕裂或执行任何不连续的动作。人们可以通过考虑曲面可能具有的所有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，如果一个圆被允许改变它的形状，但仍然是一个椭圆，那么模空间将由两个数字描述，短轴的长度和长轴的长度，因此是二维的。拓扑表面（和更高维的物体）可以通过考虑其模空间的形状或几何来研究。一个这样的几何是韦尔-彼得森几何，它在数学和物理中起着重要作用。这个NSF奖项支持一个专注于模量空间的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重整化体积梯度流。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，流动任何形状以使其尽可能对称：在圆的类比中，使椭圆成为圆形。其中一个主要的方向是要表明，这种流动是均匀化的所有空间的某一类型。这项工作是在数学和物理学的交叉点，预计将导致两个领域之间的新的连接。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工作。该项目的重点是两个主要的研究领域，1）重整化体积及其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年中发展起来的。在1）PI计划使用重整化体积来研究双曲三流形的结构。双曲流形的重正化体积与其凸核体积密切相关，但具有良好的分析性质，例如是模空间上的光滑函数。在之前的工作中，PI和合作者引入了重整化体积的Weil-Petersson梯度流来研究三维流形上凸余紧双曲结构的变形空间的几何。特别是这项工作表明，当空间是acylinertium然后流线是威尔-彼得森拟测地线和重整化体积最小化的唯一结构，具有凸核边界全测地线。此外，一个surgered版本的流是一个均匀化流，流动的每一个点的唯一结构，具有凸核边界全测地线。一个主要的项目是表明，在边界不可压缩的情况下，流动限制在约束分解沿着其窗口和acylindrical片。在更高的Teichmuller理论中，PI和合作者考虑将经典Teichmuller理论的分析和度量结构扩展到更高秩李群的几何表示。在早期的工作中，PI和合作者引入了Weil-Petersson度量的自然扩展到更高的Teichmuller理论。最近，他们推广了这种构造，以根据相关李代数的简单根来定义扩展。PI将研究这些Weil-Petersson扩展并研究它们的几何结构。这已经导致了一些刚性的结果，有关简单的光谱长度和刘维体积的希钦representations.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.