Pinching surface groups in complex hyperbolic plane

Pinching, Pontrjagin classes, and negatively curved vector bundles

We prove several finiteness results for the class $M_{a,b,G,n}$ of $n$-manifolds that have fundamental groups isomorphic to $G$ and that can be given complete Riemannian metrics of sectional curvatures within $[a,b]$ where $a\le b<0$. In particular, if $M$ is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class $M_{a,b,π_1(M), n}$ are total spaces of vector bundles over $M$. Furthermore, given a word-hyperbolic group $G$ and an integer $n$ there exists a positive $ε=ε(n,G)$ such that the tangent bundle of any manifold in the class $M_{…

Manifolds with many hyperbolic planes

We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogenous.

Isometric Immersions of the Hyperbolic Plane into the Hyperbolic Space

In this paper, we parametrize the space of isometric immersions of the hyperbolic plane into the hyperbolic 3-space in terms of null-causal curves in the space of oriented geodesics. Moreover, we characterize "ideal cones" (i.e., cones whose vertices are on the ideal boundary) by behavior of their mean curvature.

从双曲平面的乘积到双曲空间的调和映射

Harmonic maps from the product of the hyperbolic planes to the hyperbolic space

研究从双曲平面乘积到双曲空间的调和映射，证明了当边界数据给定在特异边界（无穷远圆周之积）上时，渐近 Dirichlet 问题解的存在性。

Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.

Polar actions on the complex hyperbolic plane

We classify the polar actions on the complex hyperbolic plane up to orbit equivalence. Apart from the trivial and transitive polar actions, there are five polar actions of cohomogeneity one and four polar actions of cohomogeneity two.

Strongly 2-Hopf hypersurfaces in complex projective and hyperbolic planes

We give a geometric characterization of certain hypersurfaces of cohomogeneity one in the complex projective and hyperbolic planes. We also obtain some partial classifications of austere hypersurfaces and of Levi-flat hypersurfaces with constant mean curvature in these spaces.

Non-properly Embedded H-Planes in Hyperbolic 3-Space

For any H in [0,1), we construct complete, non-proper, stable, simply-connected surfaces with constant mean curvature H embedded in hyperbolic 3-space.

通过Bäcklund变换构造到双曲平面的调和映射的新例子

New examples of harmonic maps to the hyperbolic plane via Bäcklund transformation

本文利用连接sinh-Gordon与sine-Gordon方程的Bäcklund变换，发展相关机制，构造出从复平面子集到双曲平面的新调和映射。

A dual rigidity of the sphere and the hyperbolic plane

There are several well-known characterizations of the sphere as a regular surface in the Euclidean space. By means of a purely synthetic technique, we get a rigidity result for the sphere without any curvature conditions, nor completeness or compactness. As well as a dual result for the hyperbolic plane, the spacelike sphere in the Minkowski space.

Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting structure Jacobi operators

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.

On the asymptotic geometry of the hyperbolic plane

Asymptotic subcone of an unbounded metric space is another metric space, capturing the structure of the original space at infinity. In this paper we define a functional metric space S which is an asymptotic subcone of the hyperbolic plane. This space is a real tree branching at every its point. Moreover, it is a homogeneous metric space such that any real tree with countably many vertices can be isometrically embedded into it. This implies that every such tree is also an asymptotic subcone of the hyperbolic plane.

Embedded loops in the hyperbolic plane with prescribed, almost constant curvature

Given a constant $k>1$ and a real valued function $K$ on the hyperbolic plane $\mathbb H^2$, we study the problem of finding, for any $ε\approx 0$, a closed and embedded curve $u^ε$ in $\mathbb H^2$ having geodesic curvature $k+εK(u^ε)$ at each point.

Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes

We find the first examples of real hypersurfaces with two nonconstant principal curvatures in complex projective and hyperbolic planes, and we classify them. It turns out that each such hypersurface is foliated by equidistant Lagrangian flat surfaces with parallel mean curvature or, equivalently, by principal orbits of a cohomogeneity two polar action.

The hanging chain problem in the sphere and in the hyperbolic plane

In this paper, the notion of the catenary curve in the sphere and in the hyperbolic plane is introduced. In both spaces, a catenary is defined as the shape of a hanging chain when its potential energy is determined by the distance to a given geodesic of the space. Several characterizations of the catenary are established in terms of the curvature of the curve and of the angle that its unit normal makes with a vector field of the ambient space. Furthermore, in the hyperbolic plane, we extend the concept of catenary substituting the reference geodesic by a horocycle or the hyperbolic distance by…

双曲平面中曲线缩短流的几何特性

Geometric aspects of the curve shortening flow in the hyperbolic plane

本文研究了双曲平面中曲线缩短流的翻译概念，解决了曲线缩短流方程，并探讨了古早凸解的曲率函数分离变量情况下的解法，以及闭古早解的面积估计。

双曲面上的逆曲线缩短流

The inverse curve shortening flow on the hyperbolic plane

本文研究双曲面上的逆曲线缩短流，分类了所有与双曲面的抛物线性和共形向量场相关的孤立子，并证明抛物线性孤立子在y轴上为图，共形孤立子在x轴上为图，分析了这些孤立子的凹性和接近坐标轴的行为。

Closed $1/2$-Elasticae in the Hyperbolic Plane

We study critical trajectories in the hyperbolic plane for the $1/2$-Bernoulli's bending energy with length constraint. Critical trajectories with periodic curvature are classified into three different types according to the causal character of their momentum. We prove that closed trajectories arise only when the momentum is a time-like vector. Indeed, for suitable values of the Lagrange multiplier encoding the conservation of the length during the variation, we show the existence of countably many closed trajectories with time-like momentum, which depend on a pair of relatively prime natural …

Proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane

Let $Σ$ be a compact Riemann surface and $D_1,...,D_n$ a finite number of pairwise disjoint closed disks of $Σ$. We prove the existence of a proper harmonic map into the Euclidean plane from a hyperbolic domain $Ω$ containing $Σ\backslash\cup_{j=1}^n D_j$ and of its topological type. Here, $Ω$ can be chosen as close as necessary to $Σ\backslash\cup_{j=1}^n D_j$. In particular, we obtain proper harmonic maps from the unit disk into the Euclidean plane, which disproves a conjecture posed by R. Schoen and S.T. Yau.

关于双曲平面中保持长度和面积的逆曲率流研究

On length-preserving and area-preserving inverse curvature flows in the hyperbolic plane

本文研究双曲平面中保持面积和长度的κ^α型曲率流，证明凸性在流中保持，并展示曲线光滑收敛到测地圆，同时推导了流存在性的充分条件。

Addendum to the paper "Hypersurfaces with Isometric Reeb Flow in Complex hyperbolic Two-Plane Grassmannians"

We classify all of real hypersurfaces $M$ with Reeb invariant shape operator in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m \geq 2$. Then it becomes 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 horosphere whose center at infinity is singular and of type $JN \in {\mathfrak J}N$ for a unit normal vector field $N$ of $M$.

Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves

We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hamiltonian-minimal and Willmore surfaces in terms of simple properties of the curvature of the generating curves. As applications, we provide explicitly conformal parametrizations of known and new examples of minimal, constant mean curvature, Hamiltonian-minimal and Willmore surface…

复双曲平面闭包中的托勒密不等式

The Ptolemaean Inequality in the closure of complex hyperbolic plane

本文在赋予Cygan度量的复双曲平面闭包中证明了托勒密不等式与托勒密定理，推广了经典度量几何结果。

Bas du spectre et delta-hyperbolicité en géométrie de Hilbert plane

We prove that the Hilbert geometry of a convex domain in the plane is Gromov hyperbolic, if, and only if, the bottom of its spectrum is not zero

Least Area Planes in Hyperbolic 3-Space are Properly Embedded

We show that if P is an embedded least area (area minimizing) plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded.

Hyperbolic Distance versus Quasihyperbolic Distance in Plane Domains

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct an example where the spaces are not quasiisometrically equivalent.

Smooth Knot Limit Sets of the Complex Hyperbolic Plane

It is shown that if a regular knot of class C2 is embedded in the boundary of the complex hyperbolic plane as the limit set of a discrete subgroup of PU(2, 1) then it is either a chain or an R-circle.

Octonions, triality, the exceptional Lie algebra F4, and polar actions on the Cayley hyperbolic plane

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras $\mathfrak{f}_4$ and $\mathfrak{f}^*_4$, i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we classify polar actions on the Cayley hyperbolic plane which leave a totally geodesic subspace invariant.

Number of Least Area Planes in Gromov Hyperbolic 3-Spaces

We show that for a generic simple closed curve C in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exist a unique least area plane P in X with asymptotic boundary C. This result has interesting topological applications for constructions of canonical 2-dimensional objects in 3-manifolds.

Non-properly Embedded Minimal Planes in Hyperbolic 3-Space

In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with nonpositive curvature. We show this result by constructing a non-properly embedded minimal plane in hyperbolic 3-space. Hence, this gives a counterexample to Calabi-Yau conjecture for embedded minimal surfaces in the negative curvature case.

On `A characterization of R-Fuchsian groups acting on the complex hyperbolic plane'

We indicate a C-Fuchsian counter-example to the result with the above title announced at http://www.maths.dur.ac.uk/events/Meetings/LMS/2011/GAL11/program.pdf and prove a stronger statement.

Hyperbolic mean curvature flow: Evolution of plane curves

In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. We show that there exists a class of initial velocities such that the solution of the corresponding initial value problem exists only at a finite time interval $[0,T_{\max})$ and when $t$ goes to $T_{\max}$, the solution converges to a point. We also discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time $\mathbb{R}^{1,1}$.

Properly Embedded Least Area Planes in Gromov Hyperbolic 3-Spaces

We show that for any simple closed curve in the sphere at infinity of a Gromov hyperbolic 3-space with cocompact metric, there exist a properly embedded least area plane in the space spanning the given curve. This gives a positive answer to a conjecture of Gabai. Soma has already proven this conjecture earlier. Our technique here is simpler and more general, and it can be applied to many similar settings.

复双曲平面中的外台球

Outer billiards in the complex hyperbolic plane

本文研究复双曲平面中二次凸超曲面诱导的外台球映射，证明其外部具有双重测地叶状结构，且该映射为辛同胚，推广了双曲平面和复欧氏空间中的已知几何性质。

具有双曲单值性的自行车轨迹——结果与猜想

Bicycle tracks with hyperbolic monodromy -- results and conjectures

本文研究闭平面曲线自行车单值性为双曲的条件，利用“双曲展开”方法给出新的充要条件，并基于计算机实验对严格凸闭曲线提出两个猜想。

On an isoperimetric problem on the Lobachevsky hyperbolic plane with left-invariant Finsler structure

We deal with an isoperimetric problem on the Finsler hyperbolic plane. The space is defined as the Lie group of proper affine transformations of the line with a left-invariant Finsler structure. To state the problem, we use the left-invariant volume form. The optimal isoperimetric loops are found in terms of convex trigonometry functions. We also propose a generalization of the isoperimetric inequality in parametric form.

One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula

In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane $\mathbb{E}^2$ were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite{Mul}. One-parameter planar homothetic motion was defined in the Complex plane, \cite{Kur}. In this paper, analogous to homothetic motion in the Complex plane given by \cite{Kur}, one-parameter planar homothetic motion is defined in the Hyperbolic plane. Some characteristic prope…

Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane

Let N be a complete, simply-connected surface of constant curvature κ\leq 0. Moreover, suppose that Ωand \tildeΩ are strictly convex domains in N with the same area. We show that there exists an area-preserving diffeomorphism from Ωto \tildeΩ whose graph is a minimal submanifold of N \times N.

Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane

This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. We prove that the only convex billiard without conjugate points on the Hyperbolic plane or on the Hemisphere is circular billiard.

Minimal Planes in Hyperbolic Space

We show a generic finiteness result for least area planes in 3-dimensional hyperbolic space. Moreover, we prove that the space of minimal immersions of disk into hyperbolic space is a submanifold of a product bundle over a space of immersions of circle into sphere at infinity. The bundle projection map when restricted to this submanifold is Fredholm of index zero. By using this result, we also show that the space of minimal planes with smooth boundary curve at infinity is a manifold.

The Euler and Navier-Stokes equations on the hyperbolic plane

We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Hamiltonian setting of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.

Constructing the Hyperbolic Plane as the reduction of a three-body problem

We construct the hyperbolic plane with its geodesic flow as the scale plus symmetry reduction of a three-body problem in the Euclidean plane. The potential is $-I/Δ^2$ where $I$ is the triangle's moment of inertia and $Δ$ its area. The reduction method uses the Jacobi-Maupertuis metric, following the author's earlier paper "Putting Hyperbolic Pants on a Three-body Problem".

平面中的离散双曲曲率流

Discrete hyperbolic curvature flow in the plane

该研究针对平面双曲曲率流这一几何演化方程，提出半离散有限差分逼近方法并证明离散范数误差界，还给出含光滑凸初值奇异性起始的数值模拟，为相关固液界面波动现象建模提供数值工具。

Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane

We consider a convex curve $γ$ lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the domain bounded by $γ$. We extend the result by S. Bolotin (1992) and get new obstructions on polynomial integrability in terms of the dual curve $Γ$. We follow a method which was introduced by S. Tabachnikov for Outer billiards in the plane and was applied later on in our recent paper to Birkhoff billiards with the help of a new the so called Angular billiard.

Motion in a symmetric potential on the hyperbolic plane

We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is the analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not neces- sarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with spe…

Quantum Hall Effect on the Hyperbolic Plane

In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use the pairing between $K$-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the Hall conductivity in this case.