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Pullback theorem and rigidity for Sobolev mappings on Carnot groups

Yihan Cui

TL;DR

This work extends rigidity and regularity phenomena for Sobolev mappings between Carnot groups to the low-integrability regime $p<\nu$ by developing a mollification-based pullback theorem for differential forms, enabling distributional convergence to the Pansu differential even without pointwise differentiability. It proves a generalized rigidity result for $p<\nu$ (and Hölder regularity for $p>Q$), establishes an invariance of the horizontal determinant under higher-layer directions in both a.e. Pansu differentiable and general Sobolev settings, and derives area formulas and unboundedness conclusions for images under such mappings. The paper further provides a new equivalent characterization of quasiconformal mappings under lower integrability and demonstrates the results through Heisenberg and filiform group examples, thereby broadening the scope of rigidity and quasiconformality in sub-Riemannian geometry.

Abstract

We study the pullback theorem of Sobolev mappings on Carnot groups via mollification of mappings. With the pullback theorem we extend the classical result proved by Xiangdong Xie : Rigidity of Sobolev mappings $W^{1,p}(G_1;G_2)$ for $p>ν$, to the case $p<ν$, where $ν$ is the homogeneous dimension of $G_1$. Therefore, some conclusions about continuity of Sobolev mappings on Carnot groups for $p<ν$ are found. And also, the determine of horizontal gradient $D_Hf$ is invariant under the motion related to higher layer left-invariant vector fields. At last, we find a equivalent definition of quasiconformal mappings with lower integrability $dim(g^{[1]})<p<ν$.

Pullback theorem and rigidity for Sobolev mappings on Carnot groups

TL;DR

This work extends rigidity and regularity phenomena for Sobolev mappings between Carnot groups to the low-integrability regime by developing a mollification-based pullback theorem for differential forms, enabling distributional convergence to the Pansu differential even without pointwise differentiability. It proves a generalized rigidity result for (and Hölder regularity for ), establishes an invariance of the horizontal determinant under higher-layer directions in both a.e. Pansu differentiable and general Sobolev settings, and derives area formulas and unboundedness conclusions for images under such mappings. The paper further provides a new equivalent characterization of quasiconformal mappings under lower integrability and demonstrates the results through Heisenberg and filiform group examples, thereby broadening the scope of rigidity and quasiconformality in sub-Riemannian geometry.

Abstract

We study the pullback theorem of Sobolev mappings on Carnot groups via mollification of mappings. With the pullback theorem we extend the classical result proved by Xiangdong Xie : Rigidity of Sobolev mappings for , to the case , where is the homogeneous dimension of . Therefore, some conclusions about continuity of Sobolev mappings on Carnot groups for are found. And also, the determine of horizontal gradient is invariant under the motion related to higher layer left-invariant vector fields. At last, we find a equivalent definition of quasiconformal mappings with lower integrability .
Paper Structure (6 sections, 25 theorems, 128 equations)

This paper contains 6 sections, 25 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Pullback theorem for Sobolev mappings on Carnot groups
  4. Rigidity of Sobolev mappings on Carnot group
  5. Invariance of horizontal determinant for a.e. Pansu differentiable Sobolev mappings
  6. Invariance of horizontal determinant for general Sobolev mappings

Key Result

Theorem 1.1

Suppose that $f \in W^{1,q}\left(\Omega ; G_2\right)$ , where $G_2$ is a step-$n$ Carnot group and $\nu_2$ is the topological dimension of $G_2$, $\Omega \subset G_1$ is open , $X$ is a left-invariant vector field belonging to the horizontal layer and $q>n+1$ . If $B\left(x_{o}, 2 r\right) \subset \ When $D_Pf$ exists a.e., where the convergence is in the sense of the norm on Euclidean space $\ma

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1: Metric Definition
  • Definition 2.2: Geometric Definition
  • Definition 2.3: Analytic Definition via Pansu Differential
  • ...and 37 more