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Andrea Tamburelli

Publications and source records attributed to Andrea Tamburelli.

At least 19 recordsLinked to original sources
arXiv2026

Global comparison of pseudo-Kähler structures on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component

We compare the closed $2$-form $ω_f$ constructed by Rungi-Tamburelli and Goldman's symplectic form $ω_G$ on the $\mathrm{SL}(3,\mathbb R)$-Hitchin component. In particular, we establish that the semi-pseudo-Kähler structure on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component defined by Rungi-Tamburelli is non-degenerate everywhere and, after aligning the normalization of the Killing form on $\mathfrak{sl}(3,\mathbb{R})$, coincides with the one recently found by Collier-Toulisse-Wentworth.

math.DG↗
arXiv2026

Limits of Cubic Differentials and Buildings

In the Labourie-Loftin parametrization of the Hitchin component of surface group representations into SL(3,R), we prove an asymptotic formula for holonomy along rays in terms of local invariants of the holomorphic differential defining that ray. Globally, we show that the corresponding family of equivariant harmonic maps to a symmetric space converge to a harmonic map into the asymptotic cone of that space. The geometry of the image may also be described by that differential: it is weakly convex and a (one-third) translation surface. We define a compactification of the Hitchin component in this setting for triangle groups that respects the parametrization by Hitchin differentials.

math.DG↗
arXiv2025

Para-complex geometry and cyclic Higgs bundles

We introduce para-complex and pseudo-Riemannian geometric methods for the study of representations of surface groups in $\mathrm{SL}(2m+1,\mathbb{R})$. For $m=1$ our techniques allow to recover several known results for Hitchin representations without any reference to convex projective geometry or hyperbolic affine spheres. In particular, we describe analytically the Guichard-Wienhard domain of discontinuity in the flag variety and the corresponding concave foliated flag structure of Nolte-Riestenberg. In higher rank, we obtain a one-to-one correspondence between stable cyclic $\mathrm{SL}(2m+1,\mathbb{R})$-Higgs bundles (not necessarily in the Hitchin component) and a special class of surfaces, which we call isotropic $\mathbf{P}$-alternating, in the para-complex hyperbolic space $\mathbb{H}^{2m}_τ$. As a result, we give a geometric interpretation to the holomorphic differential $q_{2m+1}$ in the Hitchin base in terms of harmonic sequences for immersions in para-complex manifolds.

math.DG↗
arXiv2024

Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations

In this paper we introduce complex minimal Lagrangian surfaces in the bi-complex hyperbolic space and study their relation with representations in $\mathrm{SL}(3,\mathbb{C})$. Our theory generalizes at the same time minimal Lagrangian surfaces in the complex hyperbolic plane, hyperbolic affine spheres in $\mathbb{R}^3$, and Bers embeddings in the holomorphic space form $\mathbb{CP}^1 \times \mathbb{CP}^1 \setminus Δ$. If these surfaces are equivariant under representations in $\mathrm{SL}(3,\mathbb{C})$, our approach generalizes the study of almost $\mathbb{R}$-Fuchsian representations in $\mathrm{SU}(2,1)$, Hitchin representations in $\mathrm{SL}(3,\mathbb{R})$, and quasi-Fuchsian representations in $\mathrm{SL}(2,\mathbb{C})$. Moreover, we give a parameterization of $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations by an open set in the product of two copies of the bundle of holomorphic cubic differentials over the Teichmüller space of $S$, from which we deduce that this space of representations is endowed with a bi-complex structure. In the process, we introduce bi-complex Higgs bundles as a new tool for studying representations into semisimple complex Lie groups.

math.DG↗
arXiv2024

Pseudo-Kähler structure on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component and Goldman symplectic form

The aim of this paper is to show the existence and give an explicit description of a pseudo-Riemannian metric and a symplectic form on the $\mathrm{S}\mathrm{L}(3,\mathbb{R})$-Hitchin component, both compatible with Labourie and Loftin's complex structure. In particular, they give rise to a mapping class group invariant pseudo-Kähler structure on a neighborhood of the Fuchsian locus, which restricts to a multiple of the Weil-Petersson metric on Teichmüller space. By comparing our symplectic form with Goldman's $\boldsymbolω_G$, we prove that the pair $(\boldsymbolω_G, \mathbf{I})$ cannot define a Kähler structure on the Hitchin component.

math.DG↗
arXiv2023

The moduli space of flat maximal space-like embeddings in pseudo-hyperbolic space

We study the moduli space of flat maximal space-like embeddings in $\mathbb{H}^{2,2}$ from various aspects. We first describe the associated Codazzi tensors to the embedding in the general setting, and then, we introduce a family of pseudo-Kähler metrics on the moduli space. We show the existence of two Hamiltonian actions with associated moment maps and use them to find a geometric global Darboux frame for any symplectic form in the above family.

math.DG↗
arXiv2022

Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R}\mathbb{P}^2$-structures in genus one

In this paper we study a broad class of complete Hamiltonian integrable systems, namely the ones whose associated Lagrangian fibration is complete and has non compact fibres. By studying the associated complete Lagrangian fibration, we show that, under suitable assumptions, the integrals of motion can be taken as action coordinates for the Hamiltonian system. As an application we find global Darboux coordinates for a new family of symplectic forms $\boldsymbolω_f$, parametrized by smooth functions $f:[0,+\infty)\to(-\infty,0]$, defined on the deformation space of properly convex $\mathbb{R}\mathbb{P}^2$-structures on the torus. Such a symplectic form is part of a family of pseudo-Kähler metrics $(\mathbf{g}_f,\mathbf{I},\boldsymbolω_f)$ defined on $\mathcal{B}_0(T^2)$ and introduced by the authors. In the last part of the paper, by choosing $f(t)=-kt, k>0$ we deduce the expression for an arbitrary isometry of the space.

math.SG↗
arXiv2021

A closed ball compactification of a maximal component via cores of trees

We show that, in the character variety of surface group representations into the Lie group $\mathrm{PSL}(2,\mathbb{R}) \times \mathrm{PSL}(2,\mathbb{R})$, the compactification of the maximal component introduced by the second author is a closed ball upon which the mapping class group acts. We study the dynamics of this action. Finally, we describe the boundary points geometrically as $(\overline{A_{1} \times A_{1}},2)$-valued mixed structures.

math.GT↗
arXiv2021

Pseudo-Kähler geometry of properly convex projective structures on the torus

In this paper we prove the existence of a pseudo-Kähler structure on the deformation space $\mathcal{B}_0(T^2)$ of properly convex $\mathbb R\mathbb P^2$-structures over the torus. In particular, the pseudo-Riemannian metric and the symplectic form are compatible with the complex structure inherited from the identification of $\mathcal{B}_0(T^2)$ with the complement of the zero section of the total space of the bundle of cubic holomorphic differentials over the Teichmüller space. We show that the $S^1$-action on $\mathcal{B}_0(T^2)$, given by rotation of the fibers, is Hamiltonian and it preserves both the metric and the symplectic form. Finally, we prove the existence of a moment map for the $\mathrm{SL}(2,\mathbb R)$-action over $\mathcal{B}_0(T^2)$.

math.DG↗
arXiv2021

Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds

In this paper we study the para-hyperKähler geometry of the deformation space of MGHC anti-de Sitter structures on $Σ\times\mathbb R$, for $Σ$ a closed oriented surface. We show that a neutral pseudo-Riemannian metric and three symplectic structures coexist with an integrable complex structure and two para-complex structures, satisfying the relations of para-quaternionic numbers. We show that these structures are directly related to the geometry of MGHC manifolds, via the Mess homeomorphism, the parameterization of Krasnov-Schlenker by the induced metric on $K$-surfaces, the identification with the cotangent bundle $T^*\mathcal{T}(Σ)$, and the circle action that arises from this identification. Finally, we study the relation to the natural para-complex geometry that the space inherits from being a component of the $\mathrm{PSL}(2,\mathbb{B})$-character variety, where $\mathbb{B}$ is the algebra of para-complex numbers, and the symplectic geometry deriving from Goldman symplectic form.

math.DG↗
arXiv2021

Boundary of the Gothen components

In this short note we describe an interesting new phenomenon about the $\mathrm{Sp}(4,\mathbb{R})$-character variety. Precisely, we show that the Hitchin component and all Gothen components share the same boundary in our length spectrum compactification.

math.DG↗
arXiv2020

Length spectrum compactification of the $\mathrm{SO}_{0}(2,3)$-Hitchin component

We find a compactification of the $\mathrm{SO}_{0}(2,3)$-Hitchin component by studying the degeneration of the induced metric on the unique equivariant maximal surface in the 4-dimensional pseudo-hyperbolic space $\mathbb{H}^{2,2}$. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic quartic differentials on a Riemann surface. As an application, we describe the behavior of the entropy of Hitchin representations along rays of quartic differentials.

math.DG↗
arXiv2020

Limits of Blaschke metrics

We find a compactification of the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component by studying the degeneration of the Blaschke metrics on the associated equivariant affine spheres. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic cubic differentials on a Riemann surface.

math.DG↗
arXiv2020

Riemannian metrics on the moduli space of GHMC anti-de Sitter structures

In this short note we explain how to adapt the construction of two Riemannian metrics on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component to the deformation space of globally hyperbolic anti-de Sitter structures: the pressure metric and the Loftin metric (studied by Qiongling Li). We show that the former is degenerate and we characterize its degenerate locus, whereas the latter is nowhere degenerate and the Fuchsian locus is a totally geodesic copy of Teichmüller space endowed with a multiple of the Weil-Petersson metric.

math.DG↗
arXiv2020

Planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4, \mathbb{R})$-symmetric space

We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4,\mathbb{R})$-symmetric space. We describe a homeomomorphism between the "Hitchin component" of wild $\mathrm{Sp}(4,\mathbb{R})$-Higgs bundles over $\mathbb{CP}^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in $\mathbb{H}^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of $\mathbb{R}^{4}$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in $\mathbb{H}^{2,2}$ associated to $\mathrm{Sp}(4,\mathbb{R})$-Hitchin representations along rays of holomorphic quartic differentials.

math.DG↗
arXiv2020

Degeneration of globally hyperbolic maximal anti-de Sitter structures along rays

Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle of the Teichmüller space of $S$, we study how some geometric quantities, such as the Lorentzian Hausdorff dimension of the limit set, the width of the convex core and the Hölder exponent, degenerate along rays of quadratic differentials.

math.DG↗
arXiv2019

Fenchel-Nielsen coordinates on the augmented moduli space of anti-de Sitter structures

In this paper we combine our recent work on regular globally hyperbolic maximal anti-de Sitter structures with the classical theory of globally hyperbolic maximal Cauchy-compact anti-de Sitter manifolds in order to define an augmented moduli space. Moreover, we introduce a coordinate system in this space that resembles the complex Fenchel-Nielsen coordinates for hyperbolic quasi-Fuchsian manifolds.

math.GT↗