Conformality and Q-harmonicity in sub-Riemannian manifolds
Copyright policy )Conformality and Q-harmonicity in sub-Riemannian manifolds
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 $p$-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.
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
- University of Pisa Italy
- UNSW Sydney Australia
- Worcester Polytechnic Institute United States
- Smith College United States
- University of Jyväskylä Finland
Mathematics - Differential Geometry, morphism property, conformal transformation, Subelliptic equations, regularity for \(p\)-harmonic functions, differentiaaligeometria, Mathematics - Analysis of PDEs, Differentiable maps on manifolds, Mathematics - Metric Geometry, Liouville Theorem, Subelliptic PDE, FOS: Mathematics, Matematiikka, Harmonic coordinates, popp measure, osittaisdifferentiaaliyhtälöt, subelliptic PDE, Harmonic coordinates; Liouville Theorem; Quasi-conformal maps; Regularity for p-harmonic functions; Sub-Riemannian geometry; Subelliptic PDE; Mathematics (all); Applied Mathematics, ta111, 53C17, 35H20, 58C25, Metric Geometry (math.MG), Quasi-conformal maps, regularity for p-harmonic functions, Sub-Riemannian geometry, sub-Riemannian geometry, Differential Geometry (math.DG), quasi-conformal maps, Regularity for p-harmonic functions, harmonic coordinates, monistot, Liouville theorem, Mathematics, Analysis of PDEs (math.AP)
Mathematics - Differential Geometry, morphism property, conformal transformation, Subelliptic equations, regularity for \(p\)-harmonic functions, differentiaaligeometria, Mathematics - Analysis of PDEs, Differentiable maps on manifolds, Mathematics - Metric Geometry, Liouville Theorem, Subelliptic PDE, FOS: Mathematics, Matematiikka, Harmonic coordinates, popp measure, osittaisdifferentiaaliyhtälöt, subelliptic PDE, Harmonic coordinates; Liouville Theorem; Quasi-conformal maps; Regularity for p-harmonic functions; Sub-Riemannian geometry; Subelliptic PDE; Mathematics (all); Applied Mathematics, ta111, 53C17, 35H20, 58C25, Metric Geometry (math.MG), Quasi-conformal maps, regularity for p-harmonic functions, Sub-Riemannian geometry, sub-Riemannian geometry, Differential Geometry (math.DG), quasi-conformal maps, Regularity for p-harmonic functions, harmonic coordinates, monistot, Liouville theorem, Mathematics, Analysis of PDEs (math.AP)
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