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Carlo Mantegazza

Publications and source records attributed to Carlo Mantegazza.

At least 19 recordsLinked to original sources
arXiv2026

Short-time existence for the Schouten flow

Let $(M^n,g_0)$ be a closed smooth Riemannian manifold, with $n\geqslant 3$. We prove short--time existence and uniqueness for the "critical" {\em Ricci--Bourguignon flow} \begin{equation} \partial_t g=-2\operatorname{Ric}_g+\frac{R_g}{n-1}g\,. \end{equation} Up to a constant rescaling of time, this is the {\em Schouten flow}, since the right--hand side of the equation is $-2(n-2)$ times the Schouten tensor. At the critical parameter, however, the principal symbol of the linearization of the operator that we obtain after the usual "DeTurck modification" still has a zero eigenvalue. We resolve this degeneracy by adjoining an independent scalar unknown $r$, intended to represent the scalar curvature, and considering an "extended" system consisting of a strictly parabolic equation for the metric coupled to a transport--reaction equation for $r$ with a curvature--square forcing. The extended system belongs to the class of nonlinear composite parabolic--hyperbolic systems studied by Vol'pert and Khudyaev~\cite{VolpertKhudyaev1972}. Then, a standard compact--manifold adaptation of their theorem yields a unique smooth solution of the extended system. Finally, the difference $R_g-r$ satisfies a homogeneous linear parabolic equation, so the scalar--curvature constraint "propagates" and the original critical Ricci--Bourguignon flow is recovered. This settles the short--time existence and uniqueness problem at the critical ``Schouten'' value $ρ=1/(2(n-1))$, which was left open by the previous Ricci--Bourguignon theory in~\cite{CatinoEtAl2017}.

math.DG↗
arXiv2026

Quasiconvexity in the Riemannian setting

We introduce a notion of quasiconvexity for continuous functions $f$ defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold $(M,g)$ and $\mathbb{R}^m$, naturally generalizing the classical Euclidean definition. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functional \[ F(u, Ω) = \int_Ω f(du) \, dμ\] with respect to the weak$^*$ topology of $W^{1,\infty}(Ω, \mathbb{R}^m)$, for every bounded open subset $Ω\subseteq M$.

math.AP↗
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↗
arXiv2026

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-Körber 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↗
arXiv2025

Stability for the Surface Diffusion Flow

We study the global existence and stability of surface diffusion flow (the normal velocity is given by the Laplacian of the mean curvature) of smooth boundaries of subsets of the $n$--dimensional flat torus. More precisely, we show that if a smooth set is ``close enough'' to a strictly stable critical set for the Area functional under a volume constraint, then the surface diffusion flow of its boundary hypersurface exists for all time and asymptotically converges to the boundary of a ``translated'' of the critical set. This result was obtained in dimension $n=3$ by Acerbi, Fusco, Julin and Morini (extending previous results for spheres of Escher, Mayer and Simonett and Elliott and Garcke in dimension $n=2$). Our work generalizes such conclusion to any dimension $n\in\mathbb N$. For sake of clarity, we show all the details in dimension $n=4$ and we list the necessary modifications to the quantities involved in the proof in the general $n$--dimensional case, in the last section.

math.AP↗
arXiv2025

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↗
arXiv2024

Uniform Sobolev, interpolation and geometric Calderón-Zygmund inequalities for graph hypersurfaces

In this note, our aim is to show that families of smooth hypersurfaces of $\mathbb R^{n+1}$ which are all $C^1$--close enough to a fixed compact, embedded one, have uniformly bounded constants in some relevant inequalities for mathematical analysis, like Sobolev, Gagliardo-Nirenberg and ``geometric'' Calderón-Zygmund inequalities. This technical result is quite useful, in particular, in the study of the geometric flows of hypersurfaces.

math.DG↗
arXiv2023

A survey of the elastic flow of curves and networks

We collect and present in a unified way several results in recent years about the elastic flow of curves and networks, trying to draw the state of the art of the subject. In particular, we give a complete proof of global existence and smooth convergence to critical points of the solution of the elastic flow of closed curves in $\mathbb{R}^2$. In the last section of the paper we also discuss a list of open problems.

math.AP↗
arXiv2022

Global Existence and Stability for the Modified Mullins-Sekerka and Surface Diffusion Flow

In this survey we present the state of the art about the asymptotic behavior and stability of the modified Mullins--Sekerka flow and the surface diffusion flow of smooth sets, mainly due to E.~Acerbi, N.~Fusco, V.Julin and M.Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the strict stability property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $W^{2,p}$-perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently "close" to a smooth strictly stable critical set $E$, both flows exist for all positive times and asymptotically "converge" to a translate of $E$.

math.AP↗
arXiv2022

Type-0 singularities in the network flow -- Evolution of trees

The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a Type-0 singularity, in contrast to the well known Type-I and Type-II ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree-like networks till the first singular time, under the assumption that all the tangents flows have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounded and we can apply the results by Ilmanen-Neves-Schulze/Lira-Mazzeo-Pluda-Saez to restart the flow after the singularity.

math.DG↗
arXiv2022

Semilinear Li & Yau inequalities

We derive an adaptation of Li & Yau estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We then apply these estimates to obtain a Harnack inequality and to discuss monotonicity, convexity, decay estimates and triviality of ancient and eternal solutions.

math.AP↗
arXiv2021

Asymptotic convergence of evolving hypersurfaces

If $ψ:M^n\to \mathbb{R}^{n+1}$ is a smooth immersed closed hypersurface, we consider the functional $\mathcal{F}_m(ψ) = \int_M 1 + |\nabla^m ν|^2 \, dμ$, where $ν$ is a local unit normal vector along $ψ$, $\nabla$ is the Levi-Civita connection of the Riemannian manifold $(M,g)$, with $g$ the pull-back metric induced by the immersion and $μ$ the associated volume measure. We prove that if $m>\lfloor n/2 \rfloor$ then the unique globally defined smooth solution to the $L^2$-gradient flow of $\mathcal{F}_m$, for every initial hypersurface, smoothly converges asymptotically to a critical point of $\mathcal{F}_m$, up to diffeomorphisms. The proof is based on the application of a Lojasiewicz-Simon gradient inequality for the functional $\mathcal{F}_m$.

math.DG↗
arXiv2021

A matrix Harnack inequality for semilinear heat equations

We derive a matrix version of Li \& Yau--type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R.~Hamilton did in~\cite{hamilton7} for the standard heat equation. We then apply these estimates to obtain some Harnack--type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.

math.AP↗
arXiv2020

The Lojasiewicz-Simon inequality for the elastic flow

We define the elastic energy of smooth immersed closed curves in $\mathbb{R}^n$ as the sum of the length and the $L^2$-norm of the curvature, with respect to the length measure. We prove that the $L^2$-gradient flow of this energy smoothly converges asymptotically to a critical point. One of our aims was to the present the application of a Lojasiewicz-Simon inequality, which is at the core of the proof, in a quite concise and versatile way.

math.AP↗
arXiv2020

A Triviality Result for Semilinear Parabolic Equations

We show a triviality result for "pointwise" monotone in time, bounded "eternal" solutions of the semilinear heat equation \begin{equation*} u_{t}=Δu + |u|^{p} \end{equation*} on complete Riemannian manifolds of dimension $n \geq 5$ with nonnegative Ricci tensor, when $p$ is smaller than the critical Sobolev exponent $\frac{n+2}{n-2}$.

math.AP↗