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On potentials for sub-Laplacians and geometric applications

Shiguang Ma, Jie Qing

TL;DR

The paper develops a potential theory framework for sub-Laplacians on homogeneous Carnot groups, introducing a completeness-based method and Riesz-type inequalities to bound polar sets. It extends Euclidean insights to polarizable Carnot groups via polar coordinates and geometric measure theory, proving a main bound that the Hausdorff dimension of polar sets is at most (Q-2)/2. The results are then deployed to geometric problems, including Yamabe-type equations in CR, quaternionic CR, and octonionic CR geometries, as well as to quotients of spherical CR boundaries by convex cocompact subgroups of rank-one groups, with implications for Yamabe constants and limit-set dimensions. Overall, the work provides a versatile bridge between sub-Riemannian potential theory and conformal/hyperbolic geometry, offering tools applicable to multiple CR-geometric contexts and rank-one symmetric spaces.

Abstract

In this paper we extend the research on potential theory and its geometric applications from Euclidean spaces to homogeneous Carnot groups. We introduce a new approach to use the geometric completeness to estimate the Hausdorff dimension of polar sets of potentials of nonnegative Radon measures for sub-Laplacians in homogeneous Carnot groups. Our approach relies on inequalities that are analogous to the classic integral inequalities about Riesz potentials in Euclidean spaces. Our approach also uses extensions of some of geometric measure theory to homogeneous Carnot groups and the polar coordinates with horizontal radial curves constructed by Balogh and Tyson for polarizable Carnot groups. As consequences, we develop applications of potentials for sub-Laplacians in CR geometry, quaternionic CR geometry, and octonionic CR geometry.

On potentials for sub-Laplacians and geometric applications

TL;DR

The paper develops a potential theory framework for sub-Laplacians on homogeneous Carnot groups, introducing a completeness-based method and Riesz-type inequalities to bound polar sets. It extends Euclidean insights to polarizable Carnot groups via polar coordinates and geometric measure theory, proving a main bound that the Hausdorff dimension of polar sets is at most (Q-2)/2. The results are then deployed to geometric problems, including Yamabe-type equations in CR, quaternionic CR, and octonionic CR geometries, as well as to quotients of spherical CR boundaries by convex cocompact subgroups of rank-one groups, with implications for Yamabe constants and limit-set dimensions. Overall, the work provides a versatile bridge between sub-Riemannian potential theory and conformal/hyperbolic geometry, offering tools applicable to multiple CR-geometric contexts and rank-one symmetric spaces.

Abstract

In this paper we extend the research on potential theory and its geometric applications from Euclidean spaces to homogeneous Carnot groups. We introduce a new approach to use the geometric completeness to estimate the Hausdorff dimension of polar sets of potentials of nonnegative Radon measures for sub-Laplacians in homogeneous Carnot groups. Our approach relies on inequalities that are analogous to the classic integral inequalities about Riesz potentials in Euclidean spaces. Our approach also uses extensions of some of geometric measure theory to homogeneous Carnot groups and the polar coordinates with horizontal radial curves constructed by Balogh and Tyson for polarizable Carnot groups. As consequences, we develop applications of potentials for sub-Laplacians in CR geometry, quaternionic CR geometry, and octonionic CR geometry.

Paper Structure

This paper contains 22 sections, 22 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. On potentials for sub-Laplacians in homogeneous Carnot groups
  3. Basic notations and definitions
  4. $\Delta_\mathbb{G}$-superharmonic functions and potentials of nonnegative Radon measures for sub-Laplacians
  5. New integral inequalities about potentials for sub-Laplacians
  6. On the size of polar sets of potentials for sub-Laplacians
  7. On Newtonian potentials in Euclidean spaces and the new idea
  8. On potentials for sub-Laplacians in polarizable Carnot groups
  9. Polarizable Carnot groups and polar coordinates
  10. Some preliminaries in geometric measure theory in homogeneous Carnot groups
  11. Proof of Theorem \ref{['Thm:potential-carnot']}
  12. Geometric applications
  13. Yamabe problems in H-type groups $\mathsf{N}$
  14. Yamabe type problems in CR cases
  15. Yamabe type problems in quaternionic CR cases
  16. ...and 7 more sections

Key Result

Theorem 2.5

(BLU) Let $\mathbb{G}= (\mathbb{R}^N, \circ, \delta_\lambda)$ be a homogeneous Carnot group. Suppose $\Omega\subset \mathbb{R}^N$ and $u$ is a $\Delta_\mathbb{G}$-superharmonic function on $\Omega$. Then, there is a nonnegative Radon measure $\mu$ in $\Omega$, which is called the $\Delta_\mathbb{G}$ for some function $h$ which is $\Delta_\mathbb{G}$-harmonic in $\Omega'$, where $\Gamma_{\Delta_\ma

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 30 more