D. Danielli, N. Garofalo, D. M. Nhieu
We prove that a family of entire intrinsic minimal graphs in the Heisenberg group are not perimeter minimizing.
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We prove a monotonicity result at specific points for the Horizontal Perimeter for a class of surfaces in the Heisenberg group.
Quantitative Centre Rigidity under Riemannian Approximation in Heisenberg Groups
该论文证明了海森堡群中满足稳定性条件的有限黎曼周长集合的定量中心刚性，给出了水平中心导数等量的界，并指出周长超出量满足相同界。
This work is an investigation of perimeter measures in the metric measure space given by the Heisenberg group with Haar measure and a Carnot-Carathéodory metric, which is in general a sub-Finsler metric. Included is a reduction of Minkowski content in any CC-metric to an integral formula in terms of Lebesgue surface area in $\mathbb{R}^3$. Using this result, I study two perimeter measures that arise from the study of Finsler normed planes, and provide evidence that Pansu's conjecture regarding the isoperimetric problem in the sub-Riemannian case appears to hold in the more general sub-Finsler …
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。
The Sharp Isoperimetric Inequality in the First Heisenberg Group
本文证明了第一海森堡群中尖锐水平等周不等式，等号条件为Pansu球的左平移和伸缩。
Let $D$ be a closed disk centered at the origin in the horizontal hyperplane $\{t=0\}$ of the sub-Riemannian Heisenberg group $\hh^n$, and $C$ the vertical cylinder over $D$. We prove that any finite perimeter set $E$ such that $D\subset E\subset C$ has perimeter larger than or equal to the one of the rotationally symmetric sphere with constant mean curvature of the same volume, and that equality holds only for the spheres using a recent result by Monti and Vittone [12].
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。
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 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 characterize convex isoperimetric sets in the Heisenberg group endowed with horizontal perimeter. We first prove Sobolev regularity for a certain class of vector fields in the plane with bounded variation, related to the curvature equations. Then, by an approximation-reparameterization argument, we show that the boundary of convex isoperimetric sets is foliated by geodesics of the Carnot-Caratheodory distance.
In this paper we prove that isoperimetric sets in three-dimensional homogeneous spaces diffeomorphic to $\mathbb{R}^3$ are topological balls. We also prove that in three-dimensional homogeneous spheres isopermetric sets are either two-spheres or symmetric genus-one tori. We then apply our first result to the three-dimensional Heisenberg group $\mathsf{H}^1$, characterizing the isoperimetric sets and constants for a family of Riemannian adapted metrics. Using $Γ$-convergence of the perimeter functionals, we also settle an isoperimetric conjecture in $\mathsf{H}^1$ posed by P.Pansu.
The Bernstein problem in higher dimensional Heisenberg groups
针对高维Heisenberg群中的Bernstein问题，证明存在边界为光滑非特征整体内禀图但非垂直超平面的全局周长极小化子。
We study maximal horizontal subgroups of Carnot groups of Heisenberg type. We classify those of dimension half of that of the canonical distribution ("lagrangians") and illustrate some notable ones of small dimension. An infinitesimal classification of the arbitrary maximal horizontal submanifolds follows as a consequence.
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。
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。
In this paper, by extending the notions of harmonic transplantation and harmonic radius in the Heisenberg group, we give an upper bound for the first eigenvalue for the following Dirichlet problem: $$(P_Ω) \left\{ \begin{array}{lllll} -Δ_{\mathbb{H}^1} u & = & λu & \mbox{in} & Ωu & = & 0 & \mbox{on} & \partial Ω, \end{array} \right.$$ where $ Ω$ is a regular bounded domain of $ \mathbb{H}^1$ with smooth boundary and $Δ_{\mathbb{H}^1}$ is the Kohn-Laplace operator. Using the results of P.Pansu which give the relation between the volume of $Ω$ and the perimeter of its boundary. we prove the foll…
We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this note is the introduction of a concept of mildly degenerate characteristic point for a smooth surface of the Heisenberg group, in a neighborhood of which the sub-Riemannian mean curvature is integrable (with respect to the perimeter measure induced by the Euclidean structure). As a consequence we partially answer to a question posed by Danielli-Garofalo-Nhieu in [Danielli D., Garofalo N., Nhieu D.M., Proc. Amer. Math. Soc.,…
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 $C^1$.
In this paper we study sets in the $n$-dimensional Heisenberg group $\hhn$ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in $\hhn$. We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones. Our main result describes which are the CMC hypersurfaces of revolution in $\hhn$. The fact that such a hypersurface is invariant under a compact group of rotations allow…
We study the isoperimetric problem for anisotropic left-invariant perimeter measures on $\mathbb R^3$, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm $ϕ$ on the horizontal distribution. We first prove a representation formula for the $ϕ$-perimeter of regular sets and, assuming some regularity on $ϕ$ and on its dual norm $ϕ^*$, we deduce a foliation property by sub-Finsler geodesics of $\mathrm C^2$-smooth surfaces with constant $ϕ$-curvature. We then prove that the characteristic set of $\mathrm C^2$-smooth surfaces that are locally extremal…
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