Higher Teichmüller理论中若干控制型问题的研究

Higher Teichmüller理论，作为Teichmüller理论的推广，由Hitchin于上世纪九十年代初引入，近些年来发展迅速，已成为热门研究领域之一。其研究对象涵盖多个数学分支，研究方法也多种多样，低维拓扑的、动力系统的、几何分析的等等。本项目主要关心其中与几何分析相关的问题。由non-Abelian Hodge理论，曲面群约化表示的模空间一一对应于多稳定Higgs丛的模空间，也一一对应于等变调和映照的模空间。Hitchin发现模空间中有自然的纤维化结构，并找到一个特殊的截面对应于Hitchin表示。另外，调和映照给出了模空间上一些来自几何的不变量，比如能量密度等等。本项目主要研究Hitchin表示关于这些几何量的控制型关系，包括Hitchin表示的负曲率性、关于能量密度与长度谱的极大性，以及能量密度沿C流的单调性等等。本项目将进一步加深人们对几何与拓扑之间深刻联系的理解。

Higher Teichmüller theory, as a generalization of Teichmüller theory, was introduced by Hitchin in the early 1990s. It has developed rapidly in recent years and has become one of the hot research areas. Its research objects cover multiple branches of mathematics and research methods are also diverse, such as low-dimensional topology, dynamic system, geometric analysis, and so on. This project focuses on issues related to geometric analysis. According to the non-Abelian Hodge theory, the moduli space of the reductive representations of the surface group is one-to-one corresponding to the moduli space of the polystable Higgs bundles, and also one-to-one corresponding to the module space of the equivariant harmonic maps. Hitchin found a natural fibration structure in the moduli space and found a special section corresponding to the Hitchin representations. In addition, the harmonic map gives some invariants from the geometry in the module space, such as energy density and so on. This project mainly researches the domination relationship of Hitchin representations about these geometric quantities, including the negative curvature property, the maximality of the energy density and the length spectrum of the Hitchin representations, and the monotonicity of the energy density along the C flow. This project will further deepen people's understanding of the deep connection between geometry and topology.