Sobolev mappings and the Rumin complex

Sobolev mappings between nonrigid Carnot groups

We consider mappings between Carnot groups. In this paper, which is a continuation of "Pansu pullback and rigidity of mappings between Carnot groups" (arXiv:2004.09271), we focus on Carnot groups which are nonrigid in the sense of Ottazzi-Warhurst. We show that quasisymmetric homeomorphisms are reducible in the sense that they preserve a special type of coset foliation, unless the group is isomorphic to R^n or a real or complex Heisenberg group (where the assertion fails). We use this to prove the quasisymmetric rigidity conjecture for such groups. The starting point of the proof is the pullba…

Pansu pullback and exterior differentiation for Sobolev maps on Carnot groups

We show that in an $m$-step Carnot group, a probability measure with finite $m^{th}$ moment has a well-defined Buser-Karcher center-of-mass, which is a polynomial in the moments of the measure, with respect to exponential coordinates. Using this, we improve the main technical result of our previous paper concerning Sobolev mappings between Carnot groups; as a consequence, a number of rigidity and structural results from recent papers hold under weaker assumptions on the Sobolev exponent. We also give applications to quasiregular mappings, extending earlier work in the $2$-step case to general …

A Gromov's dimension comparison estimate for rectifiable sets

We extend the validity of a Gromov's dimension comparison estimate for topological hypersurfaces to sufficiently large classes of rectifiable sets, arising from Sobolev mappings. Our tools are a suitably weak exterior differentiation for pullback differential forms and a new low rank property for Sobolev mappings.

Contact equations, Lipschitz extensions and isoperimetric inequalities

We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called …

Carnot群上的Korevaar-Schoen能量与分数阶Sobolev映射的插值

Korevaar-Schoen Energy and Interpolations of Fractional Sobolev mappings on Carnot groups

该研究将欧氏空间的弱接触方程结果推广到次黎曼框架下的Carnot群，探究分数阶Sobolev映射的Korevaar-Schoen能量与插值性质，证明满足弱接触方程的映射属于特定水平分数阶Sobolev空间，建立插值定理并刻画Korevaar-Schoen能量的极小化子。

A low rank property and nonexistence of higher dimensional horizontal Sobolev sets

We establish a "low rank property" for Sobolev mappings that pointwise solve a first order nonlinear system of PDEs, whose smooth solutions have the so-called "contact property". As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of the Heisenberg group. This provides a complete solution to a question raised in a paper by Z. M. Balogh, R. Hoefer-Isenegger and J. T. Tyson. Our approach d…

Trace theory for Sobolev mappings into a manifold

We review the current state of the art concerning the characterization of traces of the spaces $W^{1, p} (\mathbb{B}^{m-1}\times (0,1), \mathcal{N})$ of Sobolev mappings with values into a compact manifold $\mathcal{N}$. In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p < m$ is an integer and the homotopy group $π_p (\mathcal{N})$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group $π_1 (\mathcal{N})$ is finite and $π_2 (\mathcal{N}) \simeq \dotsb \simeq π_{\lfloor p - 1 \rfloor} (\mathcal{…

曲面上重参数化不变索伯列夫度量的完备性

Completeness of reparametrization-invariant Sobolev metrics on the space of surfaces

研究浸入曲面空间上重参数化不变索伯列夫型黎曼度量，通过重铸完备性方法并结合几何估计，得到度量和测地线完备性等结果，是浸入曲线完备性结果的首次扩展，有望用于其他映射流形。

临界Sobolev映射到完备黎曼流形的奇异延拓

Singular extension of critical Sobolev mappings with values into complete Riemannian manifolds

基于Petrunin的判据，将临界Sobolev映射的奇异延拓结果从欧氏空间推广到具有正触及完备非紧黎曼流形，并证明延拓映射满足指数弱型Sobolev-Marcinkiewicz估计。

Sobolev metrics on the manifold of all Riemannian metrics

On the manifold $\Met(M)$ of all Riemannian metrics on a compact manifold $M$ one can consider the natural $L^2$-metric as described first by \cite{Ebin70}. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics.

欧几里得空间中的Sobolev映射与积结构

Sobolev mappings of Euclidean space and product structure

研究证明在 $n \ge 2$ 时，满足特定条件的 Sobolev 映射必须分裂，但 $n=1$ 时这一结论不成立，且与之前工作相补充。

Pansu pullback and rigidity of mappings between Carnot groups

This is the first in a series of papers on geometric mapping theory in Carnot groups -- and more generally equiregular manifolds -- in which we prove a number of new structural results for Sobolev (in particular quasisymmetric) mappings, establishing (partial) rigidity or (partial) regularity theorems, depending on the context.

Sobolev spaces and $\nabla$-differential operators on manifolds I: basic properties and weighted spaces

We study {\em $\nabla$-Sobolev spaces} and {\em $\nabla$-differential operators} with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate free approach that uses connections (which are typically denoted $\nabla$). These concepts arise naturally from Partial Differential Equations, including some that are formulated on plain Euclidean domains, such as the weighted Sobolev spaces used to study PDEs on singular domains. We prove several basic properties of the $\nabla$-Sobolev spaces and of the $\nabla$-differential operators on general manifolds. For …