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Luca Benatti

Publications and source records attributed to Luca Benatti.

13 recordsLinked to original sources
arXiv2026

An isoperimetric characterization of a new ADM-like mass for $C^0$-asymptotically flat manifolds

We prove that the notion of mass recently introduced by Mazurowski and Yao for continuous metrics coincides with Huisken's isoperimetric mass on smooth Riemannian $3$-manifolds with nonnegative scalar curvature whose metric has sharp $C^0$-asymptotically flat behavior. As a consequence, we deduce a Riemannian Penrose inequality for the mass parameter of asymptotically Schwarzschildian $3$-manifolds.

math.DG↗
arXiv2026

One thousand and one higher-order geometric flows of networks

We show short-time existence and uniqueness up to reparametrization for a large class of higher-order geometric flows of curves and networks obtained as $L^2$-gradient flow of higher-order functionals involving derivatives of the curvature. Additionally, we prove a long-time existence result via energy methods.

math.AP↗
arXiv2025

Proper solutions of the $1/H$-flow and the Green kernel of the $p$-Laplacian

We show existence and optimal growth estimate for the weak inverse mean curvature flow issuing from a point, on manifolds with certain curvature and isoperimetric conditions. These theorems imply analogous ones for the flow issuing from relatively compact sets. Some of the results are obtained by proving new decay estimates for the Green kernel of the $p$-Laplacian which fix a gap in the literature. Additionally, we address the convergence of renormalized $p$-capacitary potentials to the inverse mean curvature flow with outer obstacle.

math.AP↗
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

Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas

We rigorously show that a large family of monotone quantities along the weak inverse mean curvature flow is the limit case of the corresponding ones along the level sets of $p$-capacitary potentials. Such monotone quantities include Willmore and Minkowski-type functionals on Riemannian manifolds with nonnegative Ricci curvature. In $3$-dimensional manifolds with nonnegative scalar curvature, we also recover the monotonicity of the Hawking mass and its nonlinear potential theoretic counterparts. This unified view is built on a refined analysis of $p$-capacitary potentials. We prove that they strongly converge in $W^{1,q}_{\mathrm{loc}}$ as $p\to 1^+$ to the inverse mean curvature flow and their level sets are curvature varifolds. Finally, we also deduce a Gauss-Bonnet-type theorem for level sets of $p$-capacitary potentials.

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↗
arXiv2024

Second-order estimates for the $p$-Laplacian in RCD spaces

We establish quantitative second-order Sobolev regularity for functions having a $2$-integrable $p$-Laplacian in bounded RCD spaces, with $p$ in a suitable range. In the finite-dimensional case, we also obtain Lipschitz regularity under the assumption that $p$-Laplacian is sufficiently integrable. Our results cover both $p$-Laplacian eigenfunctions and $p$-harmonic functions having relatively compact level sets.

math.MG↗
arXiv2023

Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature

We deal with suitable nonlinear versions of Jauregui's isocapacitary mass in 3-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter $1<p\leq 2$, interpolate between Jauregui's mass $p=2$ and Huisken's isoperimetric mass, as $p \to 1^+$. We derive positive mass theorems for these masses under mild conditions at infinity, and we show that these masses do coincide with the ADM mass when the latter is defined. We finally work out a nonlinear potential theoretic proof of the Penrose inequality in the optimal asymptotic regime.

math.DG↗
arXiv2022

On the Isoperimetric Riemannian Penrose Inequality

We prove that the Riemannian Penrose Inequality holds for Asymptotically Flat $3$-manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the $\mathrm{ADM}$ mass being a well-defined geometric invariant. Our proof builds on a novel interplay between the Hawking mass and a potential-theoretic version of it, recently introduced by Agostiniani, Oronzio and the third named author. As a consequence, we establish the equality between $\mathrm{ADM}$ mass and Huisken's Isoperimetric mass under the above sharp assumptions. Moreover, we establish a Riemannian Penrose Inequality in terms of the Isoperimetric mass on any $3$-manifold with nonnegative scalar curvature, connected horizon boundary, and which supports a well-posed notion of weak Inverse Mean Curvature Flow. In particular, such Isoperimetric Riemannian Penrose Inequality does not require the asymptotic flatness of the manifold. The argument is based on a new asymptotic comparison result involving Huisken's Isoperimetric mass and the Hawking mass.

math.DG↗
arXiv2021

Minkowski Inequality on complete Riemannian manifolds with nonnegative Ricci curvature

In this paper we consider Riemannian manifolds of dimension at least $3$, with nonnegative Ricci curvature and Euclidean Volume Growth. For every open bounded subset with smooth boundary we establish the validity of an optimal Minkowski Inequality. We also characterise the equality case, provided the domain is strictly outward minimising and strictly mean convex. Along with the proof, we establish in full generality sharp monotonicity formulas, holding along the level sets of $p$-capacitary potentials in $p$-nonparabolic manifolds with nonnegative Ricci curvature.

math.DG↗