arXiv research

New flows defined on Hitchin components for PGL(V).

Explicit computation of symplectic form for $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component.

The Goldman symplectic form is trivialized for a class of surfaces.

The paper studies conjugating complex representations into real ones.

New findings on cusped Borel Anosov representations and their properties.

Study the topology of spaces of pleated surfaces.

F. Labourie [arXiv:1212.5015] characterized the Hitchin components for $\operatorname{PSL}(n, \mathbb{R})$ for any $n>1$ by using the swapping algebra, where the swapping algebra should be understood as a ring equipped with a Poisson bracket. We introduce the rank $n$ swapping algebra, which is the quotient of the swap…

We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of $L$-twisted $G$-Higgs bundles, for the groups $G = GL(2,\mathbb{C})$, $SL(2,\mathbb{C})$ and $PSL(2,\mathbb{C})$. We also determine the twisted Chern class of the regula…

Given a closed, oriented surface X of genus g>1, and a semisimple Lie group G, let R_G be the moduli space of reductive representations of the fundamental group of X in G. We determine the number of connected components of R_PGL(n,R), for n>=4 even. In order to have a first division of connected components, we first cl…

The paper defines a universal Teichmüller space for PGL_d(R) and proves its properties.

Given a convex representation $ρ:Γ\to\textrm{PGL}(d,\mathbb{R})$ of a convex co-compact group $Γ$ of $\mathbb{H}^k$ we find upper bounds for the quantity $αh_ρ,$ where $h_ρ$ is the entropy of $ρ$ and $α$ is the Hölder exponent of the equivariant map $\partialΓ\to\mathbb{P}(\mathbb{R}^d).$ We also give rigidity statemen…

Study k-positive surface group representations and their degenerations.

Study of parabolic Higgs bundles on curves with special fixed points.

Surveying Hitchin representations of Fuchsian groups.

Study Fuchsian loci in ${ m PSL}_n(\mathbb{R})$-Hitchin components of a pair of pants.

We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in two different senses. First, they satisfy the requirements laid down by Strominger-Yau-Zaslow…

Researchers simplified Hitchin component computation for orbifold groups.

Symplectic coordinates found on a Hitchin component for a hyperbolic surface.

Study pressure metrics for cusped Hitchin representations.

Goldman symplectic form and complex structure compatible on $\mathrm{SL}(3,\mathbb R)$ Hitchin component.

In this article we give a geometric interpretation of the Hitchin component for PSL(4,R) in the representation variety of a closed oriented surface of higher genus. We show that representations in the Hitchin component are precisely the holonomy representations of properly convex foliated projective structures on the u…

The paper studies properties of triangle and shearing invariants in PSL(n,R) and connects them to a slice of Hitchin components.

New Teichmüller spaces found for orbifolds.

New metrics derived from Hölder distortion on Hitchin components.

Hitchin representations are identified by curve spectral radii.

Study compares two pseudo-Kähler structures on a specific mathematical component.

Paper describes a pseudo-Kähler structure on a specific Hitchin component.

Compactifies a component by studying metric degeneration.

New phenomenon found in Gothen components' boundary.

Sequences of Hitchin representations on surfaces are studied to describe their limits on trees.

Study Blaschke metrics to compactify Hitchin component.

New insights into currents of Hitchin representations with combinatorial restrictions.

We construct a geometric, real analytic parametrization of the Hitchin component Hit_n(S) of the PSL_n(R)-character variety R_{PSL_n(R)}(S) of a closed surface S. The approach is explicit and constructive. In essence, our parametrization is an extension of Thurston's shear coordinates for the Teichmueller space of a cl…

The paper establishes a correspondence between solutions of extended Bogomolny equations and Higgs bundles.

Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).

Study on representations of a specific surface group, identifying components with non-maximal Euler class.

Geometric structures defined for $G_2'$-Hitchin component on surfaces.

Study Liouville currents for Hitchin representations using root flows.

The paper analyzes cyclic Higgs bundles using elliptic systems and immersion properties.

Using the work of Bonahon-Dreyer and Fock-Goncharov, one can construct a real-analytic parameterization for the PSL(n,R) Hitchin component of a surface S, that is explicitly analogous to the Fenchel-Nielsen coordinates on the Teichmuller space of S. Given a Hitchin representation, we give a lower bound on the "length" …

Develops complex harmonic maps for Teichmüller theory, proving new theorems.

Canonical maps connect complex structures to Hitchin components.

The study describes the geometry of surfaces and their representations in SL(3,R).

We prove that given a Hitchin representation in a real split rank 2 group $\mathsf G_0$, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin components by a Hermitian bundle over Teichmüller space. The proof goes through intr…

Shearing deformations in Hitchin representations are computed for a symplectic form.

For a closed surface S, the Hitchin component Hit_n(S) is a preferred component of the character variety consisting of group homomorphisms from the fundamental group pi_1(S) to the Lie group PSL_n(R). We construct a parametrization of the Hitchin component that is well-adapted to a maximal geodesic lamination on the su…

Using Hitchin's parameterization of the Hitchin-Teichmüller component of the $SL(n,\mathbb{R})$ representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the $SL(n,\mathbb{R})$-Hitchin component, we study the asymptotics of the Hermitian metric solvin…

Constructs a new geometric structure on surfaces to generalize Teichmüller theory.