Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces

Abstract

This note is devoted to the study of sets of finite perimeter over RCD(K, N ) metric measure spaces. Its aim is to complete the picture about the generalization of De Giorgi's theorem within this framework. Starting from the results of [2] we obtain uniqueness of tangents and rectifiability for the reduced boundary of sets of finite perimeter. As an intermediate tool, of independent interest, we develop a Gauss-Green integration-by-parts formula tailored to this setting. These results are new and non-trivial even in the setting of Ricci limits.

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

“…• we will prove that reduced boundaries of sets of finite perimeter have constant dimension, positively answering to one of the questions left open in [BPS19]; • we will clarify in which sense the blow-up of a set of finite perimeter is orthogonal to its unit normal at almost every point and develop a series of useful tools suitable to treat cut and paste operations between sets of finite perimeter in this setting, by analogy with the Euclidean theory (see for instance [Ma12,Chapter 16]); • relying on the finite dimensionality assumption N < ∞, we will sharpen the Gauss-Green integration by parts formulae for essentially bounded divergence measure vector fields studied in [BCM19] on RCD(K, ∞) metric measure spaces. The class of RCD(K, N ) metric measure spaces includes as notable examples (pointed measured) Gromov-Hausdorff limits of smooth manifolds with uniform lower bounds on their Ricci curvature (the so called Ricci limit spaces) and Alexandrov spaces with sectional curvature bounded from below.…”

“…In [ABS19,BPS19] a new set of ideas was needed to develop the regularity theory for sets of finite perimeter, as it was necessary to avoid (1.1) and the use of Besicovitch differentiation theorem. However, after Theorem 1.2 it is natural to investigate if blow-ups of sets of finite perimeter are orthogonal to their unit normal, in some sense.…”

“…In this framework it is too optimistic to hope for a regularity theorem as strong as the Euclidean one. However, in [ABS19,BPS19] a counterpart of De Giorgi's regularity theorem has been obtained in the setting of RCD(K, N ) spaces, with finite N , that are a class of possibly singular metric measure spaces with Ricci curvature bounded from below and dimension bounded from above, in synthetic sense. We recall below two of the main results of [ABS19,BPS19].…”

“…However, in [ABS19,BPS19] a counterpart of De Giorgi's regularity theorem has been obtained in the setting of RCD(K, N ) spaces, with finite N , that are a class of possibly singular metric measure spaces with Ricci curvature bounded from below and dimension bounded from above, in synthetic sense. We recall below two of the main results of [ABS19,BPS19]. By Tan x (X, d, m, E) we shall denote the set of all possible blow-ups of a set of finite perimeter E at a point x, see Definition 2.19 for the precise notion.…”

“…One of the questions left open in [BPS19] was the possibility of having sets of finite perimeter E ⊂ X such that |Dχ E | (F k E) > 0 for some k < n, where F k E is as in (1.2) and n denotes the essential dimension of (X, d, m) as above. Our first main result is a negative answer to this question.…”