Searcharxiv⌕ Search

arXiv subjects

Rabah Souam

Publications and source records attributed to Rabah Souam.

16 recordsLinked to original sources
arXiv2022

On stable capillary hypersurfaces with planar boundaries

We study stable immersed capillary hypersurfaces $Σ$ in domains B of R n+1 bounded by hyperplanes. When B is a half-space, we show $Σ$ is a spherical cap. When B is a domain bounded by k hyperplanes P 1 ,. .. , P k , 2 $\le$ k $\le$ n + 1, having independent normals, and $Σ$ has contact angle $θ$ i with P i and does not touch the vertices of B, we prove there exists $δ$ > 0, depending only on P 1 ,. .. , P k , so that if $θ$ i $\in$ ($π$ 2 -- $δ$, $π$ 2 + $δ$) for each i, then $Σ$ has to be a piece of a sphere.

math.DG↗
arXiv2019

Mean curvature rigidity of horospheres, hyperspheres and hyperplanes Rabah Souam

We prove that horospheres, hyperspheres and hyperplanes in a hyperbolic space H n , n $\ge$ 3, admit no perturbations with compact support which increase their mean curvature. is is an extension of the analogous result in the Euclidean spaces, due to M. Gromov, which states that a hyperplane in a Euclidean space R n admits no compactly supported perturbations having mean curvature $\ge$ 0.

math.DG↗
arXiv2019

Stable constant mean curvature surfaces with free boundary in slabs

We study stable constant mean curvature (CMC) hypersurfaces $Σ$ in slabs in a product space $M\times\r,$ where $M$ is an orientable Riemannian manifold. We obtain a characterization of stable cylinders and prove that if $Σ$ is not a cylinder then it is locally a vertical graph. Moreover, in case $M$ is $\h^n,\r^n$ or $\s_+^n$ and each of its boundary components is embedded then $Σ$ is rotationally invariant. When $M$ has dimension 2 and Gaussian curvature bounded from below by a positive constant $κ,$ we prove there is no stable CMC with free boundary connecting the boundary components of a slab of width $l>4π/\sqrt{3κ}.$ We also show that a stable capillary surface of genus 0 in a warped product $[0,l]\times_f M$ where $M=\r^2, \h^2$ or $\s^2,$ is rotationally invariant. Finally, we prove that a stable closed CMC surface in $M\times\s^1(r),$ where $M$ is a surface with Gaussian curvature bounded from below by a positive constant $κ$ and $\s^1(r)$ the circle of radius $r,$ lifts to $M\times\r$ provided $r>4/\sqrt{3κ}.$

math.DG↗
arXiv2015

Stable CMC and index one minimal surfaces in conformally flat manifolds

Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $Σ$ be a compact connected and orientable surface immersed in $M$ which is a stable constant mean curvature (CMC) surface or an index one minimal surface. We prove that $Σ$ is homeomorphic either to a sphere or to a torus. Moreover, in case $Σ$ is homeomorphic to a torus, then it is embedded, minimal, conformal to a flat square torus and Ric$(N)=0$ where $N$ is a unit field normal to $Σ.$ The result is sharp, we can perturb the standard metric on the 3-sphere in its conformal class to obtain metrics of nonnegative Ricci curvature admitting minimal tori which are stable as CMC surfaces. As a consequence, in any 3-sphere of positive Ricci curvature which is conformally flat, the isoperimetric domains are topologically 3-balls. This proves a special case of a conjecture of A. Ros.

math.DG↗
arXiv2015

Stable capillary hypersurfaces in a half-space or a slab

We study stable immersed capillary hypersurfaces in a domain $\mathcal B$ which is either a half-space or a slab in the Euclidean space $\Bbb R^{n+1}.$ We prove that such a hypersurface $Σ$ is rotationally symmetric in the following cases: (1) $n=2$, $\mathcal B$ is a slab and $Σ$ has genus zero, (2) $n\geq 2$, $\mathcal B$ is a slab, the angle of contact is $π/2$ and each component of $\partialΣ$ is embedded, (3) $n\geq 2,$ $\mathcal B$ is a half-space, the angle of contact is $<π/2$ and each component of $\partialΣ$ is embedded. Moreover, in case (2), if not a right circular cylinder, the hypersurface has to be graphical over a domain in $\partial\mathcal B.$ In case (3), the hypersurface is a spherical cap.

math.DG↗
arXiv2014

Capillary surfaces inside polyhedral regions

In this paper we provide a large new family of embedded capillary surfaces inside polyhedral regions in the Euclidean space. The angle of contact of the examples we furnish is prescribed to be any value in $(\fracπ{2}, π]$ and it is allowed to vary from one boundary component to the other.

math.DG↗
arXiv2013

The Minkowski problem, new constant curvature surfaces in R^3, and some applications

Let $m\in\mathbb{N},$ $m\geq 2,$ and let $\{p_j\}_{j=1}^m$ be a finite subset of $\mathbb{S}^2$ such that $0\in\mathbb{R}^3$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the complete family of smooth convex bodies $K$ in $\mathbb{R}^3$ whose boundary surface consists of an open surface $S$ with constant Gauss curvature (respectively, constant mean curvature) and $m$ planar compact discs $\bar{D_1},...,\bar{D_m},$ such that the Gauss map of $S$ is a homeomorphism onto $\mathbb{S}^2-\{p_j\}_{j=1}^m$ and $D_j\bot p_j,$ for all $j.$ We derive applications to the generalized Minkowski problem, existence of harmonic diffeomorphisms between domains of $\mathbb{S}^2,$ existence of capillary surfaces in $\mathbb{R}^3,$ and a Hessian equation of Monge-Ampere type.

math.DG↗
arXiv2013

The classification of totally umbilical surfaces in homogeneous 3-manifolds

We obtain an exhaustive classification of totally umbilical surfaces in unimodular and non-unimodular simply-connected 3-dimensional Lie groups endowed with arbitrary left-invariant Riemannian metrics. This completes the classification of totally umbilical surfaces in homogeneous Riemannian 3-manifolds.

math.DG↗
arXiv2011

Harmonic diffeomorphisms between domains in the Euclidean 2-sphere

We study the existence or not of harmonic diffeomorphisms between certain domains in the Euclidean 2-sphere. In particular, we show harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for maximal graphs in the Lorentzian product $M\times\mathbb{R}_1,$ where $M$ is an arbitrary $n$-dimensional compact Riemannian manifold, $n\geq 2.$ In contrast, we show that there is no harmonic diffeomorphism from the unit complex disc onto the once punctured sphere and no harmonic diffeomeorphisms from finitely punctured spheres onto circular domains in the Euclidean 2-sphere.

math.DG↗
arXiv2010

Totally umbilical hypersurfaces of manifolds admitting a unit Killing field

We prove that a Riemannian product of type M x R (where R denotes the Euclidean line) admits totally umbilical hypersurfaces if and only if M has locally the structure of a warped product and we give a complete description of the totally umbilical hypersurfaces in this case. Moreover, we give a necessary and sufficient condition under which a Riemannian three-manifold carrying a unit Killing field admits totally geodesic surfaces and we study local and global properties of three-manifolds satisfying this condition.

math.DG↗
arXiv2009

General curvature estimates for stable H-surfaces in 3-manifolds and applications

We obtain an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds with bounded sectional curvature. Our estimate depends on the distance to the boundary of the surface and on the bounds on the geometry of the ambient manifold but not on the manifold itself. We give some applications, in particular we obtain an interior gradient estimate for H-sections in Killing submersions.

math.DG↗
arXiv2008

Totally umbilic surfaces in homogeneous 3-manifolds

We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in $H^2 \times R$, $S^2 \times R$ and the Sol group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms.

math.DG↗
arXiv2008

Higher Schl{ä}fli Formulas and Applications II. Vector-valued differential relations

The classical Schläfli formula, and its ``higher'' analogs given in [SS03], are relations between the variations of the volumes and ``curvatures'' of faces of different dimensions of a polyhedra (which can be Euclidean, spherical or hyperbolic) under a first-order deformation. We describe here analogs of those formulas which are vector-valued rather than scalar. Some consequences follow, for instance constraints on where cone singularities can appear when a constant curvature manifold is deformed among cone-manifolds.

math.DG↗
arXiv2005

The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space $ł^3$

We show that a complete embedded maximal surface in the 3-dimensional Lorentz-Minkowski space $L^3$ with a finite number of singularities is, up to a Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a vertical half catenoid or a horizontal plane and with conelike singular points. We study the space $G_n$ of entire maximal graphs over $\{x_3=0\}$ in $L^3$ with $n+1 \geq 2$ conelike singularities and vertical limit normal vector at infinity. We show that $G_n$ is a real analytic manifold of dimension $3n+4,$ and the coordinates are given by the position of the singular points in $R^3$ and the logarithmic growth at the end. We also introduce the moduli space $M_n$ of {\em marked} graphs with $n+1$ singular points (a mark in a graph is an ordering of its singularities), which is a $(n+1)$-sheeted covering of $G_n.$ We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space $M_n$ is an analytic manifold of dimension $3n-1.$

math.DG↗
arXiv2004

The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz-Minkowski space $ł^3$

We show that, up to some natural normalizations, the moduli space of singly periodic complete embedded maximal surfaces in the Lorentz-Minkowski space $ł^3=(\r^3,dx_1^2+dx_2^2-dx_3^2),$ with fundamental piece having a finite number $(n+1)$ of singularities, is a real analytic manifold of dimension $3n+4.$ The underlying topology agrees with the topology of uniform convergence of graphs on compact subsets of $\{x_3=0\}.$

math.DG↗