Bisectors in the Heisenberg group I
Gaoshun Gou, Yueping Jiang, Ioannis D. Platis
本文证明了海森堡群中科拉尼度量下的度量平分面与脊球面等价，并计算了其水平平均曲率。
我们证明，海森堡群中关于科拉尼度量的度量平分面是脊球面，反之亦然。我们还明确计算了它们的水平平均曲率。
We show that metric bisectors with respect to the Korányi metric in the Heisenberg group are spinal spheres and vice versa. We also calculate explicitly their horizontal mean curvature.
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Bisectors in the Heisenberg group II
本文在前期Korányi度量下度量角平分线研究基础上，探讨配备Carnot–Carathéodory度量的海森堡群中的角平分线问题。
In this paper, we study the Ricci-Bourguignon flow on higher dimensional classical Heisenberg nilpotent Lie groups and construct a solution of this flow on Heisenberg and quaternion nilpotent Lie groups. In the end, we investigate the deformation of spectrum and length spectrum on compact nilmanifolds obtained of Heisenberg and quaternion nilpotent Lie groups.
In this paper, we investigate the geometry of left-invariant Randers metrics on the Heisenberg group.
Minimal translation surfaces in the Heisenberg group Nil3
海森堡群$\mathrm{Nil}_3$中的平移曲面是通过将两条曲线相乘（使用群运算）而构造的曲面。本文对海森堡群$\mathrm{Nil}_3$中的极小平移曲面进行了完全分类。
In this note we present a short alternative proof for the Bernstein problem in the three-dimensional Heisenberg group ${\rm Nil}_3$ by using the loop group technique.
We classify the geodetically convex sets and geodetically convex functions on the Heisenberg group ${\mathbb H}^n$, $n\geq 1$.
Biharmonic Curves in 3-dimensional Hyperbolic Heisenberg Group
本文研究了三维双曲Heisenberg群中的非测地非零双调和曲线。证明了三维双曲Heisenberg群中所有的非测地非零双调和曲线都是螺旋线。此外，我们还分别得到了三维双曲Heisenberg群中非测地非零双调和曲线和非测地类空水平双调和曲线的显式参数方程。我们还证明了在三维双曲Heisenberg群中不存在非测地的水平类时双调和曲线。
Homogeneity of magnetic geodesics in the Heisenberg group
研究Heisenberg群中由标准接触结构导出的磁流形的齐性性质，证明其具有齐性特征。
A notable family of entire intrinsic minimal graphs in the Heisenberg group which are not perimeter minimizing
我们证明了Heisenberg群中的一类整内在极小图不是周长极小图。
We describe all Lorentzian semi-direct extensions of the Heisenberg group which are conformally Einstein. As a by side result, Bach-flat left-invariant Lorentzian metrics on semi-direct extensions of the Heisenberg group are classified, thus providing new background solutions in conformal gravity.
A left invariant Z-Randers metric on the five-dimensional Heisenberg group is a left invariant Randers metric with deformation vector from the center of the Heisenberg algebra. In this note we prove that for every left invariant Z-Randers metric on the five-dimensional Heisenberg group there exist flags of strictly negative and there exist flags of strictly positive curvatures.
A study of smooth contact quasiconformal mappings of the hyperbolic Heisenberg group is presented in this paper. Our main result is a Lifting Theorem; according to this, a symplectic quasiconformal mapping of the hyperbolic plane can be lifted to a circles preserving quasiconformal mapping of the hyperbolic Heisenberg group.
In this paper, we study some basic geometric properties of pseudohermitian submanifolds of the Heisenberg groups. In particular, we obtain the uniqueness and existence theorems, and some rigidity theorems.
We prove that the configuration space of equidistant triples on the Heisenberg group equipped with the Korányi metric, is isomorphic to a hypersurface of $\mathbb{R}^3$.
Plongements quasiisométriques du groupe de Heisenberg dans $L^p$, d'après Cheeger, Kleiner, Lee, Naor
本文简要介绍了Cheeger和Kleiner关于Heisenberg群在$L^1$中的不可嵌入性定理。
Averages are invariants defined on the $\ell^1$ cohomology of Lie groups. We prove that they vanish for abelian and Heisenberg groups. This result completes work by other authors and allows to show that the $\ell^1$ cohomology vanishes in these cases.
In this work, we give a survey on non characteristic domains of Heisenberg groups. We prove that bounded domains which are diffeomorphic to the solid torus having the center of the group as rotation axis, are non characteristic. Then, we state the following conjecture : The bounded non characteristic domains of the Heisenberg group of dimension 1 are those diffeomorphic to a solid torus having the center of the group as rotation axis.
Traslation surfaces in the Heisenberg Group
Heisenberg群中通过两个平面曲线的乘积构造平移表面，并通过高斯映射行列式分类内在曲率消失的表面。
We study immersed, connected, umbilic hypersurfaces in the Heisenberg group $H_{n}$ with $n$ $\geq $ $2.$ We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigma-k curvature up to Heisenberg translations.
Geometric structures arising from the deformation of groups of Heisenberg type
本文研究Heisenberg型群变形产生的几何结构，定义了接触结构并探讨了其与Yamabe方程的关系。
We consider Heisenberg groups equipped with a sub-Finsler metric. Using methods of optimal control theory we prove that in this geometric setting the infinite geodesics are horizontal lines under the assumption that the sub-Finsler metric is defined by a strictly convex norm. This answers a question posed in [5] and has applications in the characterisation of isometric embeddings into Heisenberg groups.
An integral formula on the Heisenberg group
设Hn表示（2n + 1）维（亚黎曼）海森堡群.本文将证明一个积分恒等式（见定理1.2），它推广了70年代Reilly得到的一个公式。第一部分的应用将在第4节中给出。
The Gauss map of Minimal graphs in the Heisenberg group
本文研究了Heisenberg群$\mathcal{H}_{3}.$中曲面的几何性质，利用李群的Gauss映射，得到了$\mathcal{H}_{3}.$中极小图的部分分类，并证明了$\mathcal{H}_{3}.$中脐曲面的不存在性
In this paper, we study the Gauss map of surfaces in 3-dimensional Heisenberg group using the Gans model of the hyperbolic plane. We establish a relationship between the tension field of the Gauss map and mean curvature of a surface in $\mathcal{H}_{3}$.
Horizontal normal map on the Heisenberg group
基于Heisenberg群上函数的次黎曼结构，研究了Heisenberg群上函数的H-次微分和H-正规映射的概念。特别地，通过H-次微分在每一点的非空性给出了函数凸性的一个特征。
Ruled minimal surfaces in the three dimensional Heisenberg group
证明了部分平面、螺旋面和双曲抛物面是三维黎曼海森堡群中仅有的由测地线刻划的极小曲面。证明了它们是三维Heisenberg群中唯一的关于标准黎曼度量和标准洛伦兹度量的中曲率均为零的曲面.
Horizontal submanifolds of groups of Heisenberg type
研究Heisenberg型Carnot群的极大水平子群。我们分类的尺寸一半的典型分布（“拉格朗日”），并说明了一些值得注意的小尺寸。一个无穷小分类的任意极大水平子流形如下的结果。
The fundamental theorems for curves and surfaces in 3d Heisenberg group
利用活动标架和李群的Cartans方法，研究了三维Heisenberg群中曲线曲面的局部等价问题，得到了曲线曲面的完备不变量集.对于曲面，根据这些不变量及其适当的导数，给出了Heisenberg群上由适应度量导出的度量的高斯曲率公式，从而形成了一个新的闭曲面的欧拉数公式.
In this paper, we define and, then, we characterize constant angle spacelike and timelike surfaces in the three-dimensional Heisenberg group, equipped with a 1-parameter family of Lorentzian metrics. In particular, we give an explicit local parametrization of these surfaces and we produce some examples.
Explicit Formulas for Non-Geodesic Biharmonic Curves of the Heisenberg Group
本文研究了黎曼流形间映射的双调和条件（见[BK]），并研究了Heisenberg群H_3中的非测地双调和曲线。首先证明它们都是螺旋，然后明确地得到它们的参数方程。
We study minimal cylinders in the three-dimensional Heisenberg group ${\rm Nil}_3$ using the generalized Weierstrass type representation, the so-called loop group method. We characterize all non-vertical minimal cylinders in terms of pairs of two closed plane curves which have the same signed area. Moreover, as a byproduct of the construction, spacelike CMC cylinders can also be obtained.
For $n\geq 2$ we define a notion of umbilicity for hypersurfaces in the Heisenberg group $H_{n}$. We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant $p$(or horizontal)-mean curvature in $H_{n}$ up to Heisenberg translations.
Lie groups locally isomorphic to generalized Heisenberg groups
我们对与广义海森堡群局部同构的连通李群进行了分类。对于给定的广义Heisenberg群$N$，在与$N$局部同构的连通李群的同构类集合与非紧黎曼对称空间的某些等价物的并之间存在一一对应.
A partial solution of the isoperimetric problem for the Heisenberg group
给出了海森堡群中等周问题的部分解。
We give a geometric criterion for a topological surface in the first Heisenberg group to be an intrinsic Lipschitz graph, using planar cones instead of the usual open cones.
Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups $H^n, n>1$
本文证明了具有$n>1$的Heisenberg群$H^n$中的Lipschitz本征图是光滑的，这些图是极小曲面方程的消失黏性解.
We prove quantitative stability of isometries on the first Heisenberg group with sub-Riemannian geometry: every $ (1+ \varepsilon)$-quasi-isometry of the John domain of the Heisenberg group $ \mathbb {H} $ is close to some isometry with order of closeness $ \sqrt{\varepsilon} + \varepsilon $ in the uniform norm and with order of closeness $ \varepsilon $ in the Sobolev norm $L_2^1$. Homogeneous dilations show the asymptotic sharpness of the results.
The optimal constant in the $L^2$ Folland-Stein inequality on the quaternionic Heisenberg group
我们确定了四元数海森堡群上的$L^2$ Folland-Stein不等式中的最佳（最佳）常数以及等式成立的非负函数。
We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group $H^1$. These spheres are conjectured to be the isoperimetric sets of $H^1$. We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.
Local control on quaternionic Heisenberg group of dimension $7$
该研究将7维四元数Heisenberg群刻画为矩阵群，探究其左不变控制系统的局部控制特性，分析次黎曼结构对称性对测地线最优性的作用。
We give a simpler proof of the sharp Frank-Lieb inequality on the Heisenberg group. The proof bypasses the sophisticated argument for existence of a minimizer and is based on the study of the 2nd variation of subcritical functionals using their fundamental techniques.
We examine the geometry of the level sets of particular horizontally $p$-harmonic functions in the Heisenberg group. We find sharp, natural geometric conditions ensuring that the level sets of the $p$-capacitary potential of a bounded annulus in the Heisenberg group are strictly starshaped.
Let $\H^n$ be the Heisenberg group of topological dimension $2n+1$. We prove that if $n$ is odd, the pair of metric spaces $(\H^n, \H^n)$ does not have the Lipschitz extension property.
We expand the class of polarizable Carnot groups by implementing a technique to polarize anisotropic Heisenberg groups.
Logarithmic Sobolev inequalities on infinite-dimensional reduced Heisenberg groups
本文在无限维约化海森堡群上构造了热核测度，并研究了其中的亚椭圆对数Sobolev不等式。
Some properties of Hölder surfaces in the Heisenberg group
这是一个民间猜想，当α> 1/2时，在亚黎曼海森堡群中不存在α-Hoelder曲面。也就是说，可以预期没有从R^2的开子集嵌入到阶严格大于大于1/2的Hoelder连续的海森堡群中。这里的海森堡群具有卡诺-卡拉西奥多利距离。我们证明，在这样的曲面存在的情况下，它不能是本质有界变差，它相交于一些垂直线在至少一个拓扑康托尔集。
Submanifolds with boundary in sub-Riemannian Heisenberg Groups
本文研究亚黎曼海森堡群中具有内在$C^1$正则性的带边界子流形，给出实例并建立涉及Rumin微分形式积分的斯托克斯定理。
Hölder continuous mappings, differential forms and the Heisenberg groups
本文开展Hölder连续映射的分析研究，涵盖其分布雅可比矩阵、微分形式拉回及相关公式定理，应用于海森堡群时，简洁证明格罗莫夫非嵌入定理的推广，并得到其Hölder同伦群的新结果。
Lorentzian problems on the Heisenberg group
针对海森堡群上的三个左不变洛伦兹问题，通过庞特里亚金最大值原理获得异常与正常极值轨迹的参数化，同时研究了可达集及最优轨迹的存在性。
Computational geometry in Heisenberg group Heis^3
本文研究海森堡群 Heis^3 中的计算几何，利用左不变度量和联络曲率结果求解测地线方程，并用 Mathematica 绘制测地线与度量球。
Compression bounds for Lipschitz maps from the Heisenberg group to $L_1$
本文证明了具有Carnot-Carathéodory度量的Heisenberg群的一个定量的双Lipschitz非嵌入定理，并应用它给出了Spartens-Cut问题的Goemans-Linial半定松弛的积分间隙的一个下界.
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