0000000000814651

AUTHOR

Luca Capogna

showing 3 related works from this author

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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Conformal equivalence of visual metrics in pseudoconvex domains

2017

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in $\C^n$ are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between pseudoconvex domains. The proofs are inspired by Mostow's proof of his rigidity theorem and are based on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings.

Mathematics - Differential GeometryComputer Science::Machine LearningPure mathematicsGeneral Mathematics32T15 32Q45 32H40 53C23 53C17Rigidity (psychology)Conformal mapMathematical proofComputer Science::Digital Libraries01 natural sciencesdifferentiaaligeometriaStatistics::Machine LearningCorollaryMathematics - Metric Geometry0103 physical sciencesFOS: MathematicsMathematics::Metric GeometryComplex Variables (math.CV)0101 mathematicsEquivalence (formal languages)kompleksifunktiotMathematicsMathematics - Complex VariablesMathematics::Complex Variables010102 general mathematicsMetric Geometry (math.MG)16. Peace & justiceDifferential Geometry (math.DG)Bounded functionComputer Science::Mathematical Software010307 mathematical physicsMathematische Annalen
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Harnack estimates for degenerate parabolic equations modeled on the subelliptic $p-$Laplacian

2014

Abstract We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype ∂ t u = − ∑ i = 1 m X i ⁎ ( | X u | p − 2 X i u ) where p ⩾ 2 , X = ( X 1 , … , X m ) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure μ, and X i ⁎ denotes the adjoint of X i with respect to μ. Our estimates are derived assuming that (i) the control distance d generated by X induces the same topology on M ; (ii) a doubling condition for the μ-measure of d-metric balls; and (iii) the validity of a Poincare inequality involving X and μ. Our results extend the recent work in [16] , [36] , to a more general setting including the model cases of (1)…

Mathematics - Differential GeometryPure mathematicsGeneral MathematicsMathematics::Analysis of PDEsPoincaré inequalityVolume formsymbols.namesakeMathematics - Analysis of PDEsMathematics - Metric GeometryFOS: MathematicsP-LAPLACIAN OPERATORBorel measureRicci curvatureMathematicsHarnack's inequalityMatematikLebesgue measureta111HORMANDER VECTOR FIELDSMetric Geometry (math.MG)Lipschitz continuity35H20Differential Geometry (math.DG)p-LaplaciansymbolsHARNACK INEQUALITYMathematicsAnalysis of PDEs (math.AP)
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