Cross Ratios, Anosov Representations and the Energy Functional on Teichmuller Space

相对阿诺索夫群的轨道计数

Orbital counting for relatively Anosov groups

该研究针对有有限临界指数线性泛函的相对阿诺索夫群，通过等分布结果得出轨道计数结果，关键依赖前期周期计数结果的等分布结论，推广了桑巴里诺在阿诺索夫群的早期工作。

Anosov表示的Falconer泛函的临界指数

The critical exponent of the Falconer functional for Anosov representations

本文研究具有Lipschitz极限集的射影Anosov表示的Falconer泛函的临界指数，在稠密性和不可约性假设下证明其等于极限集的Hausdorff维数，还构造相关反例并得到独立意义的结果。

Anosov接触度量、Dirichlet优化与熵

Anosov contact metrics, Dirichlet optimization and entropy

本文研究了Anosov接触度量中Dirichlet能量泛函的优化问题，揭示其与Reeb动力学和熵的关系，并解决了接触3流形的临界度量分类问题。

具有Lipschitz极限集的Anosov表示

Anosov representations with Lipschitz limit set

本文研究极限集为旗流形Lipschitz子流形的Anosov表示，引入不稳定Jacobian泛函证明其轨道增长率为整数，并应用于对称空间的刚性理论。

一般凯莱对应与高维 Teichmüller 空间

A general Cayley correspondence and higher Teichmüller spaces

本文引入复单李代数中的 magical sl2-三元组，利用其分解数据参数化紧黎曼曲面上相应实李群 Higgs 丛模空间的特殊分支，恢复已知分支并构造 E6、E7、E8、F4 四元数实形式的新分支，且与 Θ-正结构一一对应，猜想性地探测所有高维 Teichmüller 空间。

The Weil-Petersson Hessian of Length on Teichmuller Space

We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmuller space (due to Wolpert (1987)). We then estimate this Hessian from below in terms of local quantities and distance along the geodesic. The formula extends to proper arcs on punctured hyperbolic surfaces, and the estimate to laminations. Wolpert's result…

幽灵多边形、泊松括号与凸性

Ghost polygons, Poisson bracket and convexity

本文研究半单群中曲面群阿诺索夫表示模空间的长度与关联函数，利用幽灵代数的幽灵多边形和括号推导泊松括号公式，证明该函数集在泊松括号下稳定，应用于证明长度函数凸性及展示交换函数子代数。

Growth of quadratic forms under Anosov subgroups

Let $ρ:Γ\rightarrow PSL_d(\mathbb{K})$ be a Zariski dense Borel-Anosov representation, for $\mathbb{K}$ equal to $\mathbb{R}$ or $\mathbb{C}$. Let $o$ be a form of signature $(p,d-p)$ on $\mathbb{K}^d$ (where $0<p<d)$. Let $\mathsf{S}^o$ be the corresponding geodesic copy of the Riemannian symmetric space of $PSO(o)$, inside the Riemannian symmetric space of $PSL_d(\mathbb{K})$. For certain choices of $o$ and every $t$ large enough, we show exponential bounds for the number of $γ\inΓ$ for which the distance between $\mathsf{S}^o$ and $ργ\cdot\mathsf{S}^o$ is smaller than $t$. Under an extra as…

Comparison between Second Variation of Area and Second Variation of Energy of a Minimal Surface

The conformal parameterisation of a minimal surface is harmonic. Therefore, a minimal surface is a critical point of both the energy functional and the area functional. In this paper, we compare the Morse index of a minimal surface as a critical point of the area functional with its Morse index as a critical point of the energy functional. The difference between these indices is at most the real dimension of Teichmuller space. This comparison allows us to obtain surprisingly good upper bounds on the index of minimal surfaces of finite total curvature in Euclidean space of any dimension. We als…

Teichmüller空间上能量的非严格多重次调和性

Non-strict plurisubharmonicity of energy on Teichmüller space

该研究探讨Teichmüller空间上能量泛函的多重次调和性，分析其Levi形式的核与Higgs丛模空间$\mathbb{C}^*$作用、Hitchin纤维化临界点的关联，并证明特定亏格与维数下泛函的严格多重次调和性。

Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces

We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus $g>1$, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation for the action functional in terms of the geometry of different fiber…

Asymptotics for minimizers of a Donaldson functional and mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds

It has been shown in by Huang-Lucia-Tarantello [17] that, for given $\vert c \vert <1$, the moduli space of constant mean curvature (CMC) $c$-immersions of a closed orientable surface of genus $\mathfrak{g} \geq 2$ into a hyperbolic $3$-manifold can be parametrized by elements of the tangent bundle of the corresponding Teichmüller space. This is attained by showing the unique solvability of the Gauss-Codazzi equations governing (CMC) c-immersions. The corresponding unique solution is identified as the global minimum (and only critical point) of the Donaldson functional $D_t$ (introduced in [11…

Weil-Petersson类在$\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$中的最大圆盘

Maximal discs of Weil-Petersson class in $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$

研究Weil-Petersson类在3维反de Sitter空间中的最大圆盘，证明Mess映射在Weil-Petersson万有Teichmüller空间的余切 bundle 上的辛微分同胚，并展示反共轭能量函数作为Kähler势的作用。