Surveying Hitchin representations of Fuchsian groups.
problem Understanding representations of Fuchsian groups.
method Survey and conjectural geometric description.
result Conjectural geometric picture of an augmented Hitchin component.
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48 results for “Hitchin Component”
Study Fuchsian loci in -Hitchin components of a pair of pants.
problem Understanding Fuchsian loci in -Hitchin components.
method Using Bonahon-Dreyer parametrization, explicit parametrization of Fuchsian loci of a pair of pants.
result Explicit parametrization of Fuchsian loci of a pair of pants.
Researchers simplified Hitchin component computation for orbifold groups.
problem Computing the dimension of Hitchin components for orbifold groups.
method Reduction to orbi-curves and application of orbifold Riemann-Roch theorem.
result Dimension of Hitchin component computation simplified.
New flows defined on Hitchin components for PGL(V).
problem Understanding dynamics on Hitchin components.
method Define and analyze new flows on Hitchin components.
result Construct a global coordinate system on the Hitchin component.
Symplectic coordinates found on a Hitchin component for a hyperbolic surface.
problem Parametrizing the -Hitchin component with canonical coordinates.
method Proved global Darboux coordinates with half canonical Goldman coordinates.
result Global Darboux coordinates exist for the -Hitchin component.
Explicit computation of symplectic form for -Hitchin component.
problem Symplectic structure of -Hitchin component.
method Atiyah-Bott-Goldman symplectic form and global coordinates.
result Coefficients of the symplectic form are constant.
Study pressure metrics for cusped Hitchin representations.
problem Characterize cusped Hitchin representations of Fuchsian groups.
method Develop pressure metrics associated to fundamental weights and roots.
result New pressure metrics for Hilbert length when .
Compatibility of Goldman's symplectic form with the complex structure on the Hitchin componentmath.DG
Goldman symplectic form and complex structure compatible on Hitchin component.
problem Compatibility of Goldman's symplectic form with complex structure on Hitchin component.
method Proof of compatibility between Goldman's symplectic form and Labourie-Loftin complex structure.
result Goldman symplectic form and complex structure determine a pseudo-Kähler structure on 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.
problem Understanding invariants of PSL(n,R)-Fuchsian representations and their relationship to Hitchin components.
method Examined triangle and shearing invariants, used Bonahon-Dreyer parameterization.
result The Fuchsian locus of Hitchin components corresponds to a slice.
New Teichmüller spaces found for orbifolds.
problem Understanding geometric structures on orbifolds.
method Extending Hitchin components to orbifold groups and proving properties.
result Hitchin components are homeomorphic to open balls and have explicit dimensions.
New metrics derived from Hölder distortion on Hitchin components.
problem Deriving metrics on Hitchin components from Hölder distortion.
method Expressing Thurston's metric in terms of Hölder regularity of boundary maps, associating stratified loci, and measuring relative Hölder distortion.
result First known geometrically significant complete metrics on Hitchin components for .
Hitchin representations are identified by curve spectral radii.
problem Classifying isometries of Hitchin components.
method Establishing transversality for positive quadruples of flags.
result Hitchin representations are uniquely determined by spectral radii of curves.
The Goldman symplectic form is trivialized for a class of surfaces.
problem Trivializing the Goldman symplectic form on specific surfaces.
method Using ideal triangulations and compatible bridge systems, the authors prove the existence of a symplectic trivialization.
result A global Darboux coordinate system is established for the PGL(V)-Hitchin component.
Study compares two pseudo-Kähler structures on a specific mathematical component.
problem Comparing two pseudo-Kähler structures on the -Hitchin component.
method Examined Rungi-Tamburelli's and Goldman's forms, and aligned Killing forms.
result Rungi-Tamburelli's semi-pseudo-Kähler structure is non-degenerate and matches another structure after normalization.
Paper describes a pseudo-Kähler structure on a specific Hitchin component.
problem Existence and description of a pseudo-Kähler structure on the SL(3,R)-Hitchin component.
method Explicit construction of a pseudo-Riemannian metric and symplectic form compatible with complex structure.
result Existence of a pseudo-Kähler structure on a neighborhood of the Fuchsian locus.
Compactifies a component by studying metric degeneration.
problem Compactify the SO(2,3)-Hitchin component.
method Study metric degeneration on a surface in pseudo-hyperbolic space.
result Establish closure in projectivized geodesic currents space.
New phenomenon found in Gothen components' boundary.
problem Character variety boundaries.
method Length spectrum compactification analysis.
result Hitchin component and Gothen components share boundary.
Sequences of Hitchin representations on surfaces are studied to describe their limits on trees.
problem Understanding the limits of sequences of Hitchin representations on surfaces.
method Using Fock-Goncharov coordinates on moduli spaces of flags.
result Non-trivial sufficient conditions for describing the limit of a sequence of Hitchin representations as an action on a tree.
Limits of Blaschke metricsmath.DG
Study Blaschke metrics to compactify Hitchin component.
problem Compactify the SL(3,R)-Hitchin component.
method Study degeneration of Blaschke metrics on equivariant affine spheres.
result Establish closure in projectivized geodesic currents space.
New insights into currents of Hitchin representations with combinatorial restrictions.
problem Understanding currents associated with Hitchin representations.
method Defining dual spaces and analyzing combinatorial restrictions on self-intersection.
result Dual spaces of discrete boundary currents are polyhedral complexes with dimension at most n-1.
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.
problem Solutions of the extended Bogomolny equations on $Σ imes \RP$ with specific singularities.
method Develops a Kobayashi-Hitchin type correspondence and partial correspondence for solutions with Nahm pole and knot singularities.
result Verifies a conjecture and proves existence and uniqueness of solutions with knot singularities.
Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).
problem Classifying Anosov representations of hyperbolic triangle groups into SL(3,R).
method Proving representations are Anosov if they lie in the Hitchin component or the Barbot component, with specific conditions for eigenvalues.
result Anosov representations in SL(3,R) have non-convex boundary maps.
Geometric structures defined for -Hitchin component on surfaces.
problem Understanding the geometric structures of -Hitchin component on surfaces.
method Explicit geometric structures interpretation and moduli space construction.
result Proves is homeomorphic to a moduli space of -structures.
Study Liouville currents for Hitchin representations using root flows.
problem Understanding Liouville currents in Hitchin representations.
method Investigate simple root flows and Liouville currents.
result Derive a Liouville volume rigidity result and construct a Liouville pressure metric.
On cyclic Higgs bundlesmath.DG
The paper analyzes cyclic Higgs bundles using elliptic systems and immersion properties.
problem Analyzing cyclic Higgs bundles and their associated immersions.
method Derive a maximum principle for elliptic systems and apply it to the Hitchin equation.
result Obtain bounds on extrinsic curvature and complete the picture for specific representations.
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.
problem Analyzing complex harmonic maps in Teichmüller theory.
method Complex harmonic maps and Higgs bundles.
result Proves a Bers-type theorem for rank 2 Hitchin components.
Canonical maps connect complex structures to Hitchin components.
problem Connecting higher complex structures to Hitchin components.
method Equivariant vector bundle construction and canonical isomorphisms.
result Canonical diffeomorphisms between spaces of higher complex structures and Hitchin components.
The study describes the geometry of surfaces and their representations in SL(3,R).
problem Understanding the geometry of surface group representations into SL(3,R).
method Proving asymptotic formulas and harmonic map convergence for equivariant maps.
result The geometry of the image is weakly convex and a (one-third) translation surface.
We prove that given a Hitchin representation in a real split rank 2 group , 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.
problem Computing symplectic form pairings for Hitchin representations.
method Shearing deformations of Hitchin representations.
result Pairings of shearing deformations computed for the Atiyah-Bott-Goldman 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 representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the -Hitchin component, we study the asymptotics of the Hermitian metric solvin…
Constructs a new geometric structure on surfaces to generalize Teichmüller theory.
problem Exploring new geometric structures in Teichmüller theory.
method Uses the punctual Hilbert scheme of the plane to construct a higher complex structure and explores its properties.
result Establishes a canonical diffeomorphism between the moduli space of higher complex structures and Hitchin's component.
N. Hitchin recently introduced the notion of folded hyperKähler metrics, in relation with SL(\infty,R) Higgs bundles. We provide a construction of such metrics, and prove the local existence of the Hitchin component for SL(\infty,R).
New geometric structures on surfaces generalize complex and real Lie algebra properties.
problem Generalizing geometric structures associated with Lie algebras.
method Define and analyze generalizations of punctual Hilbert schemes for complex and real Lie algebras.
result Construct geometric structures homeomorphic to Hitchin components.
The abstract discusses rational approximations for Hitchin representations on surfaces.
problem Density of Hitchin representations in the Hitchin component for surfaces of genus .
method Dynamical proof for ; extension to other -groups.
result Density of Hitchin representations for various -groups.
Study of energy functional on Deligne-Hitchin moduli space sections.
problem Understanding energy functionals on sections of Deligne-Hitchin moduli space.
method Generalizes energy of equivariant harmonic maps to holomorphic sections, links to meromorphic connections, and uses Willmore energy analogy.
result Shows functional is essentially Willmore energy for certain sections, distinguishes new components from twistor lines.
Study local models for special Kähler metrics near discriminant locus components.
problem Analyzing singularities of special Kähler metrics along discriminant locus of Hitchin base.
method Computed Taylor expansion, defined subsystems, and analyzed asymptotics and convergence of metrics.
result Logarithmic asymptotics in transversal directions and convergence to a metric on strata.
Geodesic coordinates derived for a specific metric in surface group representations.
problem Computing geodesic coordinates for a specific metric in surface group representations.
method Using thermodynamic formalism and gauge-theoretic formulas, computing first and second derivatives of the pressure metric.
result First derivatives of the pressure metric vanish at the Fuchsian locus.
Characterizes flag geometries for Hitchin representations in SL3(R).
problem Understanding flag geometries associated with Hitchin representations in SL3(R).
method Geometric characterization based on invariant foliations and refraction flows.
result Constructs refraction flows for positive roots in general sl_n(R), with highest root flows being C^1+α.
Introduces Fock bundles for studying surface group character varieties.
problem Character varieties of surface groups without fixed complex structures.
method Introduces Fock bundles as smooth principal bundles with special adjoint-valued 1-forms, constructs canonical connections, and solves non-linear PDEs.
result Explicit solutions for Fock bundles in the Fuchsian locus map to the Hitchin component.
Defines and parametrizes -type singular fibres in symplectic and odd orthogonal Hitchin systems.
problem Characterizing and understanding singular fibres in Hitchin systems.
method Stratification by semi-abelian spectral data, study of irreducible components, global description of degenerations.
result Extension of Langlands duality to -type Hitchin fibres.
F. Labourie [arXiv:1212.5015] characterized the Hitchin components for for any by using the swapping algebra, where the swapping algebra should be understood as a ring equipped with a Poisson bracket. We introduce the rank swapping algebra, which is the quotient of the swap…
Describes spectral data for singular fibres of a specific Hitchin system.
problem Characterizing singular fibres of the -Hitchin system.
method Using Hecke transformations and analysis of parameter spaces, the paper stratifies and compactifies the singular spaces.
result Large classes of singular fibres are shown to be fibre bundles over Prym varieties.
Study of minimal surfaces in a specific symmetric space with polynomial growth.
problem Asymptotic geometry of minimal surfaces in a symmetric space.
method Homeomorphism between Hitchin components and maximal surfaces, identification of convex embeddings, local limits of equivariant surfaces.
result Identification of planar maximal surfaces as local limits of equivariant surfaces.