Search results for " 58C25"

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Conformality and $Q$-harmonicity in sub-Riemannian manifolds

2016

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.

Harmonic coordinatesMathematics - Differential GeometryPure mathematicsWork (thermodynamics)morphism propertyGeneral Mathematicsconformal transformationBoundary (topology)Conformal map01 natural sciencesdifferentiaaligeometriaMathematics - Analysis of PDEsMathematics - Metric GeometryLiouville TheoremRegularity for p-harmonic functionSubelliptic PDE0103 physical sciencesFOS: MathematicsMathematics (all)0101 mathematicspopp measureMathematicsosittaisdifferentiaaliyhtälötsubelliptic PDESmoothnessQuasi-conformal mapApplied MathematicsHarmonic coordinates; Liouville Theorem; Quasi-conformal maps; Regularity for p-harmonic functions; Sub-Riemannian geometry; Subelliptic PDE; Mathematics (all); Applied Mathematicsta111Harmonic coordinate010102 general mathematics53C17 35H20 58C25Metric Geometry (math.MG)16. Peace & justiceregularity for p-harmonic functionsSub-Riemannian geometrysub-Riemannian geometryDifferential Geometry (math.DG)quasi-conformal mapsRegularity for p-harmonic functionsharmonic coordinates010307 mathematical physicsMathematics::Differential GeometrymonistotLiouville theoremAnalysis of PDEs (math.AP)
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Equivalence of quasiregular mappings on subRiemannian manifolds via the Popp extension

2016

We show that all the common definitions of quasiregular mappings $f\colon M\to N$ between two equiregular subRiemannian manifolds of homogeneous dimension $Q\geq 2$ are quantitatively equivalent with precise dependences of the quasiregularity constants. As an immediate consequence, we obtain that if $f$ is $1$-quasiregular according to one of the definitions, then it is also $1$-quasiregular according to any other definition. In particular, this recovers a recent theorem of Capogna et al. on the equivalence of $1$-quasiconformal mappings. Our main results answer affirmatively a few open questions from the recent research. The main new ingredient in our proofs is the distortion estimates for…

Mathematics - Differential GeometryDifferential Geometry (math.DG)Mathematics::Complex VariablesMathematics - Complex VariablesFOS: MathematicsComplex Variables (math.CV)53C17 30C65 58C06 58C25
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