0000000000814652

AUTHOR

Alessandro Ottazzi

showing 2 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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Sard property for the endpoint map on some Carnot groups

2016

In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizat…

Mathematics - Differential Geometry0209 industrial biotechnologyPure mathematics53C17 22F50 22E25 14M17SubvarietyGroup Theory (math.GR)02 engineering and technologySard's property01 natural sciencesSet (abstract data type)020901 industrial engineering & automationAbnormal curves; Carnot groups; Endpoint map; Polarized groups; Sard's property; Sub-Riemannian geometry; Analysis; Mathematical PhysicsMathematics - Metric GeometryFOS: MathematicsPoint (geometry)Canonical mapAbnormal curves; Carnot groups Endpoint map Polarized groups Sard's property Sub-Riemannian geometry Analysis0101 mathematicsMathematics - Optimization and ControlMathematical PhysicsMathematicsApplied Mathematics010102 general mathematicsta111Polarized groupsCarnot groupLie groupEndpoint mapMetric Geometry (math.MG)Base (topology)ManifoldSub-Riemannian geometryDifferential Geometry (math.DG)Optimization and Control (math.OC)Carnot groupsAbnormal curvesMathematics - Group TheoryAnalysis
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