arXiv subjects

Marco Magliaro

Publications and source records attributed to Marco Magliaro.

9 recordsLinked to original sources

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

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 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

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

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↗

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.

math.DG↗

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.

math.DG↗

arXiv2010

Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group

The aim of this paper is to study the qualitative behaviour of non-negative entire solutions of certain differential inequalities involving gradient terms on the Heisenberg group. We focus our investigation on the two classes of inequalities of the form $Δ^ϕu \ge f(u)l(|\nabla u|)$ and $Δ^ϕu \ge f(u) - h(u) g(|\nabla u|)$, where $f,l,h,g$ are non-negative continuous functions satisfying certain monotonicity properties. The operator $Δ^ϕ$, called the $ϕ$-Laplacian, can be viewed as a natural generalization of the $p$-Laplace operator recently considered by various authors in this setting. We prove some Liouville theorems introducing two new Keller-Osserman type conditions, both extending the classical one which appeared long ago in the study of the prototype differential inequality $Δu \ge f(u)$ in $\erre^m$. Furthermore, we show sharpness of our conditions when we specialize to the case of the $p$-Laplacian. Needless to say, our results continue to hold, with the obvious minor modifications, also in the Euclidean space.

math.DG↗

arXiv2010

On the geometry of curves and conformal geodesics in the Mobius space

This paper deals with the study of some properties of immersed curves in the conformal sphere $\mathds{Q}_n$, viewed as a homogeneous space under the action of the M\"obius group. After an overview on general well-known facts, we briefly focus on the links between Euclidean and conformal curvatures, in the spirit of F. Klein's Erlangen program. The core of the paper is the study of conformal geodesics, defined as the critical points of the conformal arclength functional. After writing down their Euler-Lagrange equations for any $n$, we prove an interesting codimension reduction, namely that every conformal geodesic in $\mathds{Q}_n$ lies, in fact, in a totally umbilical 4-sphere $\mathds{Q}_4$. We then extend and complete the work in (Musso, "The conformal arclength functional", Math Nachr.) by solving the Euler-Lagrange equations for the curvatures and by providing an explicit expression even for those conformal geodesics not included in any conformal 3-sphere.

math.DG↗