退化双曲曲面与大亏格下的谱隙

高亏格多项对数的Enriquez连接

The Enriquez connection for higher genus polylogarithms

本文研究了高亏格多项对数的Enriquez连接在黎曼面退化时的变化，并展示了其与退化指针黎曼面族连接的关系，证明了该连接可将多项对数表示为变形参数和多重ζ值的幂级数。

Degenerating sequences of conformal classes and the conformal Steklov spectrum

Let $Σ$ be a compact surface with boundary. For a given conformal class $c$ on $Σ$ the functional $σ_k^*(Σ,c)$ is defined as the supremum of the $k-$th normalized Steklov eigenvalue over all metrics on $c$. We consider the behaviour of this functional on the moduli space of conformal classes on $Σ$. A precise formula for the limit of $σ_k^*(Σ,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander-Nadirashvili invariants of closed manifolds defined as $\inf_{c}σ_k^*(Σ,c)$, where the infimum is taken over all conformal …

大属随机双曲曲面的低音

Bass notes of random hyperbolic surfaces of large genus

该文是布尔巴基研讨会综述，先简述随机双曲曲面几何与谱，后探讨Hide - Magee等关于随机曲面近最优谱隙的结果。

模空间厚部上的谱间隙

Spectral gaps on thick part of moduli spaces

该文研究大亏格闭双曲曲面的谱间隙问题，证明当亏格趋于无穷时，闭黎曼曲面模空间任意厚部上第k个与第k-1个特征值之差的最大值趋近于1/4。

Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces

Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $λ_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto λ_k(M,g)$. We propose a computational method for finding the conformal spectrum $Λ^c_k(M,[g_0])$, which is defined by the eigenvalue optimization problem of maximizing $λ_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $[g_0]$ of fixed volume $textrm{vol}(M,g) = 1$. We also propose a computational method for the problem where $M$ is additionally allowed to vary over surfac…

具有许多尖点的随机双曲面的任意小谱隙

Arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps

本文研究了当亏格 $g$ 趋于无穷且穿孔数 $n(g)$ 增长慢于 $g$ 时，赋予 Weil-Petersson 度量的随机双曲面模空间中 Cheeger 常数与谱隙的渐近行为。