Traces in Complex Hyperbolic Triangle Groups

双曲平面中三角形平铺图上的Steklov问题

The Steklov problem on triangle-tiling graphs in the hyperbolic plane

本文引入一个与双曲平面大致等距的图，研究其带边界子图的Steklov特征值，证明第k个特征值随图规模增大按比例趋于0，并通过双曲平面有界域离散化过程完成证明。

On the type of triangle groups

We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups.

Hyperbolic triangles without embedded eigenvalues

We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degenerating family of triangles. In particular, we use a careful analysis of spectral projections near the crossings of these eigenvalue branches with the eigenvalue branches of a model operator.

双曲平面中的拿破仑构造

Napoleonic Constructions in the Hyperbolic Plane

本文研究双曲平面中的拿破仑构造问题，证明双曲平面中仅等边三角形具备拿破仑性质，且任意三角形的迭代拿破仑化会逐渐趋近等边并收敛至单点，揭示双曲情形与欧氏、球面情形的显著差异。

Ideal triangle groups, dented tori, and numerical analysis

We prove the Goldman-Parker Conjecture: A complex hyperbolic ideal triangle group is directly embedded in PU(2,1) if and only if the product of its three standard generators is not elliptic. We also prove that such a group is indiscrete if the product of its three standard generators is elliptic. A novel feature of this paper is that it uses a rigorous computer assisted proof to deal with difficult geometric estimates.

作为双曲重心的类似中线

Symmedians as Hyperbolic Barycenters

该研究提出类似中线的新型动态坐标化方法，将其推广为理想双曲多边形的双曲重心并证明其保留相似性质，同时开启相关模空间及额外最优性的研究。

The Cheeger constants of random Belyi surfaces

Brooks and Makover developed a combinatorial model of random hyperbolic surfaces by gluing certain hyperbolic ideal triangles. In this paper we show that for any $ε>0$, as the number of ideal triangles goes to infinity, a generic hyperbolic surface in Brooks-Makover's model has Cheeger constant less than $\frac{3}{2π}+ε$.

The trace of the heat kernel on a compact hyperbolic 3-orbifold

The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold $H^3/\Ga$ are evaluated in the case in which the discrete group $\Ga$ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expansion, but the Weyl term, which remains unchanged. Some physical consequences are briefly discussed in the examples.

Redundancy of triangle groups in spherical CR representations

Falbel, Koseleff and Rouillier computed a large number of boundary unipotent CR representations of fundamental groups of non compact three-manifolds. Those representations are not always discrete. By experimentally computing their limit set, one can determine that those with fractal limit sets are discrete. Many of those discrete representations can be related to (3,3,n) complex hyperbolic triangle groups. By exact computations, we verify the existence of those triangle representations, which have boundary unipotent holonomy. We also show that many representations are redundant: for n fixed, a…

退化双曲曲面上自旋Dirac算子的Selberg迹公式

The Selberg trace formula for spin Dirac operators on degenerating hyperbolic surfaces

本文研究双曲曲面族上自旋Dirac算子的谱，通过迹公式推导Weyl定律及Selberg zeta函数收敛性，核心是自旋结构决定的类函数。

Wave 0-Trace and length spectrum on convex co-compact hyperbolic manifolds

For quotients of the $n+1$-dimensional hyperbolic space by a convex co-compact group $Γ$, we obtain a formula relating the renormalized trace of the wave operator with the resonances of the Laplacian and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. By writing this trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of $Γ$ is greater than $n/2$.

第四几何 I：超越欧几里得、双曲和椭圆几何的差角几何

The Fourth Geometry I: Difference--Angle Geometry Beyond Euclid, Hyperbolic, and Elliptic

本文提出了一种基于差角的新几何，通过定义差角与相关公理，揭示了角的构造性定义，并展示了其在三角形、范数等几何结构中的独特性质。

Trace formula for the magnetic Laplacian on a compact hyperbolic surface

We compute the trace formula for the magnetic Laplacian on a compact hyperbolic surface of constant curvature with constant magnetic field for energies above the Mane critical level of the corresponding magnetic geodesic flow. We discuss the asymptotic behavior of the coefficients of the trace formula when the energy approaches the Mane critical level.

Dynamics of Asymptotically Hyperbolic Manifolds

We prove a dynamical wave trace formula for asymptotically hyperbolic (n+1) dimensional manifolds with negative (but not necessarily constant) sectional curvatures which equates the renormalized wave trace to the lengths of closed geodesics. A corollary of this dynamical trace formula is a dynamical resonance-wave trace formula for compact perturbations of convex co-compact hyperbolic manifolds which we use to prove a growth estimate for the length spectrum counting function. We next define a dynamical zeta function and prove its analyticity in a half plane. In our main result, we produce a pr…