Sub-Riemannian Calculus on Hypersurfaces in Carnot Groups

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…

Stable H-minimal hypersurfaces

We prove some stability results for smooth H-minimal hypersurfaces immersed in a sub-Riemannian k-step Carnot group G. The main tools are the formulas for the 1st and 2nd variation of the H-perimeter measure.

Flat compactness of normal currents, and charges in Carnot groups

We prove that the family of normal currents in the sense of Rumin in a Carnot group is compact in the flat topology. This result is obtained through a dual compactness argument for Rumin forms, using the pseudo-differential calculus in groups developed by Folland, Christ-Geller-G lowacki-Polin and Rumin. As an application, imitating de Pauw-Moonens-Pfeffer, we describe the space of charges on a Carnot group.

超生成的Carnot群

Hypergenerated Carnot groups

本文刻画了超生成分层群，证明其等价于非特征超曲面嵌入为局部双Lipschitz，并推广至任意余维数子流形。

关于四元数Heisenberg群的子黎曼几何

On the sub-Riemannian geometry of the quaternionic Heisenberg group

本文研究四元数Heisenberg群的子黎曼几何，通过定义一系列黎曼度量并分析其极限，刻画了测地线和距离，并推导了超曲面的水平平均曲率公式。

A Calculus for Conformal Hypersurfaces and new higher Willmore energy functionals

The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold: For a given conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem. Using existence results for asymptotic solutions to this problem, we develop the details of how to proliferate conformal hypersurface invariants. In addition we show how to compute the the solution's asymptotics. We also develop a calculus of conformal hypersurface invariant differential oper…

Natural boundary conditions in geometric calculus of variations

In this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of $Y$, the manifold of dependent and independent variables underlying a given problem, as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over $Y$. E…

Singular Yamabe Problem Willmore Energies

We develop the calculus for hypersurface variations based on variation of the hypersurface defining function. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the obstruction to smoothly solving the singular Yamabe problem for conformally compact four-manifolds. We give explicit hypersurface formulae for both the energy functional and the obstruction.

Renormalized Volume

We develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. For closed hypersurfaces or conformally compact geometries, methods from a previously developed boundary calculus for conformally compact manifolds can be applied to give explicit holographic formulae for the divergences and anomaly expressed as hypersurface integrals over local qu…

几何变分犯罪：希尔伯特复形、有限元外微积分和超曲面问题

Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces

本文建立了一个抽象希尔伯特复形中变分犯罪的分析框架，并将其应用于超曲面上的有限元外微积分，推广了Arnold、Falk和Winther的工作，恢复了Dziuk和Demlow的先验估计，并推广到任意维超曲面上的混合有限元方法。

旋转与缠绕的视界：零超曲面的微分刻画

Horizons that Gyre and Gimble: A Differential Characterization of Null Hypersurfaces

本文从卡罗尔几何视角，利用带挠率的联络发展零超曲面微积分，以刻画其内在与外在微分不变量，并给出共形不变的例子。

移动流形与庞加莱猜想

Moving Manifolds and the Poincare Conjecture

本文用移动曲面微积分给出庞加莱猜想的微分几何证明框架，通过变分原理使紧致单连通超曲面演化至常平均曲率平衡态，最终收敛于球面，统一了动力学、拓扑与几何。

Tractor calculus, BGG complexes, and the cohomology of Kleinian groups

For a compact, oriented, hyperbolic $n$-manifold $(M,g)$, realised as $M= Γ\backslash \mathbb{H}^{n}$ where $Γ$ is a torsion-free cocompact subgroup of $SO(n,1)$, we establish and study a relationship between differential geometric cohomology on $M$ and algebraic invariants of the group $Γ$. In particular for $\mathbb{F}$ an irreducible $SO(n,1)$-module, we show that the group cohomology with coefficients $H^{\bullet}(Γ, \mathbb{F})$ is isomorphic to the cohomology of an appropriate projective BGG complex on $M$. This yields the geometric interpretation that $H^{\bullet}(Γ, \mathbb{F})$ parame…

子黎曼流形上的广义曲率维数不等式

Sub-Laplacian generalized curvature dimension inequalities on Riemannian foliations

本文在不假定度量束状结构和叶的总测地或最小条件的情况下，为一般黎曼流形的水平拉普拉斯算子发展了广义曲率维数不等式，并应用该框架得到多种几何和分析结果。