Examples of area-minimizing surfaces in the subriemannian Heisenberg group H^1 with low regularity

Legendrian面积极小化流的存在性与部分正则性

Existence and partial regularity of Legendrian area-minimizing currents

本文证明接触流形中局部质量极小的Legendrian积分流具有开且稠密的正则集，并据此建立Heisenberg群中Legendrian Plateau问题及闭接触流形同调类中极小流的存在性与部分正则性。

A notable family of entire intrinsic minimal graphs in the Heisenberg group which are not perimeter minimizing

We prove that a family of entire intrinsic minimal graphs in the Heisenberg group are not perimeter minimizing.

次芬斯勒海森堡群 $\mathbb{H}^1$ 中具有低正则性的面积极小水平图

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。

Existence and structure of P-area minimizing surfaces in the Heisenberg group

We study existence and structure of $P-$area minimizing surfaces in the Heisenberg group under Dirichlet and Neumann boundary conditions. We show that there exists an underlying vector field $N$ that characterized existence and structure of $P$-area minimizing surfaces. This vector field exists even if there is no $P$-area minimizing surface satisfying the prescribed boundary conditions. We prove that if $\partial Ω$ satisfies a so called Barrier condition, it is sufficient to guarantee existence of such surfaces. Our approach is completely different from previous methods in the literature and…

Heisenberg群及其他空间上的开普勒动力学

Keplerian Dynamics on the Heisenberg Group and Elsewhere

本文以度量、Laplacian基本解及齐次、各向同性、伸缩对称性为框架，在Heisenberg群上成功建立并求解开普勒问题，并比较秩二格与整数情形。

Harmonic intrinsic graphs in the Heisenberg group

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…

Area-stationary and stable surfaces of class $C^1$ in the sub-Riemannian Heisenberg group ${\mathbb H}^1$

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…

丰富群中几乎 $\mathbb G$-周长最小化边界的Lipschitz逼近

Lipschitz approximation of almost $\mathbb G$-perimeter minimizing boundaries in plentiful groups

本文证明丰富群中内蕴周长几乎最小化边界可由内蕴Lipschitz图逼近，首次将正则性理论推广至Heisenberg群之外。

亚阿贝尔幂零群的次黎曼测地流之辛约化

Sympletic reduction of the sub-Riemannian geodesic flow for metabelian nilpotent groups

本文研究导子群为阿贝尔群的幂零李群上的次黎曼正规哈密顿流，通过辛约化建立正规轨迹与多项式哈密顿量的对应，并给出可积性判据及全局长度极小化条件。

Stable minimal graphs in Heisenberg group $\mathbb{H}^n$

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.

Some remarks on contact variations in the first Heisenberg group

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.

Area-stationary surfaces in the Heisenberg group H^1

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 …

关于 CR 不变能量 $E_1$ 的极小化曲面

On minimizing surfaces of the CR invariant energy $E_1$

本文研究三维 Heisenberg 群中 $E_1$ 零曲面的 CR 不变方程，证明其双曲性、初值问题局部唯一性及旋转对称全局曲面分类，并通过第二变分说明 CR 三维球面中 Clifford 环面并非 $E_1$ 的局部极小元。

海森堡群中极小曲面的爆破

Blow-ups of minimal surfaces in the Heisenberg group

本文修正了 Monti 在海森堡群中 H-周长最小化集合爆破理论中的偏微分方程错误，指出极限函数的水平拉普拉斯算子与坐标 y_1 无关并弱满足方程 1。

海森堡群中水平平均曲率流的半离散格式

A semi-discrete scheme for horizontal mean curvature flow in the Heisenberg group

本文针对第一海森堡群的水平平均曲率流提出极小化运动半离散格式，证明了满足特定连续性条件的平坦流存在，适用于任意有限水平周长的有界集。