Conformality and $Q$-harmonicity in sub-Riemannian manifolds
Analysis of PDEs
2017-01-06 v2 Differential Geometry
Metric Geometry
Abstract
We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic -Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.
Cite
@article{arxiv.1603.05548,
title = {Conformality and $Q$-harmonicity in sub-Riemannian manifolds},
author = {Luca Capogna and Giovanna Citti and Enrico Le Donne and Alessandro Ottazzi},
journal= {arXiv preprint arXiv:1603.05548},
year = {2017}
}
Comments
68 pages, more detailed version of a submitted paper. The update includes more details of the proof of the regularity theorem for weak solutions of the p-Laplacian