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Alessandro Cucinotta

Publications and source records attributed to Alessandro Cucinotta.

9 recordsLinked to original sources
arXiv2026

New Topological Restrictions For Spaces With Nonnegative Ricci Curvature

We obtain new topological restrictions for complete Riemannian manifolds with nonnegative Ricci curvature and RCD(0,n) spaces. Our main results are a Betti number rigidity theorem which answers a question open since work of M.-T. Anderson in 1990, and a vanishing theorem for the simplicial volume generalizing a theorem of M. Gromov from 1982. Combining such results we obtain a new proof of the classification of noncompact 3-manifolds with nonnegative Ricci curvature, originally due to G. Liu in 2011, which extends to the synthetic setting.

math.DG↗
arXiv2025

Half Space Property in RCD(K,N) spaces

The goal of this note is to prove the Half Space Property for RCD(0,N) spaces, namely that if (X,d,m) is a parabolic RCD(0,N) space and $ C \subset X \times \mathbb{R}$ is locally the boundary of a perimeter minimizing set and it is contained in a half space, then $C$ is a locally finite union of horizontal slices. The same result is proved for RCD(K,N) spaces, for any $K\in \mathbb{R}$ and $N\in (1,\infty)$, under the stronger assumption that $C$ is the boundary of a \emph{globally} perimeter minimizing set. As a consequence, we obtain oscillation estimates and a Half Space Theorem for minimal hypersurfaces in products $M \times \mathbb{R}$, where $M$ is a parabolic smooth manifold (possibly weighted and with boundary), satisfying a Ricci curvature lower bound. On the way of proving the Half Space Property, we also extend to the RCD setting some classical results on Green's functions and parabolic manifolds.

math.DG↗
arXiv2025

On perimeter minimizing sets in manifolds with quadratic volume growth

This paper studies whether the presence of a perimeter minimizing set in a Riemannian manifold $(M,g)$ forces an isometric splitting. We show that this is the case when $M$ has non-negative sectional curvature and quadratic volume growth at infinity. Moreover, we obtain that the boundary of the perimeter minimizing set is identified with a slice in the product structure of $M$.

math.DG↗
arXiv2025

On manifolds with almost non-negative Ricci curvature and integrally-positive $k^{th}$-scalar curvature

We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest $k$ eigenvalues of the Ricci tensor. If $(M^n,g)$ is a Riemannian manifold satisfying such curvature bounds for $k=2$, then we show that $M$ is contained in a neighbourhood of controlled width of an isometrically embedded $1$-dimensional sub-manifold. From this, we deduce several metric and topological consequences: $M$ has at most linear volume growth and at most two ends, it has bounded 1-Urysohn width, the first Betti number of $M$ is bounded above by $1$, and there is precise information on elements of infinite order in $\pi_1(M)$. If $(M^n,g)$ is a Riemannian manifold satisfying such bounds for $k\geq 2$, then we show that $M$ has at most $(k-1)$-dimensional behavior at large scales. If $k=n={\rm dim}(M)$, so that the integral lower bound is on the scalar curvature, assuming in addition that the $(n-2)$-Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to $n-2$. From the above results we deduce topological restrictions, such as upper bounds on the first Betti number.

math.DG↗
arXiv2025

Minimal Surface Equation and Bernstein Property on RCD spaces

We show that if $(X,d,m)$ is an RCD(K,N) space and $u \in W^{1,1}_{loc}(X)$ is a solution of the minimal surface equation, then $u$ is harmonic on its graph (which has a natural metric measure space structure). If K=0 this allows to obtain an Harnack inequality for $u$, which in turn implies the Bernstein property, meaning that any positive solution to the minimal surface equation must be constant. As an application, we obtain oscillation estimates and a Bernstein Theorem for minimal graphs in products $M \times \mathbb{R}$, where $M$ is a smooth manifold (possibly weighted and with boundary) with non-negative Ricci curvature

math.DG↗
arXiv2025

Convergence of the area functional on spaces with lower Ricci bounds and applications

We show that the heat flow provides good approximation properties for the area functional on proper $\RCD(K,\infty)$ spaces, implying that in this setting the area formula for functions of bounded variation holds and that the area functional coincides with its relaxation. We then obtain partial regularity and uniqueness results for functions whose hypographs are perimeter minimizing. Finally, we consider sequences of $\RCD(K,N)$ spaces and we show that, thanks to the previously obtained properties, Sobolev minimizers of the area functional in a limit space can be approximated with minimizers along the converging sequence of spaces. Using this last result, we obtain applications on Ricci-limit spaces.

math.DG↗
arXiv2024

A splitting theorem for manifolds with a convex boundary component and applications

We prove a warped product splitting theorem for manifolds with Ricci curvature bounded from below in the spirit of [Croke-Kleiner, \emph{Duke Math.\;J}.\;(1992)], but instead of asking that one boundary component is compact and mean-convex, we require that it is parabolic and convex. We then deduce several applications, including splitting theorems and first Betti number rigidity results for - $3$-manifolds with non-negative Ricci curvature, - $4$-manifolds with weakly bounded geometry, non-negative $2$-Ricci curvature, scalar curvature $\geq 1$. In particular, the latter aswers to a rigidity question posed by [Chodosh-Li-Stryker, \emph{JEMS},\;(2024)]. The proofs rely on a metric gluing of Riemannian manifolds with boundary, resulting in a non-smooth metric space. To address this lack of smoothness, we employ synthetic tools specifically developed for non-smooth settings, with a focus on those based on optimal transportation.

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