Search results for "RCD spaces"

showing 4 items of 4 documents

Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below

2013

We prove existence and uniqueness of optimal maps on $RCD^*(K,N)$ spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation.

Mathematics - Differential GeometryExponentiationLower Ricci bounds; Optimal maps; Optimal transport; RCD spaces01 natural sciencesMeasure (mathematics)Combinatoricssymbols.namesakeMathematics - Metric GeometryRCD spacesSettore MAT/05 - Analisi MatematicaFOS: MathematicsOptimal transportMathematics::Metric GeometryUniqueness0101 mathematicsLower Ricci bounds[MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG]Ricci curvatureMathematicsDiscrete mathematics010102 general mathematicsMetric Geometry (math.MG)Absolute continuity16. Peace & justice010101 applied mathematicsMathematics::LogicDifferential geometryDifferential Geometry (math.DG)Fourier analysisBounded functionsymbolsOptimal mapsGeometry and Topology
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On the notion of parallel transport on RCD spaces

2019

We propose a general notion of parallel transport on RCD spaces, prove an unconditioned uniqueness result and existence under suitable assumptions on the space. peerReviewed

Pure mathematicsParallel transportparallel transportGeneral Mathematics010102 general mathematicsSpace (mathematics)metriset avaruudet01 natural sciencesfunktioteoriaRCD spacesSettore MAT/05 - Analisi MatematicaParallel transportMathematics::Metric GeometryUniqueness0101 mathematicsMathematicsRevista Matemática Iberoamericana
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Quasi-Continuous Vector Fields on RCD Spaces

2021

In the existing language for tensor calculus on RCD spaces, tensor fields are only defined $\mathfrak {m}$ -a.e.. In this paper we introduce the concept of tensor field defined ‘2-capacity-a.e.’ and discuss in which sense Sobolev vector fields have a 2-capacity-a.e. uniquely defined quasi-continuous representative.

Quasi-continuityPure mathematics01 natural sciencesPotential theoryTensor fielddifferentiaaligeometria010104 statistics & probabilityRCD spacesSettore MAT/05 - Analisi MatematicaFOS: Mathematics0101 mathematicsMathematicsFunctional analysisDifferential calculus; Quasi-continuity; RCD spaces010102 general mathematicsRCD spaceFunctional Analysis (math.FA)Mathematics - Functional AnalysisSobolev spaceDifferential calculusdifferential calculusVector fieldTensor calculusfunktionaalianalyysiquasi-continuityAnalysis
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Monotonicity Formulas for Harmonic Functions in RCD(0,N) Spaces

2023

We generalize to the RCD(0,N) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with nonnegative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in Agostiniani et al. (Invent. Math. 222(3):1033–1101, 2020), we also introduce the notion of electrostatic potential in RCD spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in RCD(K,N) spaces and on a new functional version of the ‘(almost) outer volume cone implies (almost) outer metric…

differentiaaligeometriaRCD spacesmonotonicity formulaalmost rigidityharmonic functions
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