Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

Abstract

We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K, N ) metric measure spaces, regularity is understood with respect to a newly defined quasi-metric built from the Green function of the Laplacian. Its main application is that RCD(K, N ) spaces have constant dimension. In this way we generalize to such abstract framework a result proved by Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting. * Scuola Normale Superiore, elia.brue@sns.it. † Scuola Normale Superiore, daniele.semola@sns.it.

Collapsed Ricci Limit Spaces as Non-Collapsed RCD Spaces

“…Assume that Conjecture 4.1 is true. If (a) holds, then it follows from a result of [12], which confirms a conjecture raised in [21], that (X, d, m) is an RCD(K, k) space, where k = dim(X, d, m). In particular, Conjecture 4.1 yields (b).…”

“…Note that the implication from Conjectures 4.2 to 3.1 is trivial by letting n = N . Combining a result proved in [12] with Conjectures 4.1 and 4.2, we also propose: holds for m-a.e. x ∈ X, where the Hessian, Hess f , of f is defined in [26] as a (0, 2)-type L 2 tensor.…”

“…Thanks to recent quick developments on the study of RCD(K, N ) spaces, most of the well-known properties on Ricci limit spaces can be covered by the RCD theory. For example, it is proved in [12] that the essential dimension, denoted by dim(X, d, m), 2 whose definition is the unique k such that the k-dimensional regular set has positive m-measure, is well-defined. This gives a generalization of a result proved in [19] to RCD(K, N ) spaces.…”

Weakly Noncollapsed RCD Spaces with Upper Curvature Bounds

“…It is known that for a general RCD(K, N ) space (Y, d, m) with N < ∞ there exists unique n ≤ N such m(R n ) > 0 [BS18]. The number n is called the geometric dimension of Y .…”

“…By the structure theory for RCD(K, N ) spaces it is known that for a general RCD(K, N ) space (Y, d, m) with N < ∞ it holds that m is absolutely continuous with respect to H n where n = dim geom X (for instance [MN19,KM18,BS18]). We show Theorem 1.3 (Subsection 6.2).…”