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Camillo Brena

Publications and source records attributed to Camillo Brena.

At least 19 recordsLinked to original sources
arXiv2026

Improvement of the dimension bound for unweighted RCD spaces

We show that if $(X,d,\mathscr H^n)$ is an $RCD(K,N)$ space for some $K\in\mathbb{R}$ and $N\in [1,\infty)$, then it is also a (non-collapsed) $RCD(K,n)$ space. In other words, unweighted finite-dimensional $RCD$ spaces enjoy an automatic improvement of the dimension bound to the Hausdorff dimension of the space. More generally, we prove that this holds also when $N=\infty$, provided that we assume in addition the $n$-rectifiability of the space.

math.MG↗
arXiv2026

Ollivier--Ricci Curvature on Groups of Polynomial Growth

We study Ollivier--Ricci curvature on Cayley graphs of groups of polynomial growth. Our main result shows that non-negative Ollivier--Ricci curvature forces the group to be virtually abelian. As an application, we prove that connected vertex-transitive graphs of polynomial growth and non-negative Ollivier--Ricci curvature are quasi-isometric to $\mathbb{Z}^k$, for some $k\in\mathbb{N}$.

math.DG↗
arXiv2025

Transitive graphs of non-negative Ollivier-Ricci curvature have polynomial growth

We prove that every transitive graph of non-negative Ollivier-Ricci curvature has polynomial growth. We deduce from this and a theorem of Brena and Brue that a finitely generated group admits a Cayley graph of non-negative Ollivier-Ricci curvature if and only if it is virtually abelian. Beyond the transitive setting, we also prove a quasi-polynomial $r^{O(\sqrt{\log r})}$ volume bound for balls around typical vertices in finite bounded-degree graphs of non-negative Ollivier-Ricci curvature, greatly strengthening a theorem of Salez (GAFA 2022).

math.DG↗
arXiv2025

Instability of the fundamental group for non-collapsed Ricci-limits

We construct two sequences of closed $4$-dimensional manifolds with non-negative Ricci curvature, diameter bounded from above by $1$, and volume bounded from below by $v>0$, with different fundamental groups but with the same Gromov-Hausdorff limit. This provides a negative answer to the question posed in [J. Pan. Ricci Curvature and Fundamental Groups of Effective Regular Sets. Journal of Mathematical Study, 58(1):3--21, 2025].

math.DG↗
arXiv2025

Perelman's entropy and heat kernel bounds on RCD spaces

We study Perelman's W-entropy functional on finite-dimensional RCD spaces, a synthetic generalization of spaces with Bakry-\'{E}mery Ricci curvature bounded from below. We rigorously justify the formula for the time derivative of the W-entropy and derive its monotonicity and rigidity properties. Additionally, we establish bounds for solutions of the heat equation, which are of independent interest.

math.DG↗
arXiv2025

$C^\infty$ rectifiability of stationary varifolds

In this paper we prove that, for every integers $m\geq 2$ and $n\geq 1$, the support of any stationary $m$-dimensional integer rectifiable varifold $V$ in an open set $U\subset \mathbb R^{m+n}$ is $C^\infty$ rectifiable, namely it can be covered, up to an $H^m$-null set, with countably many $C^\infty$ $m$-dimensional graphs.

math.AP↗
arXiv2025

Remarks and Conjectures on Stationary Varifolds

In this paper, we revisit some known results about stationary varifolds using simpler arguments. In particular, we obtain the height bound and the Lipschitz approximation along with its estimates, and as a consequence, the excess decay

math.AP↗
arXiv2024

Lower Ricci Curvature Bounds and the Orientability of Spaces

We study orientability in spaces with Ricci curvature bounded below. Building on the theory developed by Honda, we establish equivalent characterizations of orientability for Ricci limit and RCD spaces in terms of the orientability of their manifold part. We prove a new stability theorem and, as a corollary, we deduce that four-manifolds with Ricci curvature bounded below and volume non-collapsing are uniformly locally orientable. As a global counterpart of the latter, we show that four-manifolds with nonnegative Ricci curvature and Euclidean volume growth are orientable.

math.DG↗
arXiv2024

Unique continuation for area minimizing currents

The main goal of this work is to prove an instance of the unique continuation principle for area minimizing integral currents. More precisely, consider an $m$-dimensional area minimizing integral current and an $m$-dimensional minimal surface, both contained in $\mathbb{R}^{n+m}$ with $n\geq 1$. We show that if, in an integral sense, the current has infinite order of contact with the minimal surface at a point, then the current and the minimal surface coincide in a neighborhood of that point.

math.DG↗
arXiv2023

Fine representation of Hessian of convex functions and Ricci tensor on RCD spaces

It is known that on $\mathrm{RCD}$ spaces one can define a distributional Ricci tensor ${\bf Ric}$. Here we give a fine description of this object by showing that it admits the polar decomposition $${\bf Ric}=ω\,|{\bf Ric}|$$ for a suitable non-negative measure $|{\bf Ric}|$ and unitary tensor field $ω$. The regularity of both the mass measure and of the polar vector are also described. The representation provided here allows to answer some open problems about the structure of the Ricci tensor in such singular setting. Our discussion also covers the case of Hessians of convex functions and, under suitable assumptions on the base space, of the Sectional curvature operator.

math.MG↗
arXiv2023

About the general chain rule for functions of bounded variation

We give an alternative proof of the general chain rule for functions of bounded variation ([ADM90]), which allows to compute the distributional differential of $φ\circ F$, where $φ\in \mathrm{LIP}(\mathbb{R}^m)$ and $F\in\mathrm{BV}(\mathbb{R}^n,\mathbb{R}^m)$. In our argument we build on top of recently established links between `closability of certain differentiation operators' and `differentiability of Lipschitz functions in related directions' ([ABM23]): we couple this with the observation that `the map that takes $φ$ and returns the distributional differential of $φ\circ F$ is closable' to conclude. Unlike previous results in this direction, our proof can directly be adapted to the non-smooth setting of finite dimensional RCD spaces.

math.FA↗
arXiv2023

Maps of bounded variation from PI spaces to metric spaces

We study maps of bounded variation defined on a metric measure space and valued into a metric space. Assuming the source space to satisfy a doubling and Poincar\'e property, we produce a well-behaved relaxation theory via approximation by simple maps. Moreover, several equivalent characterizations are given, including a notion in weak duality with test plans.

math.FA↗
arXiv2023

Nguyen's approach to Sobolev spaces in metric measure spaces with unique tangents

We extend Nguyen's characterization of Sobolev spaces $W^{1,p}$ to the setting of PI-metric measure spaces such that at a.e. point the tangent space (in the Gromov-Hausdorff sense) is unique and Euclidean with a fixed dimension. We also generalize [CLL14] to PI-metric measure spaces such that at a.e. point the tangent space is unique and equal to the Heisenberg group with a fixed homogeneous dimension. The approach is easier and completely different from the one in [Ngu06, Ngu08].

math.FA↗
arXiv2023

Functions with bounded Hessian-Schatten variation: density, variational and extremality properties

In this paper we analyze in detail a few questions related to the theory of functions with bounded $p$-Hessian-Schatten total variation, which are relevant in connection with the theory of inverse problems and machine learning. We prove an optimal density result, relative to the $p$-Hessian-Schatten total variation, of continuous piecewise linear (CPWL) functions in any space dimension $d$, using a construction based on a mesh whose local orientation is adapted to the function to be approximated. We show that not all extremal functions with respect to the $p$-Hessian-Schatten total variation are CPWL. Finally, we prove existence of minimizers of certain relevant functionals involving the $p$-Hessian-Schatten total variation in the critical dimension $d=2$.

math.FA↗
arXiv2022

Weakly non-collapsed RCD spaces are strongly non-collapsed

We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties $\quad$- $\mathrm{tr}(\mathrm{Hess}f)=Δf$ on $U\subset\mathsf{X}$ for every $f$ sufficiently regular, $\quad$- $\mathfrak{m}=c\mathscr{H}^n$ on $U\subset\mathsf{X}$ for some $c>0$, where $U\subset \mathsf{X}$ is open and $\mathsf{X}$ is a - possibly collapsed - RCD space of essential dimension $n$.

math.DG↗
arXiv2022

Linear Inverse Problems with Hessian-Schatten Total Variation

In this paper, we characterize the class of extremal points of the unit ball of the Hessian-Schatten total variation (HTV) functional. The underlying motivation for our work stems from a general representer theorem that characterizes the solution set of regularized linear inverse problems in terms of the extremal points of the regularization ball. Our analysis is mainly based on studying the class of continuous and piecewise linear (CPWL) functions. In particular, we show that in dimension $d=2$, CPWL functions are dense in the unit ball of the HTV functional. Moreover, we prove that a CPWL function is extremal if and only if its Hessian is minimally supported. For the converse, we prove that the density result (which we have only proven for dimension $d = 2$) implies that the closure of the CPWL extreme points contains all extremal points.

math.FA↗
arXiv2022

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

In this work we extend classical results for subgraphs of functions of bounded variation in $\mathbb{R}^n\times\mathbb{R}$ to the setting of $\mathsf{X}\times\mathbb{R}$, where $\mathsf{X}$ is an ${\rm RCD}(K,N)$ metric measure space. In particular, we give the precise expression of the push-forward onto $\mathsf{X}$ of the perimeter measure of the subgraph in $\mathsf{X}\times\mathbb{R}$ of a $\rm BV$ function on $\mathsf{X}$. Moreover, in properly chosen good coordinates, we write the precise expression of the normal to the boundary of the subgraph of a $\rm BV$ function $f$ with respect to the polar vector of $f$, and we prove change-of-variable formulas.

math.MG↗