On a conjecture of Cheeger

By Guido De Philippis, Andrea Marchese, Filip Rindler. 2016

67 citations recorded by the source corpus.

This note details how a recent structure theorem for normal $1$-currents proved by the first and third author allows to prove a conjecture of Cheeger concerning the structure of Lipschitz differentiability spaces. More precisely, we show that the push-forward of the measure from a Lipschitz differentiability space under a chart is absolutely continuous with respect to Lebesgue measure.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bbb277d8f25758e5e626dde25ece82e021e013b1

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the volume measure of non-smooth spaces with Ricci curvature bounded below (2016, background): “We briefly mention that in order to build the bi-Lipschitz charts we make use of a construction of P. Mattila [24], and in order to control the measure we combine the work of G. Alberti and A. Marchese [2] with the recent paper of G. De Philippis and F. Rindler [19] on the structure of Radon measure”
Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, background): “After [42, 38, 21, 32] we know that they are rectifiable as metric measure spaces.”
Behaviour of the reference measure on $\sf RCD$ spaces under charts (2016, result): “ibility of studying the ‘concrete and geometric’ notion of tangent space as pmGH-limit via the ‘abstract and analytic’ one proposed in [9]. Finally we remark that other two independent recent papers ([7] and [14]) cover results overlapping with ours; let us brieﬂy describe those and the relations wi”
Dimensional estimates and rectifiability for measures satisfying linear PDE constraints (2018, background): “heorem [1] (see also [24] for a diﬀerent proof based on a geometrical argument). • If A = div, combining Theorem 1.1 with the result of [2], one obtains the weak converse of Rademacher’s theorem (see [14, 19, 21] for other consequences in metric geometry). The main results of this paper is to show h”
Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, background): “ stronger property that m-almost all of Rk can be covered by bi-Lipschitz charts with biLipschitz constant arbitrarily close to 1, deﬁned in subsets of the k-dimensional Euclidean space). See [KM16], [DePhMR16] and [GP16] for the proof of the absolute continuity statement. 4 Weyl’s law In a metric m”