arXiv subjects

Farid Diaf

Publications and source records attributed to Farid Diaf.

8 recordsLinked to original sources

arXiv2026

Harmonic extension of Weil-Petersson circle homeomorphisms

In this paper, we study Weil--Petersson circle homeomorphisms from the viewpoint of harmonic maps. We prove that a homeomorphism $\varphi:\mathbb S^1\to\mathbb S^1$ is Weil--Petersson if and only if its unique quasiconformal harmonic extension to the hyperbolic disk $\mathbb D$ has square-integrable Beltrami differential. Our approach is based on the anti-holomorphic $L^2$-energy of harmonic maps. We show that this energy is finite for the quasiconformal harmonic extension of every Weil--Petersson circle homeomorphism, and that, among suitable quasiconformal extensions, the harmonic extension minimizes this energy.

math.DG↗

arXiv2026

Weil--Petersson homeomorphisms, minimal lagrangian diffeomorphisms, and maximal surfaces in anti-de Sitter space

In this paper, we study the class of Weil--Petersson circle homeomorphisms from the point of view of three-dimensional anti-de Sitter space $\mathbf{AdS}^{2,1}$. We show that a homeomorphism $φ:\mathbf{RP}^1\to\mathbf{RP}^1$ is Weil--Petersson if and only if its graph, viewed as a curve in the boundary at infinity of $\mathbf{AdS}^{2,1}$, is the asymptotic boundary of a complete maximal spacelike surface in $\mathbf{AdS}^{2,1}$ with finite renormalized area. As an application, we obtain the following AdS-independent result in Teichmüller theory: a homeomorphism is Weil--Petersson if and only if its minimal lagrangian extension to $\mathbf{H}^2$ has square-integrable Beltrami differential. We also provide two further new technical characterizations, which we believe to be of independent interest, and which are essential for the proofs of our main results.

math.DG↗

arXiv2026

Completeness of closed Kleinian flat Pseudo-Riemannian Manifolds of Signature (2,2)

Let $\mathbb{R}^{2,2}$ denote the model space of flat pseudo-Riemannian manifolds of signature $(2,2)$. We prove that the only domain divisible by a discrete subgroup of the isometry group of $\mathbb{R}^{2,2}$ is $\mathbb{R}^{2,2}$ itself. In the Kleinian setting, this provides the first completeness theorem of closed flat pseudo-Riemannian manifolds beyond the Euclidean and Lorentzian cases. Along the proof, we show two results of independent interest. The first is a geometric reduction for certain divisible domains of affine space. The second concerns the existence of syndetic hulls in semidirect products $R \ltimes G$, where $G$ is a homothety Lie group. This construction generalizes earlier constructions in affine geometry due to Carrière and Dal'bo.

math.DG↗

arXiv2025

Domination between non-Fuchsian representations and anti-de Sitter geometry

Motivated by work of various authors on domination between surface group representations, harmonic maps, and $3$-dimensional anti-de Sitter geometry, we study a new domination problem between non-Fuchsian representations of closed surface groups. We solve the problem for representations that admit branched harmonic immersions, and we show that, outside of this case, the problem cannot always be solved. We then show that a dominating pair gives rise to an anti-de Sitter $3$-manifold with singularities, and we construct large families of branched anti-de Sitter $3$-manifolds.

math.DG↗

arXiv2024

Mean surfaces in Half-Pipe space and infinitesimal Teichmüller theory

We study a correspondence between smooth spacelike surfaces in Half-Pipe space $\mathbb{HP}^3$ and divergence-free vector fields on the hyperbolic plane $\mathbb{H}^2$. We show that a particular case involves harmonic Lagrangian vector fields on $\mathbb{H}^2$, which are related to mean surfaces in $\mathbb{HP}^3$. Consequently, we prove that the infinitesimal Douady-Earle extension is a harmonic Lagrangian vector field that corresponds to a mean surface in $\mathbb{HP}^3$ with prescribed boundary data at infinity. We establish both existence and, under certain assumptions, uniqueness results for harmonic Lagrangian extension of a vector field on the circle. Finally, we characterize the Zygmund and little Zygmund conditions and provide quantitative bounds in terms of the Half-Pipe width.

math.DG↗

arXiv2023

The infinitesimal earthquake theorem for vector fields on the circle

We prove that any continuous vector field on a circle is the extension in a suitable sense, of a unique infinitesimal earthquake of the hyperbolic plane. Furthermore, we obtain other extension results when the vector field is assumed only to be upper or lower semicontinuous. This leads to a generalization of Kerckhoff's and Gardiner's infinitesimal earthquake theorems to a broader setting, using a completely novel approach. The proof is based on the geometry of the dual of Minkowski three-space, also called Half-pipe three-geometry. In this way, we obtain a simple characterization of Zygmund vector fields on the circle in terms of width of convex hulls.

math.GT↗

arXiv2023

Transition of convex core doubles from hyperbolic to Anti-de sitter geometry

Let $Σ$ be a surface of negative Euler characteristic, homeomorphic to a closed surface, possibly with a finite number of points removed. In this paper, we present a construction method for a wide range of examples of geometric transition from hyperbolic to Anti-de Sitter structures via Half-pipe geometry on $Σ\times\mathbb{S}^1$, with cone singularities along a link. The main ingredient lies in studying the deformation of a convex core structure as the bending laminations of the upper and lower boundary components of the convex core uniformly collapse to zero.

math.GT↗

arXiv2023

The Anti-de Sitter proof of Thurston's earthquake theorem

Thurston's earthquake theorem asserts that every orientation-preserving homeomorphism of the circle admits an extension to the hyperbolic plane which is a (left or right) earthquake. The purpose of these notes is to provide a proof of Thurston's earthquake theorem, using the bi-invariant geometry of the Lie group $\mathrm{PSL}(2,\mathbb R)$, which is also called Anti-de Sitter three-space. The involved techniques are elementary, and no background knowledge is assumed apart from some two-dimensional hyperbolic geometry.

math.GT↗