Tangents and Rectifiability of Ahlfors Regular Lipschitz Differentiability Spaces

David, Guy C.

doi:10.1007/s00039-015-0325-8

Cited by 21 publications

(56 citation statements)

References 31 publications

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“…Since the Heisenberg group is a Lipschitz differentiability space and purely 2-unrectifiable [4], we know that the tangent lines do not always span an n-rectifiable set. However under the additional assumption that the space is Ahlfors n-regular with n being the dimension of the chart, at almost every point there is a tangent space biLipschitz equivalent to R n , see [12]. We are interested in finding other conditions that would provide information on the tangents.…”

Section: Introductionmentioning

confidence: 99%

Tangent Lines and Lipschitz Differentiability Spaces

Cavalletti

Rajala

2016

Analysis and Geometry in Metric Spaces

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Abstract:We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We rst revisit the almost everywhere metric di erentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric di erentiability and of density one for the domain of the curve gives a tangent line. Metric di erentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz di erentiability spaces. We show that any tangent space of a Lipschitz di erentiability space contains at least n distinct tangent lines, obtained as the blow-up of n Lipschitz curves, where n is the dimension of the local measurable chart. Under additional assumptions on the space, such as curvature lower bounds, these n distinct tangent lines span an n-dimensional part of the tangent space.

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Section: Introductionmentioning

confidence: 99%

Tangent Lines and Lipschitz Differentiability Spaces

Cavalletti

Rajala

2016

Analysis and Geometry in Metric Spaces

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“…We speculate that, (5.11) being true, would imply that m restricted to R k is an Ahlfors k-regular measure. (also see the related work by David [18]).…”

Section: Further Information On the Measuresmentioning

confidence: 90%

Characterization of Low Dimensional RCD*(K, N) Spaces

Kitabeppu

Lakzian

2016

Analysis and Geometry in Metric Spaces

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Abstract:In this paper, we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called RCD * (K, N) spaces) with non-empty one dimensional regular sets.In particular, we prove that the class of Ricci limit spaces with Ric ≥ K and Hausdor dimension N and the class of RCD * (K, N) spaces coincide for N < (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality (that is ,roughly speaking, a converse to the Lévy-Gromov's isoperimetric inequality and was previously only known for Ricci limit spaces) which might be also of independent interest.

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“…13] where the surjectivity of the map E was proven for the case in which (X, µ) is a PI-space. The surjectivity of the map E in the case in which (X, µ) is a di erentiability space has already been proven in [22,42].…”

Section: In Nitesimal Structure Of DI Erentiability Spaces and Metrimentioning

confidence: 92%