arXiv subjects

Luciano Mari

Publications and source records attributed to Luciano Mari.

At least 19 recordsLinked to original sources

arXiv2026

Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$

We describe a method to prove new integral inequalities for stable minimal hypersurfaces in Euclidean space. As an application, we give a simple proof that complete, two sided, stable minimal hypersurfaces in $\mathbb{R}^4$ are hyperplanes. A core part of the argument hinges on the fact that stable minimal hypersurfaces in non-negatively curved spaces are examples of manifolds with a spectral Ricci curvature lower bound; in particular, we prove a sharp pointwise gradient estimate for the Green kernel on non-parabolic manifolds with spectral Ricci lower bounds, extending previous work by Colding.

math.DG↗

arXiv2025

Prescribing the mean curvature of an achronal hypersurface as a measure: the case of 3D spacetimes

We study the existence problem for achronal hypersurfaces $M \hookrightarrow \overline{M}$ in a globally hyperbolic spacetime, whose mean curvature is a prescribed -- possibly singular -- source, and whose boundary is a given smooth spacelike submanifold. Since $M$ is allowed to go null somewhere, the mean curvature prescription is to be understood in the distributional sense. We prove a general existence and regularity theorem for surfaces in ambient dimension $3$. Although most of our estimates hold in any dimension, recent counterexamples show that some of our conclusions fail in ambient dimension at least $5$. The case of $4$D-spacetimes is an open problem. Our theorems have application to Born-Infeld electrostatics in general static spacetimes.

math.DG↗

arXiv2025

On the classification of capillary graphs in Euclidean and non-Euclidean spaces

We prove some rigidity and classification results for graphs with prescribed mean curvature and locally constant Dirichlet and Neumann data, for instance as they appear in capillarity problems. We consider domains in Riemannian manifolds, with emphasis on $\mathbb{R}^2$ and $\mathbb{R}^3$. We classify both the underlying domain and the resulting solution, providing general splitting theorems in this setting.

math.DG↗

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↗

arXiv2025

Harmonic maps from $S^3$ to $S^2$ and the rigidity of the Hopf fibration

It was conjectured by Eells that the only harmonic maps $f : S^3 \to S^2$ are Hopf fibrations composed with conformal maps of $S^2$. We support this conjecture by proving its validity under suitable conditions on the Hessian and the singular values of $f$. Among the results, we obtain a pinching theorem in the spirit of that of Simons, Lawson and Chern, do Carmo and Kobayashi for minimal hypersurfaces in the sphere.

math.DG↗

arXiv2025

On linking numbers and Biot-Savart kernels

On a compact manifold, the solutions of the initial value problem for the heat equation with currential initial conditions are smooth families of forms for $t>0$. If the initial condition is an exact submanifold $L$ then the integral in $t$ of this family gives a smooth form $Ω$ on the complement of $L$ such that $ω:=d^*Ω$ is a solution for the exterior derivative equation $dω=L$. We introduce, for small $t$, an asymptotic approximation of these solutions in order to show that $d^*Ω$ is extendible to the oriented blow-up of $L$ in codimension $1$ and $3$ and also $2$ when $L$ is minimal. When $L$ is the diagonal in $M\times M$ we adapt these ideas to obtain a differential linking form for any compact, ambient Riemannian manifold $M$ of dimension $3$. This coincides up to sign with the kernel of the Biot-Savart operator $d^*G$ and recovers the well-known Gauss formula for linking numbers in $\mathbb{R}^3$.

math.DG↗

arXiv2025

On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature

In this paper, we study the principal eigenvalue $μ(\mathscr{F}_k^-,E)$ of the fully nonlinear operator \[ \mathscr{F}_k^-[u] = \mathcal{P}_k^-(\nabla^2 u) - h |\nabla u| \] on a set $E \Subset \mathbb{R}^n$, where $h \in [0,\infty)$ and $\mathcal{P}_k^-(\nabla^2 u)$ is the sum of the smallest $k$ eigenvalues of the Hessian $\nabla^2 u$. We prove a lower estimate for $μ(\mathscr{F}_k^-,E)$ in terms of a generalized Hausdorff measure $\mathscr{H}_Ψ(E)$, for suitable $Ψ$ depending on $k$, moving some steps in the direction of the conjecturally sharp estimate \[ μ(\mathscr{F}_k^-,E) \ge C \mathscr{H}^k(E)^{-2/k}. \] The theorem is used to study the spectrum of bounded submanifolds in $\mathbb{R}^n$, improving on our previous work in the direction of a question posed by S.T. Yau. In particular, the result applies to solutions of Plateau's problem for CMC surfaces.

math.AP↗

arXiv2025

Spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum

We investigate the spectrum of the Laplacian on complete, non-compact manifolds $M^n$ whose Ricci curvature satisfies $\mathrm{Ric} \geq -(n-1)\mathrm{H}(r)$, for some continuous, non-increasing $\mathrm{H}$ with $\mathrm{H}-1 \in L^1(\infty)$. We prove that if the bottom spectrum attains the maximal value $\frac{(n-1)^2}{4}$ compatible with the curvature bound, then the spectrum of $M$ coincides with that of hyperbolic space $\mathbb{H}^n$, namely, $σ(M) = \left[ \frac{(n-1)^2}{4}, \infty \right)$. The result can be localized to an end $E$ with infinite volume.

math.DG↗

arXiv2025

On mean curvature flow solitons in the sphere

In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere $S^{2n+1}$ moving along the integral curves of the Hopf unit vector field. While such solitons must necessarily be minimal if compact, we produce a non-minimal, complete example with topology $S^{2n-1} \times R$. The example wraps around a Clifford torus $S^{2n-1} \times S^1$ along each end, it has reflection and rotational symmetry and its mean curvature changes sign on each end. Indeed, we prove that a complete 2-dimensional soliton with non-negative mean curvature outside a compact set must be a covering of a Clifford torus. Concluding, we obtain a pinching theorem under suitable conditions on the second fundamental form.

math.DG↗

arXiv2024

Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces

In this paper we prove general criticality criteria for operators $\Delta + V$ on manifolds with more than one end, where $V$ bounds the Ricci curvature, and a related spectral splitting theorem extending Cheeger-Gromoll's one. Our results give new insight on Li-Wang's theory of manifolds with a weighted Poincar\'e inequality. We apply them to study stable and $\delta$-stable minimal hypersurfaces in manifolds with non-negative bi-Ricci or sectional curvature, in ambient dimension up to $5$ and $6$, respectively. In the special case where the ambient space is $\mathbb{R}^4$, we prove that a $1/3$-stable minimal hypersurface must either have one end or be a catenoid, and that proper, $\delta$-stable minimal hypersurfaces with $\delta > 1/3$ must be hyperplanes.

math.DG↗

arXiv2024

Some splitting and rigidity results for sub-static spaces

In this paper we study the rigidity problem for sub-static systems with possibly non-empty boundary. First, we get local and global splitting theorems by assuming the existence of suitable compact minimal hypersurfaces, complementing recent results in the literature. Next, we prove some boundary integral inequalities that extend works by Chr\'usciel and Boucher-Gibbons-Horowitz to non-vacuum spaces. Even in the vacuum static case, the inequalities improve on known ones. Lastly, we consider the system arising from static solutions to the Einstein field equations coupled with a $\sigma$-model. The Liouville theorem we obtain allows for positively curved target manifolds, generalizing a result by Reiris.

math.DG↗

arXiv2024

Compactness via monotonicity in nonsmooth critical point theory, with application to Born-Infeld type equations

In this paper, we prove new existence and multiplicity results for critical points of lower semicontinuous functionals in Banach spaces, complementing the nonsmooth critical point theory set forth by Szulkin and avoiding the need of the Palais-Smale condition. We apply our abstract results to get entire solutions with finite energy to Born-Infeld type autonomous equations. More precisely, under almost optimal conditions on the nonlinearity, we construct a positive solution and infinitely many solutions both in the classes of radially symmetric functions and nonradiallly symmetric ones.

math.AP↗

arXiv2024

Sharp pinching theorems for complete submanifolds in the sphere

We prove that every complete, minimally immersed submanifold $f\: M^n \to \mathbb{S}^{n+p}$ whose second fundamental form satisfies $|A|^2 \le np/(2p-1)$, is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface in $\mathbb{S}^4$, thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete $M^n$. We also obtain the corresponding result for complete hypersurfaces with nonvanishing constant mean curvature, due to Alencar & do Carmo in the compact case, under the optimal bound on the umbilicity tensor. In dimension $n \le 6$, a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work of Santos. Our approach is inspired by the conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni.

math.DG↗

arXiv2023

Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born-Infeld model

Given a measure $ρ$ on a domain $Ω\subset \mathbb{R}^m$, we study spacelike graphs over $Ω$ in Minkowski space with Lorentzian mean curvature $ρ$ and Dirichlet boundary condition on $\partial Ω$. The graph function $u_ρ: Ω\rightarrow \mathbb{R}$ also represents the electric potential generated by a charge $ρ$ in electrostatic Born-Infeld theory. While $u_ρ$ minimizes the action $$ I_ρ(ψ) = \int_Ω \Big( 1 - \sqrt{1-|Dψ|^2} \Big) \mathrm{d} x - \langle ρ, ψ\rangle $$ among competitors with $|Dψ| \le 1$, because of a lack of smoothness of the Lagrangian density when $|Dψ| = 1$ a direct approach via minimization may not produce a solution to the Euler-Lagrange equation (BI). In this paper, we study existence and regularity of $u_ρ$ for general $ρ$, in a bounded domain and in the entire $\mathbb{R}^m$. In particular, we find sufficient conditions to guarantee that $u_ρ$ solves (BI) and enjoys log-improved $W^{2,2}_{\mathrm{loc}}$ estimates, and we construct examples helping to identify sharp thresholds for the regularity of $ρ$ to ensure the validity of (BI). One of the main difficulties is the possible presence of light segments in the graph of $u_ρ$, which will be discussed in detail.

math.AP↗

arXiv2023

On minimal graphs of sublinear growth over manifolds with non-negative Ricci curvature

We prove that entire solutions of the minimal hypersurface equation \[ \mathrm{div}\left(\frac{Du}{\sqrt{1+|Du|^2}}\right) = 0 \] on a complete manifold with $\mathrm{Ric} \ge 0$, whose negative part grows like $\mathcal{O}(r/\log r)$ ($r$ the distance from a fixed origin), are constant. This extends the Bernstein Theorem for entire positive minimal graphs established in recent years. The proof depends on a new technique to get gradient bounds by means of integral estimates, which does not require any further geometric assumption on $M$.

math.DG↗

arXiv2023

On splitting complete manifolds via infinity harmonic functions

In this paper, we prove some splitting results for manifolds supporting a non-constant infinity harmonic function which has at most linear growth on one side. Manifolds with non-negative Ricci or sectional curvature are considered. In dimension 2, we extend Savin's theorem on Lipschitz infinity harmonic functions in the plane to every surface with non-negative sectional curvature.

math.DG↗

arXiv2023

Conformal solitons for the mean curvature flow in hyperbolic space

In this paper we study conformal solitons for the mean curvature flow in hyperbolic space $\mathbb{H}^{n+1}$. Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field $-\partial_0$. We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability, and address the case of noncompact boundaries contained between two parallel hyperplanes of $\partial_{\infty}\mathbb{H}^{n+1}$. We conclude by proving rigidity results for bowl and grim-reaper cylinders.

math.DG↗

arXiv2023

On the $1/H$-flow by $p$-Laplace approximation: new estimates via fake distances under Ricci lower bounds

In this paper we show the existence of weak solutions $w : M \rightarrow \mathbb{R}$ of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of $w$ and for the mean curvature of its level sets, that are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the $p$-Laplace equation, and relies on new gradient and decay estimates for $p$-harmonic capacity potentials, notably for the kernel $\mathcal{G}_p$ of $Δ_p$. These bounds, stable as $p \rightarrow 1$, are achieved by studying fake distances associated to capacity potentials and Green kernels. We conclude by investigating some basic isoperimetric properties of the level sets of $w$.

math.DG↗