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Francesca Oronzio

Publications and source records attributed to Francesca Oronzio.

11 recordsLinked to original sources
arXiv2026

Weighted decomposition of vector fields, $X$-ADM mass, and higher-dimensional mass-charge inequalities

We study the $X$-ADM mass on complete, one-ended asymptotically flat Riemannian manifolds without boundary in arbitrary dimension $n\geqslant 3$. Starting from a weighted gradient--divergence-free decomposition of the vector field $X$, we construct a conformal metric whose scalar curvature is governed by the critical modified scalar curvature $\mathrm{R}_{X}^{(1-n)}$ associated with $X$. This yields an exact decomposition of the $X$-ADM mass into the ADM mass of the conformal metric plus a nonnegative defect term, which vanishes precisely when $X$ is a gradient, thereby giving an alternative decomposition-based proof of positivity with a full rigidity statement. Explicit negative-mass examples show that the range of parameters $k$ in the modified scalar curvature $\mathrm{R}_{X}^{(k)}$ condition is sharp. We then apply the $X$-positive mass theorem to systems of vector fields, obtaining higher-dimensional multi-charge mass inequalities with rigidity and global alignment in the equality case. This yields the classical three-dimensional electric-magnetic inequality and its higher-dimensional analogue. In higher dimensions, the magnetic datum $\beta$ is a $2$-form and does not have a canonical scalar charge on the asymptotically flat end. We prove that a selected tensorial flux vanishes when $d\beta \in L^1$. Under a prescribed critical asymptotic ansatz, this flux defines a non-trivial tensorial magnetic charge and we formulate a conditional reduction to the $X$-positive mass theorem.

math.DG↗
arXiv2026

Distance Functions, Curvature and Topology

We discuss some properties of the distance functions on Riemannian manifolds and we relate their behavior to the geometry of the manifolds. This leads to alternative proofs of some "classical" theorems connecting curvature and topology.

math.DG↗
arXiv2026

$X$-ADM Mass and $X$-Positive Mass Theorem

For a given admissible vector field $X$, we define a geometric quantity for asymptotically flat $3$--manifolds, called $X$--ADM mass and we establish a relative positive mass theorem via a monotonicity formula along the level sets of a suitable Green's function. Under different assumptions on $X$, we obtain generalizations of the ``classical'' positive mass theorem, like the one for weighted manifolds and the one ``with charge'' under some topological restrictions. Finally, we also discuss the rigidity cases.

math.DG↗
arXiv2025

Green functions and a positive mass theorem for asymptotically hyperbolic $3$-manifolds

We prove a new positive mass theorem for three-dimensional manifolds which are asymptotically hyperboloidal of order greater than $1$. The mass quantity under consideration is the volume-renormalized mass recently introduced in a paper by Dahl, McCormick and the first author. The proof is based on a monotonicity formula holding along the level sets of the Green function for the Laplace operator centered at an arbitrary point. In order for this argument to work out, we require that the second homology of the manifold does not contain any spherical classes.

math.DG↗
arXiv2025

Area, Volume and Capacity in non--compact $3$--manifolds with non--negative scalar curvature

Let $(M,g)$ be a $3$--dimensional, complete, one--ended Riemannian manifold, with a minimal, compact and connected boundary. We assume that $M$ has a simple topology and that the scalar curvature of $(M,g)$ is non--negative. Moreover, we suppose that $(M,g)$ admits a $2$--capacitary potential $v$ with $v,\,\vert \nabla v\vert\to 0$ at infinity. In this note, we provide a gradient integral estimate for the level sets of the function $u=1-v$. This estimate leads to a sharp volume comparison for the sub--level sets of $u$, and a sharp area comparison of the level sets of $u$. From this last comparison it follows a sharp area--capacity inequality, originally derived by Bray and Miao, thereby extending its cases of validity. This work is based on a recent paper by Colding and Minicozzi. Finally, for completeness, we also show the same type of area and volume comparison, in the case where $(M,g)$ has no boundary, replacing the function $u$ with one related to the minimal positive Green's function. This volume comparison leads to a more geometric proof of the positive mass inequality than the one given in \cite{Ago_Maz_Oro}.

math.DG↗
arXiv2024

Nonlinear potential theory and Ricci-pinched 3-manifolds

In this paper, we focus on Hamilton's pinching conjecture formulated in Hamilton's paper "Three-manifolds with positive Ricci curvature". Let $(M, g)$ be a complete, connected, noncompact Riemannian $3$-manifold satisfying the Ricci-pinching condition. Then, it is flat. Here, we give an alternative proof, based on nonlinear potential theory, under the extra hypothesis of superquadratic volume growth.

math.DG↗
arXiv2024

A Note on Ricci-pinched three-manifolds

Let $(M, g)$ be a complete, connected, non-compact Riemannian $3$-manifold. Suppose that $(M,g)$ satisfies the Ricci--pinching condition $\mathrm{Ric}\geq\varepsilon\mathrm{R} g$ for some $\varepsilon>0$, where $\mathrm{Ric}$ and $\mathrm{R}$ are the Ricci tensor and scalar curvature, respectively. In this short note, we give an alternative proof based on potential theory of the fact that if $(M,g)$ has Euclidean volume growth, then it is flat. Deruelle-Schulze-Simon and Huisken-K\"{o}rber have already shown this result and together with the contributions by Lott and Lee-Topping led to a proof of the so-called Hamilton's pinching conjecture.

math.DG↗
arXiv2022

ADM mass, area and capacity in asymptotically flat $3$-manifolds with nonnegative scalar curvature

We show an improvement of Bray sharp mass-capacity inequality and Bray-Miao sharp upper bound of the capacity of the boundary in terms of its area, for three-dimensional, complete, one-ended asymptotically flat manifolds with compact, connected boundary and with nonnegative scalar curvature, under appropriate assumptions on the topology and on the mean curvature of the boundary. Our arguments relies on two monotonicity formulas holding along level sets of a suitable harmonic potential, associated to the boundary of the manifold. This work is an expansion of the results contained in the PhD thesis of the author.

math.DG↗
arXiv2022

Riemannian Penrose inequality via Nonlinear Potential Theory

We provide a new proof of the Riemannian Penrose inequality for time-symmetric asymptotically flat initial data with a single black-hole horizon. The proof proceeds through a newly established monotonicity formula holding along the level sets of the $p$-capacitary potential of the horizon boundary, in any asymptotically flat $3$-manifold with nonnegative scalar curvature.

math.DG↗
arXiv2020

A geometric capacitary inequality for sub-static manifolds with harmonic potentials

In this paper, we prove that associated with a sub-static asymptotically flat manifold endowed with a harmonic potential there is a one-parameter family $\{F_β\}$ of functions which are monotone along the level-set flow of the potential. Such monotonicity holds up to the optimal threshold $β=\frac{n-2}{n-1}$ and allows us to prove a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.

math.AP↗