Study Fuchsian loci in mPSLn(R)-Hitchin components of a pair of pants.
problem Understanding Fuchsian loci in mPSLn(R)-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.
Paper extends Kobayashi-Hitchin correspondence to non-Kähler manifolds.
problem Applying Kobayashi-Hitchin correspondence to non-Kähler manifolds.
method Continuity method for vortex equation, Kobayashi-Hitchin correspondence for holomorphic pairs.
result Proved solvability of vortex equation on holomorphic vector bundles over compact Hermitian manifolds.
Abstract reviews applications of parabolic structures to nodal curve sheaves.
problem Understanding torsion-free sheaves and Hitchin pairs on nodal curves.
method Examines the relation between nodal curve fundamental groups and moduli spaces of parabolic bundles.
result Establishes connections between representations and moduli spaces.
We prove a very general Kobayashi-Hitchin correspondence on arbitrary compact Hermitian manifolds. This correspondence refers to moduli spaces of "universal holomorphic oriented pairs". Most of the classical moduli problems in complex geometry (e. g. holomorphic bundles with reductive structure groups, holomorphic pair…
Generic Hitchin representations generate dense subgroups.
problem Understanding dense subgroups in SL_n(R) representations.
method Using a theorem by Rapinchuk, Benyash-Krivetz, and Chernousov.
result Generic Hitchin representations are strongly dense.
Study on harmonic metrics for rank 3 Higgs bundles in Hitchin section.
problem Finding compatible harmonic metrics for rank 3 Higgs bundles in the Hitchin section.
method Defined a symmetric pairing and studied spectral curves as 2-sheeted branched coverings.
result Gave a condition for Higgs bundles on C or C∗ to have compatible harmonic metrics. Identifies spectral curves for SU(3) coadjoint orbits.
problem Understanding the geometry of coadjoint orbits in SU(3).
method Using Hitchin pairs and spectral curves, identifies a Hamiltonian circle action and finds Darboux coordinates.
result Identifies a differential equation for the Hamiltonian.
Develops correlation number for specific potentials and Hitchin representations.
problem Analyzing correlation numbers for potentials with entropy gaps and Hitchin representations.
method Defines a correlation number for pairs of cusped Hitchin representations and explores its connection to the Manhattan curve.
result Establishes a connection between the correlation number and the Manhattan curve, revealing rigidity properties.
We develop a complete Hitchin-Kobayashi correspondence for twisted pairs on a compact Riemann surface X. The main novelty lies in a careful study of the the notion of polystability for pairs, required for having a bijective correspondence between solutions to the Hermite-Einstein equations, on one hand, and polystable …
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.
The paper studies sequences of solutions to Hitchin-Simpson equations on Kähler manifolds.
problem Behavior of sequences of solutions to Hitchin-Simpson equations on Kähler manifolds.
method Compactness result for connections and renormalized Higgs fields.
result Every Z/2 harmonic 1-form can be deformed into a sequence of solutions. A principal pair consists of a holomorphic principal G-bundle together with a holomorphic section of an associated Kaehler fibration. Such objects support natural gauge theoretic equations coming from a moment map condition, and also admit a notion of stability based on Geometric Invariant Theory. The Hitchin--Kobaya…
Study automorphism equivariant Hitchin index for Riemann surfaces.
problem Define and study an index for Riemann surfaces under automorphisms.
method Define automorphism equivariant Hitchin index and prove a formula in terms of cohomological pairings.
result Prove a formula for the automorphism equivariant Hitchin index.
The paper studies the moduli space of Higgs pairs and their geometric properties.
problem The moduli space of Higgs pairs and its geometric properties.
method Introduced τ-stability of Higgs pairs and established the Kobayashi-Hitchin correspondence. result Proved that the moduli space is a non-singular complex manifold for a suitable choice of τ. A twisted Higgs bundle on a Kähler manifold X is a pair (E,φ) consisting of a holomorphic vector bundle E and a holomorphic bundle morphism φ:M⊗E→E for some holomorphic vector bundle M. Such objects were first considered by Hitchin when X is a curve and M is the tangent bundle of X, and…
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.
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.
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.
Study real structures on Higgs pairs over Klein surfaces, proving a correspondence.
problem Real structures on Higgs pairs over Klein surfaces.
method Establish Hitchin-Kobayashi correspondence, homeomorphism between moduli spaces.
result Real G-Higgs bundles appear as fixed points of involutions. New equations reveal moduli space rigidity in geometric deformations.
problem Understanding moduli space rigidity in geometric deformations.
method Coupled Hitchin-He equations, Lax pair, nonlinear embedding.
result Moduli space is analytically isomorphic to the classical case for small deformations.
This work concerns the study of certain finite-energy solutions of the anti-self-dual Yang-Mills equations on Euclidean 4-dimensional space which are periodic in two directions, so-called doubly-periodic instantons. We establish a circle of ideas involving equivalent analytical and algebraic-geometric descriptions of t…
Hitchin shows that half-flat SU(3)-structures on a 6-dimensional manifold M can be lifted to parallel G_{2}-structure on the product M×R. We show that Hitchin's approach can also be used to construct nearly parallel G_{2}-structures by lifting so-called nearly half-flat structures. These SU(3)-structure…
We consider a version of Hermitian-Einstein equation but perturbed by a Higgs field with a solution called a Donaldson-Thomas instanton on compact Kähler threefolds. The equation could be thought of as a generalization of the Hitchin equation on Riemann surfaces to Kähler threefolds. In the appendix of arXiv:0805.2192,…
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.
This note proves combinatorially that the intersection pairing on the middle dimensional compactly supported cohomology of a smooth toric hyperkaehler variety is always definite, providing a large number of non-trivial L^2 harmonic forms for toric hyperkaehler metrics on these varieties. This is motivated by a result o…
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.
Develops correspondence for Bogomolny equations with specific boundary conditions.
problem Solving the extended Bogomolny equations with generalized Nahm pole boundary conditions.
method Kobayashi-Hitchin correspondence for stable Higgs pairs and holomorphic line bundles.
result Corroborates Gaiotto and Witten's prediction and extends previous work to higher dimensions.
Introduces generalized hyperpolygons and their geometric and algebraic properties.
problem Understanding moduli spaces of generalized hyperpolygons.
method Representation of a comet-shaped quiver, associated meromorphic Higgs bundles, Hitchin systems, and integrable Hamiltonian systems.
result Generalized hyperpolygons admit the structure of a completely integrable Hamiltonian system.
Holomorphic map connects Hitchin components to character varieties.
problem Complex affine spheres and their properties.
method Mapping class group equivariant holomorphic map from Hitchin components to character varieties.
result Holomorphic map includes holonomies of SL(3,C)-opers.
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.
The paper studies the correlation of Hilbert lengths for convex projective surfaces.
problem Understanding the correlation of Hilbert lengths for convex projective surfaces.
method Asymptotic formula for free homotopy classes with renormalized Hilbert length.
result The correlation number is not uniformly bounded away from zero but can be larger than a uniform strictly positive constant.
The classes of Monge-Ampère systems, decomposable and bi-decomposable Monge-Ampère systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of …
Surveying recent developments in Hitchin moduli space geometry.
problem Understanding the asymptotic geometry of Hitchin moduli space.
method Introduction to Hitchin moduli space and hyperkähler geometry.
result Recent developments in asymptotic geometry of Hitchin moduli space.
The main result is an explicit expression for the Pressure Metric on the Hitchin component of surface group representations into PSL(n,R) along the Fuchsian locus. The expression is in terms of a parametrization of the tangent space by holomorphic differentials, and it gives a precise relationship with the Petersson pa…
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.
Defines and parametrizes sl(2)-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 sl(2)-type Hitchin fibres. Real projective surfaces with Hitchin holonomy can be related via grafting.
problem Real projective surfaces with Hitchin holonomy.
method Defining graftable curves and constructing them in the Hitchin case.
result Real projective structures with the same Hitchin holonomy are related via multi-graftings.
Study shows Hitchin's metric converges to a semiflat metric on the Hitchin section.
problem Understanding the asymptotic behavior of Hitchin's metric on the Hitchin section.
method Analyzing the hyperkähler metric on moduli spaces of Higgs bundles.
result The Hitchin metric converges exponentially to a semiflat metric on the Hitchin section.
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.
We define a nondegenerate Monge-Ampère structure on a 6-dimensional manifold as a pair (Ω,ω), such that Ω is a symplectic form and ω is a 3-differential form which satisfies ω∧Ω=0 and which is nondegenerate in the sense of Hitchin. We associate with such a pair a generalized almost (pseudo) Calabi-Yau stru…
Study asymptotics of hyperkähler geometry on singular fibers of Hitchin moduli space.
problem Asymptotic hyperkähler geometry of SL2(C)-Hitchin moduli space over singular fibers. method Extension of exponential convergence results to locally fiducial Higgs bundles and subintegrable systems.
result Hyperkähler metric converges exponentially to semi-flat metric on subintegrable systems.
New representations of hyperbolic 3-manifold groups into larger groups.
problem Finding representations of hyperbolic 3-manifold groups into larger matrix groups.
method Holonomy representations from projective deformations of hyperbolic structures.
result First examples of strongly dense representations into SL(4,R) and SU(3,1). Study shows rapid decay of Hitchin metric from semi-flat metric on Higgs bundles.
problem Analyzing the asymptotic behavior of the Hitchin metric on moduli spaces of Higgs bundles.
method Examined the decay rate of the difference between Hitchin and semi-flat metrics on smooth spectral curves.
result Exponential decay of the difference between Hitchin and semi-flat metrics as t approaches infinity.
New estimates for Hitchin's equations at high energy.
problem Solutions to Hitchin's self-duality equations at high energy.
method New estimates and asymptotic decoupling phenomenon.
result Generalization to arbitrary Higgs bundles.
Study of a basic Hitchin equation on Sasakian 3-folds, showing hyperKähler metric.
problem Understanding moduli spaces of the basic Hitchin equation on Sasakian 3-folds.
method Construction of moduli space and calculation of its dimension.
result Moduli space admits a hyperKähler metric.
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 d=3. 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.
Survey on 2k-Hitchin equations and Higgs bundles from geometric perspective.
problem Understanding 2k-Hitchin equations through Higgs bundles and complex geometry. method Review of Higgs bundles, holomorphic vector bundles, and Hermite-Yang-Mills equations; geometric tools applied to simplify equations.
result Simplified 2k-Hitchin equations to a set of two equations for Higgs bundles.