Wensheng Cao, Krishnendu Gongopadhyay
Let $H^n$ denote the complex hyperbolic space of dimension $n$. The group $U(n,1)$ acts as the group of isometries of $H^n$. In this paper we investigate when two isometries of the complex hyperbolic space commute. Along the way we determine the centralizers.
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We give a classification of the matrices in the unitary group U(1,1;H),where H is the division ring of the real quaternions. To this end, we consider the complex representation phi(P) for P in U(1,1;H). Next, we compute the characteristic polynomial f of the 4x4 complex matrix phi(P) and then study the sign of the resultant of f and its derivative f'.
In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_ξϕT=TR_ξϕ$, where $T=A$ or $T=S$ for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators, we give a complete classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting condition respectively.
We consider the Lie group PSL(2) (the group of orientation preserving isometries of the hyperbolic plane) and a left-invariant Riemannian metric on this group with two equal eigenvalues that correspond to space-like eigenvectors (with respect to the Killing form). For such metrics we find a parametrization of geodesics, the conjugate time, the cut time and the cut locus. The injectivity radius is computed. We show that the cut time and the cut locus in such Riemannian problem converge to the cut time and the cut locus in the corresponding sub-Riemannian problem as the third eigenvalue of the m…
In this paper we first introduce the full expression of the curvature tensor of a real hypersurface $M$ in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m{\ge}2$ from the equation of Gauss. Next we derive a new formula for the Ricci tensor of $M$ in $SU_{2,m}/S(U_2{\cdot}U_m)$. Finally we give a complete classification of Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$ with commuting Ricci tensor. Each can be described as a tube over a totally geodesic $SU_{2,m-1}/S(U_2{\cdot}U_{m-1})$ in $SU_{2,m}/S(U_2{\cdot}U_m)$ or a h…
We study relations between special elliptic isometries in the complex hyperbolic plane. Relations of lengths 2, 3, and 4 are fully classified. Some relative SU(2,1)-character varieties of the quadruply punctured sphere are described and applied to the study of length 5 relations.
Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces
双曲空间中的富克斯多面体是在等距富克斯群作用下不变的多面体曲面（即等距群留下全局不变的全测地曲面，它在其上作用余紧）。凸Fuchsian多面体上的诱导度量等距于亏格大于1的紧致曲面上具有正奇曲率的锥奇点的双曲度量。我们证明，这些指标实际上是实现了一个凸Fuchsian多面体（全球等距）。这扩展了A.D.亚历山德罗夫的一个著名定理。
In this note we prove that a complex hyperbolic triangle group of type (m,m,infinity), i.e. a group of isometries of the complex hyperbolic plane, generated by complex reflections in three complex geodesics meeting at angles Pi/m, Pi/m and 0, is not discrete if the product of the three generators is regular elliptic.
We prove a Milnor-Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodesic representations, thereby establishing a global rigidity theorem for totally geodesic representations.
Martin boundaries of diffusions and random walks on hyperbolic spaces
本文研究双曲空间上特定扩散过程及其在等距群上关联随机游走的Martin边界。
Parabolic Weingarten surfaces in hyperbolic space
在双曲空间$\h^3$中，一个不被一组抛物等距所改变的曲面叫做抛物曲面。本文研究了$\h^3$的抛物曲面满足形式为$aκ_1+bκ_2=c$或$aH+bK=c$的线性Weingarten关系，其中$a,b,c\in \r$和通常的$κ_i$是主曲率，$H$是平均曲率，$K$是de高斯曲率.我们对双曲空间中的所有抛物线性Weingarten曲面进行了分类。
Weil-Petersson isometries via the pants complex
本文推广了Masur和Wolf的一个定理，即给定一个双曲曲面S，S的Teichmuller空间中具有Weil-Petersson度量的每一个等距都是由S的映射类群中的一个元素导出的.我们的论证处理了以前未处理过的四孔球面、单孔环面和双孔环面的情形。
Horo-shrinkers in the hyperbolic space
本文研究双曲空间中由等距单参数群不变的 horo-shrinkers，刻画全测地平面为双曲平移不变情形，证明 grim reapers 的周期性，并完整分类球面旋转不变情形。
These are lectures on discrete groups of isometries of complex hyperbolic spaces, aimed to discuss interactions between the function theory on complex hyperbolic manifolds and the theory of discrete groups.
Hodge theory on hyperbolic manifolds of infinite volume
设$Y=Γ\backslash H^n$是双曲空间在离散凸余紧等距群作用下的商.我们描述了某些空间的$Γ$不变电流的球体上的无穷远$H^n$的极限集上的支持$Γ$。这些空间是有限维的。主要结果确定了$Y$的上同调与商的空间。我们解释了这个结果在什么意义上推广了经典的Hodge定理。我们得到类似的结果为上同调群$H^p(Γ,F)$，其中$F$是一个有限维表示的方向保持等距$H^n$的完全群。
Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and $\varepsilon$-spaces exhaust the class of $n$-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least ${1/2} n(n-1)+1$, for almost all values of $n$ [Patrangenaru V., Geom. Dedicata 102 (2003), 25-33]. We shall prove that the curvature tensor of these spaces satisfy several interesting algebraic properties. In particular, we will show that Egorov spaces are Ivanov-Petrova manifolds, curvature-Ricci commuting (indee…
Lie Ball as Tangent Space to Poincare Space
我们将整个切空间$TM$内禀地赋以一个自然度量的双曲流形$M$（截面曲率为-1），使得$M$的等距通过全纯延拓推广到$TM$的等距.测地线的切空间的像等价于双曲圆盘。在双曲空间的情形下，我们证明了$TM$与E的第四对称复域之间的等变同构。卡尔丹，也被称为谎言球。李球的闭包表现为切丛到双曲空间的一个时球紧化，其Bergmann度量给出切空间$TM$上的一个内禀自然kähler度量。等变映射有一个简单的几何解释。
We study the isometry group of a globally hyperbolic spatially compact Lorentz surface. Such a group acts on the circle, and we show that when the isometry group acts non properly, the subgroups of $\mathrm{Diff}(\mathbb{S}^1)$ obtained are semi conjugate to subgroups of finite covers of $\mathrm{PSL}(2,\mathbb{R})$ by using convergence groups. Under an assumption on the conformal boundary, we show that we have a conjugacy in $\mathrm{Homeo}(\mathbb{S}^1)$.
In this paper we review some author's results about Weingarten surfaces in Euclidean space $\r^3$ and hyperbolic space $\h^3$. We stress here in the search of examples of linear Weingarten surfaces that satisfy a certain geometric property. First, we consider Weingarten surfaces in $\r^3$ that are foliated by circles, proving that the surface is rotational, a Riemann example or a generalized cone. Next we classify rotational surfaces in $\r^3$ of hyperbolic type showing that there exist surfaces that are complete. Finally, we study linear Weingarten surfaces in $\h^3$ that are invariant by a g…
On differentiable compactifications of the hyperbolic space
实n维双曲空间的直向等距群G=SOo（n，1）。这个等距作用容许许多可微紧化成为闭球上的作用。我们证明了所有这样的紧化是拓扑共轭的，但不一定是可微共轭的。给出了实解析紧化和光滑紧化的分类。
We study the varieties of invariant totally geodesic submanifolds of isometries of the spherical, Euclidean and hyperbolic spaces in each finite dimension. We show that the dimensions of the connected components of these varieties determine the orbit type (or the z-class) of the isometry. For this purpose, we introduce the Segre symbol of an isometry, a discrete invariant encoding the structure of its normal form, which parametrizes z-classes. We then provide a description of the isomorphism type of the varieties of invariant subspaces in terms of the Segre symbol.
Isometries, rigidity, and universal covers
我们对所有闭的非球面黎曼流形M进行了分类，M的万有覆盖具有不离散等距群。一个例子应用是定理，任何这样的M字双曲基本群必须等距到一个负弯曲，局部对称流形。另一个应用是所有可收缩的黎曼流形的分类覆盖紧和（非紧的，完整的）有限体积流形。也有应用到霍普夫猜想，一个新的证明Kazhdan猜想（弗兰克尔定理）在复杂的几何等思想的证明来自李理论，同调理论的变换群，调和映射，和大规模的几何。非非球面的情况下，也给出了一个扩展。
Let $Γ$ be a discrete group of isometries acting on the complex hyperbolic $n$-space $\mathbb{H}^n_\mathbb{C}$. In this note, we prove that if $Γ$ is convex-cocompact, torsion-free, and the critical exponent $δ(Γ)$ is strictly lesser than $2$, then the complex manifold $\mathbb{H}^n_\mathbb{C}/Γ$ is Stein. We also discuss several related conjectures.
Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space
假设$f(s) = F'(s)$，其中$F$是双阱势。在$[-1,1]$上的$f$的Lipschitz常数满足一定条件的情况下，证明了半线性方程$Δu = f(u)$在双曲空间$\HH^n$上的任意有界整体解，只要它们在$\HH^n$的无穷边界上的渐近边值在$\HH^n$的等距群的一个上齐次子群下不变，就必约化为一元函数.我们还证明了这些一维解的存在性。
Geometric rigidity of quasi-isometries in horospherical products
本文证明了超几何空间的horospherical products的准等距在几何上是刚性的，即它们接近于乘积映射。研究了可解李群R ___ (N 1 x N 2)的准等距不变量和分类。
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