Structure theory of metric measure spaces with lower Ricci curvature bounds

By Andrea Mondino, A. Naber. Journal of the European Mathematical Society (Print), 2014

212 citations recorded by the source corpus.

We prove that a metric measure space (X,d,m) satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space W1,2 is Hilbert is rectifiable. That is, a RCD∗(K,N)-space is rectifiable, and in particular for m-a.e. point the tangent cone is unique and euclidean of dimension at most N. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. We also show a sharp integral Abresh–Gromoll type inequality on the excess function and an Abresh–Gromoll-type inequality on the gradient of the excess. The argument is new even in the smooth setting

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/902334aa3169f89738cf72a80f8f47441cf12629

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

Citations in the corpus

Non-collapsed spaces with Ricci curvature bounded from below (2017, unknown): “…i.e. Lipschitz and with bounded Laplacian - in the context of Ricci-limit spaces has been proved in [16] and frequently used as important technical tool in [17, 18, 19, 20, 21, 22]; their existence in the RCD setting has been proved in [8, Lemma 6.7] and [35, Theorem 3.12], see also [45, Lemma 3.1]”
Geodesically complete spaces with an upper curvature bound (2018, background): “sets of Rk−1. In particular, the set of not (k,0)-strained points is countably (k −1)-rectiﬁable. For similar rectiﬁable stratiﬁcations on diﬀerent classes of metric spaces we refer, for instance, to [MN17] and the literature therein. Remark 1.7. Parts of Theorems 1.2, 1.3, 1.5, 1.6 have analogues i”
Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, background): “Also the main ideas underlying the proofs of the uniqueness of tangents and the rectifiability result are quite similar to those implemented in [42].”
Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, background): “This conclusion was greatly improved in [56], where it was proved that RCD.”
Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, background): “ference measure m, but rather to the Hausdorﬀ measure Hℓ, where ℓis the Hausdorﬀ dimension of (X,d). 2 In order to introduce the precise statement of our criterion let us recall MondinoNaber’s result [MN14]: m(X\ [[N] i=1 Ri) = 0, where [N] is the integer part of N and Ri is the i-dimensional regula”
On the topology and the boundary of N–dimensional RCD(K,N) spaces (2019, result): “plying Theorem 6.2 to the blow up sequence pmGH converging to (Y,d,mY ). (4) ⇒ (3) Follows by combining the next results: m-a.e. x ∈ X has unique tangent cone which is isomorphic to a Euclidean space [MN19], m-a.e. uniqueness of the dimension of Euclidean tangent spaces [BS18], on the k-regularstrat”
Second order differentiation formula on RCD*$(K,N)$ spaces (2018, background): “1 Introduction In the last ten years there has been a great interest in the study of metric measure spaces with Ricci curvature bounded from below, see for instance [44], [55], [56], [29], [5], [6], [27], [7], [48], [49], [30], [24], [26], [38], [9], [47], [13], [12].”