Optimal Transport: A Semi-Discrete Approach

We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov-Hausdorff convergence.

On the topology and the boundary of N-dimensional RCD(K,N) spaces

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.

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

The main result of the present paper is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension $k\leq n$ in sub-Riemannian Heisenberg groups $\mathbb H^n$. For the purpose of proving such a result we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: these results stem both from a new definition (equivalent to the one introduced by F. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher's Theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated Constancy Theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin's complex of differential forms, we also provide a convenient basis of Rumin's spaces. Eventually, we provide some applications of Rademacher's Theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between $\mathbb H$-rectifiability and ``Lipschitz'' $\mathbb H$-rectifiability, and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.

Lipschitz graphs and currents in Heisenberg groups

This paper studies the structure and stability of boundaries in noncollapsed $\text{RCD}(K,N)$ spaces, that is, metric-measure spaces $(X,\mathsf{d},\mathscr{H}^N)$ with lower Ricci curvature bounded below. Our main structural result is that the boundary $\partial X$ is homeomorphic to a manifold away from a set of codimension 2, and is $N-1$ rectifiable. Along the way we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov-Hausdorff limits $(M_i^N,\mathsf{d}_{g_i},p_i) \rightarrow (X,\mathsf{d},p)$ of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary $\partial X$. 
The key local result is an $\epsilon$-regularity theorem, which tells us that if a ball $B_{2}(p)\subset X$ is sufficiently close to a half space $B_{2}(0)\subset \mathbb{R}^N_+$ in the Gromov-Hausdorff sense, then $B_1(p)$ is biHolder to an open set of $\mathbb{R}^N_+$. In particular, $\partial X$ is itself homeomorphic to $B_1(0^{N-1})$ near $B_1(p)$. Further, the boundary $\partial X$ is $N-1$ rectifiable and the boundary measure $\mathscr{H}^{N-1}_{\partial X}$ is Ahlfors regular on $B_1(p)$ with volume close to the Euclidean volume. 
Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $X_i\to X$. Specifically, we show a boundary volume convergence which tells us that the $N-1$ Hausdorff measures on the boundaries converge $\mathscr{H}^{N-1}_{\partial X_i}\to \mathscr{H}^{N-1}_{\partial X}$ to the limit Hausdorff measure on $\partial X$. We will see that a consequence of this is that if the $X_i$ are boundary free then so is $X$.

Boundary regularity and stability for spaces with Ricci bounded below

The aim of this note is to generalize to the class of non collapsed RCD(K,N) metric measure spaces the volume bound for the effective singular strata obtained by Cheeger and Naber for non collapsed Ricci limits in \cite{CheegerNaber13a}. The proof, which is based on a quantitative differentiation argument, closely follows the original one. As a simple outcome we provide a volume estimate for the enlargement of Gigli-DePhilippis' boundary (\cite[Remark 3.8]{DePhilippisGigli18}) of ncRCD(K,N) spaces.

/pdf/volume-bounds-for-the-quantitative-singular-strata-of-non-25oqxl8f1f.pdf

Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces

This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a C ∞ -hypersurface S without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure H 12 . As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorff dimension 12, with values in S , has negligible image with respect to the Hausdorff measure H 12 . In particular, we deduce that S cannot be Lipschitz parametrizable by countably many maps each defined on some subset of some Carnot group of Hausdorff dimension 12. As main consequence we have that a notion of rectifiability proposed by S. Pauls is not equivalent to one proposed by B. Franchi, R. Serapioni and F. Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset U of a homogeneous subgroup of Hausdorff dimension 12 of a Carnot group, every bi-Lipschitz map f : U → S satisfies H 12 ( f ( U ) ) = 0 . Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all C ∞ -hypersurfaces in H n with n ≥ 2 are countably H n − 1 × R -rectifiable according to Pauls’ definition, even with bi-Lipschitz maps.

Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces

In this paper we prove the one-dimensional Preiss' theorem in the first Heisenberg group $\mathbb H^1$. More precisely we show that a Radon measure $\phi$ on $\mathbb H^1$ with positive and finite one-density with respect to the Koranyi distance is supported on a one-rectifiable set in the sense of Federer, i.e., it is supported on the countable union of the images of Lipschitz maps $A\subseteq \mathbb R\to\mathbb H^1$. The previous theorem is a consequence of a Marstrand-Mattila type rectifiability criterion, which we prove in arbitrary Carnot groups for measures with tangent planes that admit a normal complementary subgroup. Namely, in this co-normal case, even if we a priori ask that the tangent planes at a point might rotate at different scales, a posteriori the measure has a unique tangent almost everywhere. Since every horizontal subgroup has a normal complement, our criterion applies in the particular case of one-dimensional horizontal subgroups. These results are the outcome of a detailed study of a new notion of rectifiability: we say that a Radon measure on a Carnot group is $\mathscr{P}_h$-rectifiable, for $h\in\mathbb N$, if it has positive $h$-lower density and finite $h$-upper density almost everywhere, and, at almost every point, it admits as tangent measures only (multiple of) the Haar measure of a homogeneous subgroup of Hausdorff dimension $h$. We also prove several structure properties of $\mathscr{P}_h$-rectifiable measures. First, we compare $\mathscr{P}_h$-rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups and we realize that it is strictly weaker than them. Furthermore, we show that a $\mathscr{P}_h$-rectifiable measure has almost everywhere positive and finite $h$-density whenever the tangents admit at least one complementary subgroup.

On rectifiable measures in Carnot groups: structure theory

. This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on N -dimensional RCD( K, N ) spaces ( X, d , H N ). Moreover, we obtain almost regularity theorems formulated in terms of the isoperimetric proﬁle and enhanced consequences at the level of several functional inequalities. Most of our statements are new even in the classical setting of smooth, non compact manifolds with lower Ricci curvature bounds. The synthetic theory plays a key role via compactness and stability arguments.

Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

We provide a Rademacher theorem for intrinsically Lipschitz functions $\phi\colon U\subseteq\mathbf{W}\to\mathbf{L}$, where $U$ is a Borel set, $\mathbf{W}$ and $\mathbf{L}$ are complementary subgroups of a Carnot group, where we require that $\mathbf{L}$ is a normal subgroup. Our hypotheses are satisfied for example when $\mathbf{W}$ is a horizontal subgroup. Moreover, we provide an area formula for this class of intrinsically Lipschitz functions.

/pdf/intrinsically-lipschitz-functions-with-normal-target-in-3z8hmj7jg0.pdf

Gioacchino Antonelli

Papers

Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces

Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces

On rectifiable measures in Carnot groups: structure theory

Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

Intrinsically Lipschitz functions with normal target in Carnot groups