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Luciano Mari

Publications and source records attributed to Luciano Mari.

At least 37 records · Page 2Linked to original sources
arXiv2023

Non-negative Ricci curvature and Minimal graphs with linear growth

We study minimal graphs with linear growth on complete manifolds $M^m$ with $\mathrm{Ric} \ge 0$. Under the further assumption that the $(m-2)$-th Ricci curvature in radial direction is bounded below by $C r(x)^{-2}$, we prove that any such graph, if non-constant, forces tangent cones at infinity of $M$ to split off a line. Note that $M$ is not required to have Euclidean volume growth. We also show that $M$ may not split off any line. Our result parallels that obtained by Cheeger, Colding and Minicozzi for harmonic functions. The core of the paper is a new refinement of Korevaar's gradient estimate for minimal graphs, together with heat equation techniques.

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arXiv2022

Einstein-type structures, Besse's conjecture and a uniqueness result for a $φ$-CPE metric in its conformal class

In this paper, we study an extension of the CPE conjecture to manifolds $M$ which support a structure relating curvature to the geometry of a smooth map $φ: M \to N$. The resulting system, denoted by $(φ-\mathrm{CPE})$, is natural from the variational viewpoint and describes stationary points for the integrated $φ$-scalar curvature functional restricted to metrics with unit volume and constant $φ$-scalar curvature. We prove both a rigidity statement for solutions to $(φ-\mathrm{CPE})$ in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting $(φ-\mathrm{CPE})$ with $φ$ a harmonic map.

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arXiv2022

Bernstein and half-space properties for minimal graphs under Ricci lower bounds

In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold with Ricci curvature bounded from below. In particular, we show that positive, entire minimal graphs on manifolds with non-negative Ricci curvature are constant, and that complete, parabolic manifolds with Ricci curvature bounded from below have the half-space property. We avoid the need of sectional curvature bounds on $M$ by exploiting a form of the Ahlfors-Khas'minskii duality in nonlinear potential theory.

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arXiv2021

Remarks on mean curvature flow solitons in warped products

We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various geometric conditions, ranging from the stability of the soliton to the fact that the image of its Gauss map be contained in suitable regions of the sphere. We also investigate the case of entire graphs.

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arXiv2021

A barrier principle at infinity for varifolds with bounded mean curvature

Our work investigates varifolds $Σ\subset M$ in a Riemannian manifold, with arbitrary codimension and bounded mean curvature, contained in an open domain $Ω$. Under mild assumptions on the curvatures of $M$ and on $\partial Ω$, also allowing for certain singularities of $\partial Ω$, we prove a barrier principle at infinity, namely we show that the distance of $Σ$ to $\partial Ω$ is attained on $\partial Σ$. Our theorem is a consequence of sharp maximum principles at infinity on varifolds, of independent interest.

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arXiv2021

A splitting theorem for capillary graphs under Ricci lower bounds

In this paper, we study capillary graphs defined on a domain $Ω$ of a complete Riemannian manifold $M$, where a graph is said to be capillary if it has constant mean curvature and locally constant Dirichlet and Neumann conditions on $\partial Ω$. Our main result is a splitting theorem both for $Ω$ and for the graph function on a class of manifolds with nonnegative Ricci curvature. As a corollary, we classify capillary graphs over domains that are globally Lipschitz epigraphs or slabs in a product space $M = N \times \mathbb{R}$, where $N$ has slow volume growth and non-negative Ricci curvature, including the case $M = \mathbb{R}^2,\mathbb{R}^3$. A technical core of the paper is a new gradient estimate for positive CMC graphs on manifolds with Ricci lower bounds.

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arXiv2021

Density and non-density of $C^\infty_c \hookrightarrow W^{k,p}$ on complete manifolds with curvature bounds

We investigate the density of compactly supported smooth functions in the Sobolev space $W^{k,p}$ on complete Riemannian manifolds. In the first part of the paper, we extend to the full range $p\in [1,2]$ the most general results known in the Hilbertian case. In particular, we obtain the density under a quadratic Ricci lower bound (when $k=2$) or a suitably controlled growth of the derivatives of the Riemann curvature tensor only up to order $k-3$ (when $k>2$). To this end, we prove a gradient regularity lemma that might be of independent interest. In the second part of the paper, for every $n \ge 2$ and $p>2$ we construct a complete $n$-dimensional manifold with sectional curvature bounded from below by a negative constant, for which the density property in $W^{k,p}$ does not hold for any $k \ge 2$. We also deduce the existence of a counterexample to the validity of the Calderón-Zygmund inequality for $p>2$ when $\mathrm{Sec} \ge 0$, and in the compact setting we show the impossibility to build a Calderón-Zygmund theory for $p>2$ with constants only depending on a bound on the diameter and a lower bound on the sectional curvature.

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arXiv2020

Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations

In recent years, the study of the interplay between (fully) non-linear potential theory and geometry received important new impulse. The purpose of this work is to move a step further in this direction by investigating appropriate versions of parabolicity and maximum principles at infinity for large classes of non-linear (sub)equations $F$ on manifolds. The main goal is to show a unifying duality between such properties and the existence of suitable $F$-subharmonic exhaustions, called Khas'minskii potentials, which is new even for most of the "standard" operators arising from geometry, and improves on partial results in the literature. Applications include new characterizations of the classical maximum principles at infinity (Ekeland, Omori-Yau and their weak versions by Pigola-Rigoli-Setti) and of conservation properties for stochastic processes (martingale completeness). Applications to the theory of submanifolds and Riemannian submersions are also discussed.

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arXiv2020

Spacetime positive mass theorems for initial data sets with noncompact boundary

In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positive mass inequalities under the assumption that the underlying manifold is spin. In the asymptotically flat case, we also prove a rigidity statement when the energy-momentum vector is lightlike. Our treatment aims to underline both the common features and the differences between the asymptotically Euclidean and hyperbolic settings, especially regarding the boundary DECs.

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arXiv2020

Recent rigidity results for graphs with prescribed mean curvature

This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain $Ω\subset M$, namely, for CMC graphs satisfying an overdetermined boundary condition.

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arXiv2020

Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian

In this paper, we study some potential theoretic aspects of the eikonal and infinity Laplace operator on a Finsler manifold $M$. Our main result shows that the forward completeness of $M$ can be detected in terms of Liouville properties and maximum principles at infinity for subsolutions of suitable inequalities, including $Δ^N_\infty u \ge g(u)$. Also, an $\infty$-capacity criterion and a viscosity version of Ekeland principle are proved to be equivalent to the forward completeness of $M$. Part of the proof hinges on a new boundary-to-interior Lipschitz estimate for solutions of $Δ^N_\infty u = g(u)$ on relatively compact sets, that implies a uniform Lipschitz estimate for certain entire, bounded solutions without requiring the completeness of $M$.

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arXiv2018

Maximum principles at infinity and the Ahlfors-Khas'minskii duality: an overview

This note is meant to introduce the reader to a duality principle for nonlinear equations that recently appeared in the literature. Motivations come from the desire to give a unifying potential-theoretic framework for various maximum principles at infinity appearing in the literature (Ekeland, Omori-Yau, Pigola-Rigoli-Setti), as well as to describe their interplay with properties coming from stochastic analysis on manifolds. The duality involves an appropriate version of these principles formulated for viscosity subsolutions of fully nonlinear inequalities, called the Ahlfors property, and the existence of suitable exhaustion functions called Khas'minskii potentials. We discuss applications, also involving the geometry of submanifolds, in the last sections, as well as the stability of these maximum principles when we remove polar sets.

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arXiv2018

On the interplay among maximum principles, compact support principles and Keller-Osserman conditions on manifolds

This paper is about the influence of Geometry on the qualitative behaviour of solutions of quasilinear PDEs on Riemannian manifolds. Motivated by examples arising, among others, from the theory of submanifolds, in particular by the study of entire graphs with prescribed mean curvature, we consider classes of coercive differential inequalities of the form $$ \mathrm{div}\left( \frac{φ(|\nabla u|)}{|\nabla u|} \nabla u\right) \ge b(x)f(u) l(|\nabla u|) \qquad \text{(respectively, $\le$ or $=$)} $$ on domains of a manifold $M$, for suitable $φ,b,f,l$, with emphasis on mean curvature type operators. We investigate the validity of strong maximum principles, compact support principles and Liouville type theorems; in particular, the goal is to identify sharp thresholds, involving curvatures or volume growth of geodesic balls in $M$, to guarantee the above properties under appropriate Keller-Osserman type conditions, and to discuss the geometric reasons behind the existence of such thresholds. The paper also aims to give a unified view of recent results in the literature. The bridge with Geometry is realized by studying the validity of weak and strong maximum principles at infinity, in the spirit of Omori-Yau's Hessian and Laplacian principles and subsequent improvements.

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arXiv2016

Density and spectrum of minimal submanifolds in space forms

Let $M^m$ be a minimal properly immersed submanifold in an ambient space close, in a suitable sense, to the space form $\mathbb{N}^n_k$ of curvature $-k\le 0$. In this paper, we are interested in the relation between the density function $Θ(r)$ of $M^m$ and the spectrum of the Laplace-Beltrami operator. In particular, we prove that if $Θ(r)$ has subexponential growth (when $k<0$) or sub-polynomial growth ($k=0$) along a sequence, then the spectrum of $M^m$ is the same as that of the space form $\mathbb{N}^m_k$. Notably, the result applies to Anderson's (smooth) solutions of Plateau's problem at infinity on the hyperbolic space $\mathbb{H}^n$, independently of their boundary regularity. We also give a simple condition on the second fundamental form that ensures $M$ to have finite density. In particular, we show that minimal submanifolds of $\mathbb{H}^n$ with finite total curvature have finite density.

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arXiv2016

Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold $(M, \langle \, , \, \rangle)$, namely the existence of a conformal deformation of the metric $\langle \, , \, \rangle$ realizing a given function $\widetilde s(x)$ as its scalar curvature. In particular, the work focuses on the case when $\widetilde s(x)$ changes sign. Our main achievement are two new existence results requiring minimal assumptions on the underlying manifold, and ensuring a control on the stretching factor of the conformal deformation in such a way that the conformally deformed metric be quasi-isometric to the original one. The topological-geometrical requirements we need are all encoded in the spectral properties of the standard and conformal Laplacians of $M$. Our techniques can be extended to investigate the existence of entire positive solutions of quasilinear equations of the type $$ Δ_{p} u + a(x)u^{p-1} - b(x)u^σ= 0 $$ where $Δ_p$ is the $p$-Laplacian, $σ>p-1>0$, $a,b \in L^\infty_{\mathrm{loc}}(M)$ and $b$ changes sign, and in the process of collecting the material for the proof of our theorems, we have the opportunity to give some new insight on the subcriticality theory for the Schrödinger type operator $$ Q_V' \ : \ φ\longmapsto -Δ_p φ- a(x)|φ|^{p-2}φ. $$ In particular, we prove sharp Hardy-type inequalities in some geometrically relevant cases, notably for minimal submanifolds of the hyperbolic space.

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arXiv2015

On the role of gradient terms in quasilinear coercive differential inequalities on Carnot Groups

In the sub-Riemannian setting of Carnot groups, this work investigates a-priori estimates and Liouville type theorems for solutions of coercive, quasilinear differential inequalities of the type $$ Δ_{\mathbb{G}}^φu \ge b(x) f(u) l(|\nabla u|). $$ Prototype examples of $Δ_{\mathbb{G}}^φ$ are the (subelliptic) $p$-Laplacian and the mean curvature operator. The main novelty of the present paper is that we allow a dependence on the gradient $l(t)$ that can vanish both as $t \rightarrow 0^+$ and as $t \rightarrow +\infty$. Our results improve on the recent literature and, by means of suitable counterexamples, we show that the range of parameters in the main theorems are sharp.

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arXiv2012

On the spectrum of bounded immersions

In this paper, we investigate the relationship between the discreteness of the spectrum of a non-compact, extrinsically bounded submanifold $φ\colon M^m \ra N^n$ and the Hausdorff dimension of its limit set $\limφ$. In particular, we prove that if $φ\colon \!M^2 \ra D \subseteq \R^3$ is a minimal immersion into an open, bounded, strictly convex subset $D$ with $C^2$-boundary, then $M$ has discrete spectrum provided that $\haus_Ψ(\limφ\cap D)=0$, where $\haus_Ψ$ is the generalized Hausdorff measure of order $Ψ(t) = t^2|\log t|$. Our theorem applies to a number of examples recently constructed by various authors in the light of N. Nadirashvili's discovery of complete, bounded minimal disks in $\R^3$, as well as to solutions of Plateau's problems, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures. Suitable counter-examples show the sharpness of our results: in particular, we develop a simple criterion for the existence of essential spectrum which is suited for the techniques developed after Jorge-Xavier and Nadirashvili's examples.

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