A partial solution of the isoperimetric problem for the Heisenberg group

A proof by calibration of an isoperimetric inequality in the Heisenberg group H^n

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].

On the isoperimetric quotient over scalar-flat conformal classes

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) $n\ge 12$ and $\partial M$ has a nonumbilic point; or (ii) $n\ge 10$, $\partial M$ is umbilic and the Weyl tensor does not vanish at some boundary point.

An existence theorem on the isoperimetric ratio over scalar-flat conformal classes

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) $9\le n\le 11$ and $\partial M$ has a nonumbilic point; or (ii) $7\le n\le 9$, $\partial M$ is umbilic and the Weyl tensor does not vanish identically on the boundary. This is a continuation of the work \cite{Jin-Xiong} b…

Some inequalities related to isoperimetric inequalities with partial free boundary

The main purpose of this paper is to prove a sharp Sobolev inequality in an exterior of a convex bounded domain. There are two ingredients in the proof: One is the observation of some new isoperimetric inequalities with partial free boundary, and the other is an integral inequality (due to Duff [9]) for any nonnegative function under Schwarz equimeasurable rearrangement. These ingredients also allow us to establish some Moser-Trudinger type inequalities, and obtain some estimates on the principal frequency of a membrane with partial free boundary, which extend early results of Nehari [15] and …

Isoperimetric inequality on CR-manifolds with nonnegative $Q'$-curvature

In this paper, we study contact forms on the three- dimensional Heisenberg manifold with its standard CR structure. We discover that the $Q'$-curvature, introduced by Branson, Fontana and Morpurgo [BFM13] on the CR three-sphere and then generalized to any pseudo-Einstein CR three manifold by Case and Yang [CY95], controls the isoperimetric inequality on such a CR-manifold. To show this, we first prove that the nonnegative Webster curvature at infinity deduces that the metric is normal, which is analogous to the behavior on a Riemannian four-manifold.

Higher codimension isoperimetric problems

We consider a variational problem for submanifolds Q $\subset$ M with nonempty boundary $\partial$Q = K. We propose the definition that the boundary K of any critical point Q have constant mean curvature, which seems to be a new perspective when dim Q \textless{} dim M . We then construct small nearly-spherical solutions of this higher codimension CMC prob-lem; these concentrate near the critical points of a certain curvature function.

CMC spheres in the Heisenberg group

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.

子洛伦兹海森堡群中的常平均曲率曲面

Constant mean curvature surfaces in the sub-Lorentzian Heisenberg group

本文研究了子洛伦兹海森堡群中具有常水平平均曲率的曲面，通过体积保持径向变分推导了水平面积的第一变分公式，并完整分类了光滑的boost对称常平均曲率曲面，发现了一族非零常平均曲率的光滑非因果boost对称曲面，推测其为等周问题的解。

Exhaustion of isoperimetric regions in asymptotically hyperbolic manifolds with scalar curvature $R\geq -6$

In this paper, aimed at exploring the fundamental properties of isoperimetric region in $3$-manifold $(M^3,g)$ which is asymptotic to Anti-de Sitter-Schwarzschild manifold with scalar curvature $R\geq -6$, we prove that connected isoperimetric region $\{D_i\}$ with $\mathcal{H}_g ^3(D_i)\geq δ_0>0$ cannot slide off to the infinity of $(M^3,g)$ provided that $(M^3,g)$ is not isometric to the hyperbolic space. Furthermore, we prove that isoperimetric region $\{D_i\}$ with topological sphere $\{\partial D_i\}$ as boundary is exhausting regions of $M$ if Hawking mass $m_H(\partial D_i)$ has unifor…

Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space

Let $Σ$ be a $k$-dimensional complete proper minimal submanifold in the Poincaré ball model $B^n$ of hyperbolic geometry. If we consider $Σ$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold $Σ$ and the ideal boundary $\partial_\infty Σ$, say $\rvol(Σ)$ and $\rvol(\partial_\infty Σ)$, respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if $\rvol(\partial_\infty Σ) \geq \rvol(\mathbb{S}^{k-1})$, then $Σ$ satisfies the classical isoperimetric inequality. By proving the…

Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

We study the generalization of the Willmore functional for surfaces in the three-Heisenberg group. Its construction is based on the spectral theory of the Dirac operator coming to the Weierstrass representation of surfaces (see math.DG/0503707). By using surfaces of revolution we demonstrate that it resembles the Willmore functional for surfaces in the Euclidean space in many geometrical respects. We also observe the relation of these functionals to the isoperimetric problem.

一个尖锐的等周不等式与最高阶\(Q\)曲率

A sharp isoperimetric inequality and the top order $Q$-curvature

研究\(\mathbb{R}^n\)上特定度量的\(n\)阶\(Q\)曲率与截面曲率关系，基于此在\(Q_g^{(n)}\)非负时得到紧域\(\Omega\)的尖锐等周不等式，在\(Q_g^{(n)}\)非正且Cartan - Hadamard猜想成立时也得到相关尖锐不等式。

Isoperimetric domains of large volume in homogeneous three-manifolds

Given a non-compact, simply connected homogeneous three-manifold $X$ and a sequence $\{Ω_n\}_n$ of isoperimetric domains in $X$ with volumes tending to infinity, we prove that as $n\to \infty $: 1. The radii of the $Ω_n$ tend to infinity. 2. The ratios $\{Area} (\partial Ω_n)/\{Vol}(Ω_n)$ converge to the Cheeger constant Ch$(X)$, which we also prove to be equal to $2H(X)$ where $H(X)$ is the critical mean curvature of $X$. 3. The values of the constant mean curvatures $H_n$ of the boundary surfaces $\partial Ω_n$ converge to $\frac{1}{2}\{Ch}(X)$. Furthermore, when Ch$(X)$ is positive, we prov…

Isoperimetric, Sobolev and Poincaré inequalities on hypersurfaces in sub-Riemannian Carnot groups

In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or without - boundary $\partial S$; S and $\partial S$ are endowed with their homogeneous measures, actually equivalent to the intrinsic (Q-1)-dimensional and (Q-2)-dimensional Hausdorff measures with respect to some homogeneous metric $\varrho$ on G; see Section 5. This generalizes a classical inequality, involving the mean curvature of the hypersurface, prov…

The isoperimetric problem for regular and crystalline norms in $\mathbb H^1$

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…

海森堡群中的稳定刚性

Stable Rigidity in the Heisenberg Groups

本文证明海森堡群中 $C^2$ 边界保体积稳定等价于 Pansu 球面，并由此推出等周区域的刚性分类。

The Hyperplane is the Only Stable, Smooth Solution to the Isoperimetric Problem in Gaussian Space

We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces $Σ^n \subset \mathbb R^{n+1}$ that are second order stable critical points of compact variations that minimize Gaussian weighted area and preserve Gaussian weighted volume. We show that such $Σ$ satisfy a curvature condition, and derive the Jacobi operator $L$ for the second variation of such $Σ$. Our first main result is that for non-planar $Σ$, bounds on the index of $L$, acting on volume preserving variations, gives us that $Σ$ splits off a linear spac…

Isoperimetric and stable sets for log-concave perturbations of Gaussian measures

Let $Ω$ be an open half-space or slab in $\mathbb{R}^{n+1}$ endowed with a perturbation of the Gaussian measure of the form $f(p):=\exp(ω(p)-c|p|^2)$, where $c>0$ and $ω$ is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to $\partialΩ$. In this work we follow a variational approach to show that half-spaces perpendicular to $\partialΩ$ uniquely minimize the weighted perimeter in $Ω$ among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to $\partialΩ$ as…

Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group H^n

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…

亚洛伦兹海森堡群中的豪斯多夫维数与合成曲率界的失效

Hausdorff dimension and failure of synthetic curvature bounds in the sub-Lorentzian Heisenberg group

本文通过初等变分法证明亚洛伦兹海森堡群的极大测地线为双曲线提升，确定其洛伦兹豪斯多夫维数为4，并揭示该群不满足任何参数的类时曲率维数条件与测度收缩性质。

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…

Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group

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.,…

Feeling the heat in a group of Heisenberg type

In this paper we use the heat equation in a group of Heisenberg type $\mathbb{G}$ to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators $\mathscr L^s$ and $\mathscr L_s$, $0< s\leq 1$. Here, $\mathscr L^s$ is the fractional power of the horizontal Laplacian, and $\mathscr L_s$ is the conformal fractional power of the horizontal Laplacian on $\mathbb{G}$. One of our main objective is compute explicitly the fundamental solutions of these nonlocal operators by a new approach exclusively based on partial differential equa…

Convex Cylinders and the Symmetric Gaussian Isoperimetric Problem

Let $Ω$ be a measurable Euclidean set in $\mathbb{R}^{n}$ that is symmetric, i.e. $Ω=-Ω$, such that $Ω\times\mathbb{R}$ has the smallest Gaussian surface area among all measurable symmetric sets of fixed Gaussian volume. We conclude that either $Ω$ or $Ω^{c}$ is convex. Moreover, except for the case $H(x)=\langle x,N(x)\rangle+λ$ with $H\geq0$ and $λ<0$, we show there exist a radius $r>0$ and an integer $0\leq k\leq n-1$ such that after applying a rotation, the boundary of $Ω$ must satisfy $\partialΩ= rS^{k}\times\mathbb{R}^{n-k-1}$, with $\sqrt{n-1}\leq r\leq\sqrt{n+1}$ when $k\geq1$. Here $S…

The Gauss map of Minimal graphs in the Heisenberg group

In this paper we study some geometric properties of surfaces in the Heisenberg group, $\mathcal{H}_{3}.$ We obtain, using the Gauss map for Lie groups, a partial classification of minimal graphs in $\mathcal{H}_{3}.$ We also proof the non existence of umbilical surfaces in $\mathcal{H}_{3}.$

在Heisenberg群中内在图的Plateau问题

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。

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 …

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。

Heisenberg群中的不变$\lambda$-平移子

Invariant $λ$-translators in the Heisenberg group

研究Heisenberg空间$\mathrm{Nil}_3$中平均曲率$H=\langle N,\partial_z\rangle+\lambda$的定向曲面，分类在左平移、旋转和螺旋运动下不变的$\lambda$-平移子。

Regularity of Lipschitz boundaries with prescribed sub-Finsler mean curvature in the Heisenberg group $\mathbb{H}^1$

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 $C^1$.

Minimal surfaces in the Heisenberg group

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we provide a link between our minimal surfaces and Riemannian constant mean curvature surfaces in H equipped with different Riemannian metrics approximating the Carnot-Caratheodory metric. We generate a large library of examples of minimal surfaces a…

Higher Divergence Functions for Heisenberg Groups

We prove a Filling Theorem for the Heisenberg Groups $H^{2n+1}$: For a given $k$-cycle $a$ we construct a $(k+1)$-chain $b$ (the filling) with boundary $\partial b=a$ and controlled volume. For this filling $b$ we prove a uniform bound on the distance of points in $b$ to its boundary $a$. Using this we compute the higher divergence functions for the Heisenberg Groups $H^{2n+1}$. Further we generalise these results to the Jet-Groups $J^m(\mathbb R^n)$ for dimension less or equal $n$ .

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。

Boundary Regularity for the \bar{\partial}_b-Neumann problem, Part 2

We adapt the results of Part 1 to include the unit ball in the Heisenberg group, the model domain with characteristic boundary points. In particular, we construct function spaces on which the Kohn Laplacian with the \bar{\partial}_b-Neumann boundary conditions is an isomorphism. As an application, we establish sharp regularity for a canonical solution to the inhomogenous \bar{\partial}_b equation on the unit ball.

Tubular neighborhoods in the sub-Riemannian Heisenberg groups

We consider the Carnot-Carathéodory distance $δ_E$ to a closed set $E$ in the sub-Riemannian Heisenberg groups $\mathbb{H}^n$, $n\ge 1$. The $\mathbb{H}$-regularity of $δ_E$ is proved under mild conditions involving a general notion of singular points. In case $E$ is a Euclidean $C^k$ submanifold, $k\ge 2$, we prove that $δ_E$ is $C^k$ out of the singular set. Explicit expressions for the volume of the tubular neighborhood when the boundary of $E$ is of class $C^2$ are obtained, out of the singular set, in terms of the horizontal principal curvatures of $\partial E$ and of the function $\langl…

Kepler–Heisenberg问题中的自相似性

Self-similarity in the Kepler-Heisenberg problem

本文研究Kepler–Heisenberg问题的零能量轨道，证明在温和条件下其具有自相似性，并据此将轨道分为未来碰撞、过去碰撞和无碰撞拟周期三类，且碰撞必在有限时间发生。