Constancy of the dimension in codimension one and locality of the unit normal on $\RCD(K,N)$ spaces

Abstract

The aim of this paper is threefold. We first prove that, on RCD(K, N ) spaces, the boundary measure of any set with finite perimeter is concentrated on the n-regular set Rn, where n ≤ N is the essential dimension of the space. After, we discuss localization properties of the unit normal providing representation formulae for the perimeter measure of intersections and unions of sets with finite perimeter. Finally, we study Gauss-Green formulae for essentially bounded divergence measure vector fields, sharpening the analysis in [21]. These tools are fundamental for the development of a regularity theory for local perimeter minimizers on RCD(K, N ) spaces in [50].

An upper bound on the revised first Betti number and a torus stability result for RCD spaces

“…In this section we prove the first part of the main Theorem 1.2, by combining Theorem 4.1 with the structure theory of RCD .K; N / spaces [6,28,35,38]. More precisely we show the following result, which in turn immediately implies the first claim of Theorem 1.2 by a standard scaling argument.…”

“…The third property is contained in the proof of [38,Theorem 6.8]. Thanks to the constancy of the dimension of RCD .K; N / spaces proved by Brué-Semola [6], the following holds.…”

“…The existence of "-mGH approximations into the Euclidean space yields the first claim of Theorem 1.2: for ı > 0 small enough, .X; d; m/ has essential dimension equal to bN c, it is bN c-rectifiable as a metric measure space and moreover, if N is an integer, the measure coincides with the Hausdorff measure H N , up to a positive constant. This will be proved in Theorem 5.1 by combining Theorem 4.1 with an "-regularity result by Naber and the second named author [38], revisited in the light of the constancy of dimension of RCD .K; N / spaces by Brué-Semola [6] and a measure-rigidity result by Honda [33] for non-collapsed RCD .K; N / spaces. When f.X i ; d i ; m i /g i2N is a sequence of spaces as in the assumptions of Theorem 1.2 with ı i # 0, Theorem 4.1 yields pmGH convergence for .…”

“…A subsequent refinement by the independent works [19,28,35] showed that the measure m restricted to R k is absolutely continuous with respect to H k . Moreover, in [6], E. Bruè and D. Semola showed that there exists exactly one regular set R k having positive measure. It is then possible to define the essential dimension of an RCD .K; N / space as follows.…”

Subgraphs of $\rm BV$ functions on $\rm RCD$ spaces

“…The geometric structure of these spaces up to m-negligible sets is pretty well-understood after the works [MN19, GP21, DPMR17, KM18, BS20]. Recently, the research on these spaces has been focusing also on the study of structure results for sets of (locally) finite perimeter, and of fine properties of functions of (locally) bounded variation, see [ABS19,BPS22b,BPS22a,BG22]. We stress that this theory has recently found interesting applications in the study of the isoperimetric problem on non-compact smooth Riemannian manifolds with Ricci curvature bounded from below, see [APPS22], and in the proof of the Rank-One Theorem in this low regularity setting [ABP22].…”