I have co-authored 30 papers in collaboration with 18 colleagues. A full list of my papers is here. You can also find all my papers on arXiv.

Some selected papers and preprints

1. A sharp spectral splitting theorem (with M. Pozzetta and K. Xu), 2024.
   Submitted (arXiv)
   We extend the Cheeger-Gromoll splitting theorem allowing a spectral lower bound on the Ricci curvature. Specifically, if a manifold M^n with two ends satisfies the condition −γΔ + Ric ≥ 0 and γ < 4/(n−1), then the manifold splits. The upper bound 4/(n−1) is sharp, and the two-ends assumption is essential for γ > 0 (i.e., the existence of geodesic lines alone does not suffice).

2. Vertical curves and vertical fibers in the Heisenberg group (with R. Young), 2024.
   Submitted (arXiv)
   In the first Heisenberg group, equipped with any left-invariant homogeneous distance, the center Z has Hausdorff dimension 2. Unlike the Euclidean setting, we construct curves that satisfy a Lipschitz-cone condition with respect to Z but have Hausdorff dimension that is either greater or less than 2.

3. Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature (with M. Pozzetta and D. Semola), 2024.
   Submitted (arXiv)
   On manifolds (≠ Rⁿ) with Ric ≥ 0, Euclidean volume growth, and quadratic curvature decay, we show that there is a set G of positive real numbers with density one at infinity such that, for volumes in G, isoperimetric sets are unique.

4. Ergodic maps and the cohomology of nilpotent Lie groups (with R. Young), 2024.
   Submitted (arXiv)
   We introduce the concept of ergodic map and show that an ergodic map between simply connected nilpotent Lie groups naturally induces a homomorphism between their cohomology algebras. As a result, we provide a simplified proof of the fact that the cohomology algebras of simply connected nilpotent Lie groups are quasi-isometric invariants, a result originally established by Shalom, Sauer, and Gotfredsen—Kyed.

5. New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry (with K. Xu), 2024.
   Submitted (arXiv)
   We prove a sharp spectral version of the Bishop-Gromov volume comparison. Specifically, if a closed manifold M^n of dimension n ≥ 3 satisfies −γΔ + Ric ≥ n − 1 and γ < (n − 1)/(n − 2), then the volume of M is less than or equal to the volume of the n-dimensional round sphere of radius 1, and π₁(M) is finite. The upper bound (n − 1)/(n − 2) is sharp. This result has been used by L. Mazet to show that stable minimal hypersurfaces in R⁶ are flat.

6. Carnot rectifiability and Alberti representations (with E. Le Donne and A. Merlo), 2024.
   Proc. Lond. Math. Soc. (arXiv)
   A metric measure space is called Carnot-rectifiable if it is a.e. covered by countably many bi-Lipschitz images of compact sets of a fixed Carnot group. We provide a characterization of this notion through Alberti representations of the measure and the differentiability of Lipschitz maps taking values in Carnot groups. We develop an analogue of the notion of Lipschitz differentiability space (Cheeger) using Carnot groups and Pansu derivatives as models.

7. Nonexistence of isoperimetric sets in spaces of positive curvature (with F. Glaudo), 2023.
   J. Reine Angew. Math. (arXiv)
   We construct the first examples of complete, noncompact Riemannian manifolds M^n, with n ≥ 3, that have Sec > 0, Euclidean volume growth, and lack isoperimetric regions with volume less than 1.

8. Sharp isoperimetric comparison on noncollapsed spaces with lower Ricci bounds (with E. Pasqualetto and M. Pozzetta and D. Semola), 2023.
   Accepted, Ann. Sci. Éc. Norm. Supér. (arXiv)
   The isoperimetric profile I is a real-valued function that maps a volume v to the infimum of the perimeter of sets with volume v. As a corollary of our results,  which are established in the general metric measure setting, we show that on complete noncompact Riemannian manifolds M^n with Ric ≥ 0, the function I^(n/(n-1)) is concave.

9. Asymptotic isoperimetry on noncollapsed spaces with lower Ricci bounds (with E. Pasqualetto and M. Pozzetta and D. Semola), 2023.
   Math. Ann. (arXiv)
   Among other results, we provide an alternative proof of the sharp and rigid isoperimetric inequality for complete noncompact Riemannian manifolds M^n with Ric ≥ 0 and Euclidean volume growth. One of the contributions of our work is that the rigidity statement does not require any additional regularity conditions on the finite perimeter set.

10. The Rank-One Theorem on RCD spaces (with C. Brena and E. Pasqualetto), 2023.
    Anal. PDE. (arXiv)
    We extend Alberti’s Rank-One Theorem to the setting of metric measure spaces that satisfy a weak notion of Ric ≥ K and dim ≤ N, i.e., RCD(K, N) spaces.

11. On rectifiable measures in Carnot groups: Marstrand--Mattila rectifiability criterion (with A. Merlo), 2022.
    J. Funct. Anal. (arXiv)
    We extend the Marstrand-Mattia rectifiability criterion to the context of Carnot groups. Using this, we prove a generalization of Preiss' Theorem: in the first Heisenberg group, equipped with the Koranyi norm, every locally finite measure with positive and finite 1-density almost everywhere is absolutely continuous with respect to H^1 and is supported on Lipschitz fragments.

12. On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth (with E. Bruè and M. Fogagnolo and M. Pozzetta), 2022.
    Calc. Var. PDE. (arXiv)
    Among other results, we show that on complete noncompact Riemannian manifolds M^n with Sec ≥ 0 and Euclidean volume growth, isoperimetric regions exist for all sufficiently large volumes.
    
    Notes

13. A note on the stable Bernstein theorem (with K. Xu), 2024.
    We offer an alternative perspective on the solution to the Stable Bernstein problem up to R⁶. Our approach does not rely on the concept of biRicci curvature, but rather on direct computations.

14. Isoperimetric problems, 25 pp., (2024).
    These are the Lecture notes of the 4 hours minicourse I gave in Granada (July 2024) for the European Doctorate School of Differential Geometry.

Academic works

PhD Thesis: Rectifiability in Carnot groups
Supervisors: Prof. Luigi Ambrosio, Prof. Enrico Le Donne
Master thesis: Limits of Riemannian manifolds with Ricci curvature bounded from below
Supervisors: Prof. Luigi Ambrosio, Prof. Luciano Mari
Bachelor thesis: Regolarità dei minimi del problema di Lagrange (Language: Italian)
Supervisor: Prof. Luigi Ambrosio