Ana Hurtado, Manuel Ritoré, César Rosales
We prove that any $C^2$ complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group $\mathbb{H}^1$ is either a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We consider surfaces of class $C^1$ in the $3$-dimensional sub-Riemannian Heisenberg group ${\mathbb H}^1$. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In case the surface is complete and oriented, without singular points, and stable, i.e., a second order minimum of perimeter, we prove that the surface must be a vertical plane. This implies the following Bernstein type result: a complete locally area-minimizing intrinsic graph of a $C…
We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carathéodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero (or constant if a volume-preserving condition is assumed) and such that the characteristic curves meet orthogonally the singular curves. Moreover, a Minkowski type formula relating the area, the mean curvature, and the volume is obtained for volume-preserving area-stationary surfaces enclosing a given region. As …
We classify the geodetically convex sets and geodetically convex functions on the Heisenberg group ${\mathbb H}^n$, $n\geq 1$.
This paper aims to define and study a notion of orientability in the Heisenberg sense ($\mathbb{H}$-orientability) for the Heisenberg group $\mathbb{H}^n$. In particular, we define such notion for $\mathbb{H}$-regular $1$-codimensional surfaces. Analysing the behaviour of a Möbius Strip in $\mathbb{H}^1$, we find a $1$-codimensional $\mathbb{H}$-regular, but not Euclidean-orientable, subsurface. Lastly we show that, for regular enough surfaces, $\mathbb{H}$-orientability implies Euclidean-orientability. As a consequence, we conclude that non-$\mathbb{H}$-orientable $\mathbb{H}$-regular surface…
For a strictly convex set $K\subset \mathbb{R}^2$ of class $C^2$ we consider its associated sub-Finsler $K$-perimeter $|\partial E|_K$ in $\mathbb{H}^1$ and the prescribed mean curvature functional $|\partial E|_K-\int_E f$ associated to a function $f$. Given a critical set for this functional with Euclidean Lipschitz and intrinsic regular boundary, we prove that their characteristic curves are of class $C^2$ and that this regularity is optimal. The result holds in particular when the boundary of $E$ is of class $C^1$.
The notion of $Γ$-symmetric space is a natural generalization of the classical notion of symmetric space based on $\z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $\g$ of $G$ admits a $Γ$-grading where $Γ$ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group $\mathbb{H}_3$ adapted to the symmetries of a $Γ$-symmetric structure on $\mathbb{H}_3$. We prove that the classification of $\z_2^2$-symmetric Riemannian and Lorentzian metrics on $\mathbb{H}_3$ corresponds to the class…
This paper aims to define and study currents and slices of currents in the Heisenberg group $\mathbb{H}^n$. Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces and, assuming their support to be compact, we can work with currents of finite mass, define the notion of slices of Heisenberg currents and show some important properties for them. While some such properties are similarly true in Riemannian settings, others carry deep consequences because they do not include the slices of the middle dimension $n$, which opens new challen…
Area-minimizing horizontal graphs with low-regularity in the sub-Finsler Heisenberg group $\mathbb{H}^1$
研究次芬斯勒海森堡群$\mathbb{H}^1$中低正则性面积极小水平图，构造以水平线或共点水平半直线有限并为奇异集的欧氏局部利普希茨整面积极小水平图，还给出多类面积极小锥例子。
The purpose of this study is to analyse two related topics: the Rumin cohomology and the $\mathbb{H}$-orientability in the Heisenberg group $\mathbb{H}^n$. In the first three chapters we carefully describe the Rumin cohomology with particular emphasis at the second order differential operator $D$, giving examples in the cases $n=1$ and $n=2$. We also show the commutation between all Rumin differential operators and the pullback by a contact map and, more generally, describe pushforward and pullback explicitly in different situations. Differential forms can be used to define the notion of orien…
We prove that a strictly stable minimal $C^2_h$ intrinsic graph G is locally area-minimizing, i.e. given any $C^1_h$ graph $S$ with the same boundary, $\text{Area}(G)<\text{Area}(S)$ unless $G=S$. As a consequence we show the existence and the uniqueness of $C^\infty$ minimal graphs with prescribed small boundary datum.
Submanifolds with boundary and Stokes' Theorem in Heisenberg groups
本文在海森堡群中引入带边 $C^1_\mathbb{H}$-正则子流形，并结合 Rumin 微分形式复形证明了相应版本的斯托克斯定理。
It is proved that the Heisenberg group $\operatorname*{Nil}\nolimits_{3}$ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product $\mathbb{T\times Z}$, where $\mathbb{T}$ is a totally geodesic surface and $\mathbb{Z}$ the center of $\operatorname*{Nil}% \nolimits_{3}.$ It is then proved the existence of complete properly embedded minimal surfaces in $\operatorname*{Nil}\nolimits_{3}$ by solving the asymptotic Dirichlet problem for the minimal surface equation on $\mathbb{T}$. It is also proved the existence of complete properly embedded minimal s…
We show the analogy of Cauchy's surface area formula for the Heisenberg groups $\mathbb{H}_n$ for $n\geq 1$, which states that the p-area of any compact hypersurface $Σ$ in $\mathbb{H}_n$ with its p-normal vector defined almost everywhere on $Σ$ is the average of its projected p-areas onto the orthogonal complements of all p-normal vectors of the Pansu spheres (up to a constant). The formula provides a geometric interpretation of the p-areas defined by Cheng-Hwang-Malchiodi-Yang [9] in $\mathbb{H}_1$ and Cheng-Hwang-Yang [7] in $\mathbb{H}_n$ for $n\geq 2$. We also characterize the projected a…
This paper studies the geometry of bilipschitz maps $f \colon \mathbb{W} \to \mathbb{H}$, where $\mathbb{H}$ is the first Heisenberg group, and $\mathbb{W} \subset \mathbb{H}$ is a vertical subgroup of co-dimension $1$. The images $f(\mathbb{W})$ of such maps are called Rickman rugs in the Heisenberg group. The main theorem states that a Rickman rug in the Heisenberg group admits a corona decomposition by intrinsic bilipschitz graphs. As a corollary, Rickman rugs are countably rectifiable by intrinsic bilipschitz graphs. Here, an intrinsic bilipschitz graph is an intrinsic Lipschitz graph, whi…
We prove that, in the first Heisenberg group $\mathbb{H}$, an entire locally Lipschitz intrinsic graph admitting vanishing first variation of its sub-Riemannian area and non-negative second variation must be an intrinsic plane, i.e., a coset of a two dimensional subgroup of $\mathbb{H}$. Moreover two examples are given for stressing result's sharpness.
Minimal surfaces in $\mathbb{R}^n$ can be locally approximated by graphs of harmonic functions, i.e., functions that are critical points of the Dirichlet energy, but no analogous theorem is known for $H$-minimal surfaces in the three-dimensional Heisenberg group $\mathbb{H}$, which are known to have singularities. In this paper, we introduce a definition of intrinsic Dirichlet energy for surfaces in $\mathbb{H}$ and study the critical points of this energy, which we call contact harmonic graphs. Nearly flat regions of $H$-minimal surfaces can often be approximated by such graphs. We give a cal…
We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called {\it contact stationary legendrian immersions}) in the Heisenberg Group ${\mathbb H}^2$ . From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. This last result happens to be in particular a central ingredient in the analysis part of the proof of the Willmore conjecture given by the author in arXiv:2007.05467
The Folland-Stein spectrum of some Heisenberg Bieberbach manifolds
本文考察三维 Heisenberg Bieberbach 紧商流形上 Folland-Stein 算子 $\mathscr{L}_α$ 的谱，给出其特征值与特征函数的研究。
We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss-Bonnet theorem. An application to Steiner's formula for the Carnot-Carathéodory distance in $\mathbb{H}$ is provided.
The paper is devoted to the large scale geometry of the Heisenberg group $\mathbb H$ equipped with left-invariant Riemannian distances. We prove that two such distances have bounded difference if and only if they are asymptotic, i.e., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant Riemannian distance $d$ on $\mathbb H$ there is a unique subRiemanniann metric $d'$ for which $d-d'$ goes to zero at infinity, and we estimate the rate of convergence. As a first immediate consequence we get that the Riemannian Heisenberg group is at bounded distance from its as…
For the ``Hopf bundle'' $S^1\to S^{2n,1} \to {\mathbb H}^n$, horizontal lifts of simple closed curves are studied. Let $γ$ be a piecewise smooth, simple closed curve on a complete totally geodesic surface $S$ in the base space. Then the holonomy displacement along $γ$ is given by $$ V(γ)=e^{λA(γ) i} $$ where $A(γ)$ is the area of the region on the surface $S$ surrounded by $γ$; $λ=1/2 $ or 0 depending on whether $S$ is a complex submanifold or not. We also carry out a similar investigation for the complex Heisenberg group ${\mathbb R} \to {\mathcal H}^{2n+1} \to {\mathbb C}^n$.
We prove that the configuration space of equidistant triples on the Heisenberg group equipped with the Korányi metric, is isomorphic to a hypersurface of $\mathbb{R}^3$.
We study the Gauss map of minimal surfaces in the Heisenberg group $\mathrm{Nil}_3$ endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane $\mathbb{H}^2$. Conversely, any nowhere antiholomorphic harmonic map into $\mathbb{H}^2$ is the Gauss map of a nowhere vertical minimal surface. Finally, we study the image of the Gauss map of complete nowhere vertical minimal surfaces.
We verify if Gausssian curvature of surfaces and normal curvature of curves in surfaces introduced by Diniz-Veloso arXiv:1210.7110 and by Balogh-Tyson-Vecchi arXiv:1604.00180 to prove Gauss-Bonnet theorems in Heisenberg space $\mathbb H^1$ are equal. The authors in arXiv:1604.00180 utilize a limit of Gaussian and normal curvatures defined in the Riemannian approximations scheme $(\mathbb R^3,g_L)$ introduced by Gromov to study sub-Riemannian spaces. They show that these limits exist (unlike the limit of Riemannian surface area form or length form), and they obtain Gauss-Bonnet theorem in $\mat…
A Partial $ε$-Regularity Theorem for Integral Stationary Geodesic Networks in $\mathbb{R}^n$
研究\(\mathbb{R}^n\)中整数重数平稳测地线网络序列，通过证明部分\(\epsilon\)-正则性定理，得出其收敛到重数为\(k\)直线的结论。
Bergman space, Conformally flat 2-disk operads and affine Heisenberg vertex algebra
本文研究了Bergman空间的对称代数作为共形平坦二维盘操作符的代数结构，并将其与仿射Heisenberg顶点代数联系起来。
In this paper we study areas (called p-areas) and volumes for parametric surfaces in the 3D-Heisenberg group $\mathbb{H}_1$, which is considered as a flat model of pseudo-hermitian manifolds. We derive the formulas of p-areas and volumes for parametric surfaces in $\mathbb{H}_1$ and show that the classical result of Pappus-Guldin theorems for surface areas and volumes hold if the surfaces satisfy some geometric properties. Some examples are also provided, including the surfaces with constant p-mean curvatures.
We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, we show that if $f:\mathbb{R}^2\rightarrow\mathbb{R}$ is a $C^1$-intrinsic function, and $\nabla^f\nabla^ff=0$, then the first contact variation of the sub-Riemannian area of its intrinsic graph is zero and the second contact variation is positive.
We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(κ,τ)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts $\mathbb{E}(κ,τ)$, and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire m…
Polyhomogeneous symbols, defined by Kohn-Nirenberg and Hörmander in the 60's, play a central role in the symbolic calculus of most pseudodifferential calculi. We prove a simple characterisation of polyhomogeneous functions which avoids the use of asymptotic expansions. Specifically, if $U$ is open subset of $\mathbb{R}^d$, then a polyhomogeneous symbol on $U \times \mathbb{R}^d$ is precisely the restriction to $t=1$ of a function on $U \times \mathbb{R}^{d+1}$ which is homogeneous for the dilations of $\mathbb{R}^{d+1}$ modulo Schwartz class functions. This result holds for arbitrary graded di…
Symmetry and classification of solutions to an integral equation in the Heisenberg group $\mathbb{H}^n$
本文用移动平面法和Heisenberg群上的Hardy-Littlewood-Sobolev不等式，证明一类积分方程非负解的对称性、次临界无解性及临界解的柱形分类，并推出相应偏微分方程结果。
Plateau's Problem for intrinsic graphs in the Heisenberg Group
本文利用几何构造，在Heisenberg群H^1中解决内在图的Plateau问题，通过小性条件证明了边界或Lipschitz边界数据的最小性，并应用这些技术获得H-周长最小化者的新的正则性结果。
We study constant mean curvature graphs in the Riemannian 3-dimensional Heisenberg spaces ${\cal H}={\cal H}(τ)$. Each such ${\cal H}$ is the total space of a Riemannian submersion onto the Euclidean plane $\mathbb{R}^2$ with geodesic fibers the orbits of a Killing field. We prove the existence and uniqueness of CMC graphs in ${\cal H}$ with respect to the Riemannian submersion over certain domains $Ω\subset\mathbb{R}^2$ taking on prescribed boundary values.
Overdetermined problems for gauge balls in the Heisenberg group
本文证明Heisenberg群中规范球是唯一能解出Serrin型超定问题的区域，并针对不同维度建立了唯一性结果。
The prescribed mean curvature equation for $t$-graphs in the sub-Finsler Heisenberg group $\mathbb{H}^n$
本文研究了海森堡群中有界域上 $t$-图的次芬斯勒预定平均曲率方程，在预定数据为常数且小于边界芬斯勒平均曲率的条件下，通过芬斯勒逼近格式证明了次芬斯勒 CMC 方程 Dirichlet 问题 Lipschitz 解的存在性。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。