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 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.
Exponential decay observed between two metrics on a moduli space.
problem Analyzing the difference between two metrics on a moduli space.
method Proving exponential decay along generic rays in the Hitchin moduli space.
result Exponential decay of the difference between gL2 and gsf. 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 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. 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.
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 n>3. 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.
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.
Quantizes metrics for Higgs bundles over projective manifolds.
problem Quantizing Hermitian metrics for Higgs bundles.
method Geometric Invariant Theory and Gieseker stability.
result Balanced metrics converge to solutions of Hitchin equation at quantum limit.
Study on Higgs bundles and hyperpolygon spaces using Hitchin metrics.
problem Investigating the Hitchin metric on moduli spaces of Higgs bundles.
method Using Hitchin hyperkähler metric and parabolic Deligne-Hitchin moduli space.
result Rescaled Hitchin metric converges to hyperpolygon space's hyperkähler metric in the semiclassical limit.
Study on the Hitchin moduli space metric approximates it by a semiflat metric.
problem Understanding the asymptotic behavior of the Hitchin moduli space metric.
method Analyzing the L2 metric on the Hitchin moduli space for G=SU(2), proving polynomial convergence rates. result The Hitchin moduli space metric is well-approximated by the semiflat metric with polynomial decay.
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.
Numerical experiments support conjecture about opers and nonabelian Hodge.
problem Testing predictions of Gaiotto-Moore-Neitzke and Gaiotto conjectures.
method Numerical experiments on polynomial holomorphic differentials.
result Supports conjectural formulas for Stokes data and Hitchin metric tensor.
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.
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.
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.
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).
Establishes Kobayashi-Hitchin correspondence for nef and big classes.
problem Analyzing stability and positivity in algebraic geometry.
method Introducing adapted currents and metrics to establish correspondence.
result Equality cases of Bogomolov-Gieseker and Miyaoka-Yau inequalities.
The Kobayashi-Hitchin correspondence is proven for twisted vector bundles on Kähler manifolds.
problem Proving the Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles.
method Proved the correspondence and approximate correspondence for twisted holomorphic vector bundles on compact Kähler manifolds.
result A twisted holomorphic vector bundle is g−polystable if and only if it is g−Hermite-Einstein, and g−semistable if and only if it is approximate g−Hermite-Einstein. Study local models for special Kähler metrics near discriminant locus components.
problem Analyzing singularities of special Kähler metrics along discriminant locus of SL2(C) 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.
We study hyperkahler metrics and hyperholomorphic connections of Hitchin's moduli spaces after Gaiotto, Moore and Neitzke. Their construction via the twistor technique produces intricate wall crossing behaviors. For certain four dimensional Hitchin's moduli spaces local models and degeneration to local models near sing…
Study on the asymptotic geometry of Higgs bundles over projective line.
problem Understanding the asymptotic behavior of Hitchin's metric on moduli spaces of rank two irregular Higgs bundles.
method Analysis of Hitchin's hyperkähler metric and comparison with semiflat and ALG/ALG∗ models. result Hitchin's metric is asymptotic to semiflat and ALG/ALG∗ models at polynomial and exponential rates. Harmonic metrics are established for SO0(n,n)-Higgs bundles on non-compact hyperbolic surfaces.
problem Establishing harmonic metrics for SO0(n,n)-Higgs bundles.
method Using canonical line bundle and holomorphic differentials, the existence and uniqueness of harmonic metrics are proven.
result Existence and uniqueness of harmonic metrics compatible with SO0(n,n) structure on non-compact hyperbolic surfaces.
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.
This note finds metrics with specific curvature pinching on Hirzebruch surfaces.
problem Finding metrics with controlled curvature pinching on Hirzebruch surfaces.
method Proving a general pinching result for product metrics and applying it to Hirzebruch surfaces.
result Existence of Hodge metrics with controlled curvature pinching on Hirzebruch surfaces.
Study of Hitchin moduli spaces over Teichmüller space.
problem Metric aspects of Hitchin moduli spaces over varying complex structures.
method Gauge theoretical approach, Kähler fibrations, moment map interpretation, symplectic reduction.
result Establishes natural complex and pseudo-Kähler structures on universal Hitchin moduli spaces.
The Legendre transform and its generalizations, originally found in supersymmetric sigma-models, are techniques that can be used to give constructions of hyperkahler metrics. We give a twistor space interpretation to the generalizations of the Legendre transform construction. The Atiyah-Hitchin metric on the moduli spa…
In 1992, Hitchin used his theory of Higgs bundles to construct an important family of representations of the fundamental group of a closed, oriented surface of genus at least two into the split real form of a complex adjoint simple Lie group. These Hitchin representations comprise a component of the space of conjugacy …
The dynamics of representations into PSL_d(R) are studied for surfaces of genus at least 3.
problem Dynamics of representations into PSL_d(R) for surfaces of genus at least 3.
method Showed quasi-convex subsets of infinite diameter for the Weil--Petersson metric have finite diameter for the path metric of the pressure metric through controlled bounded length of biinfinite paths of bending deformations.
result Biinfinite paths of bending deformations have controlled bounded length.
The paper quantizes the Hitchin system using geometric quantization.
problem Quantizing the Hitchin system using geometric methods.
method Geometric quantization of the Hitchin moduli space by modifying the Quillen metric.
result The modified Kahler form is integral and descends as a prequantum line bundle.
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.
The Kobayashi-Hitchin correspondence is established for mini-holomorphic bundles on certain 3-folds.
problem Establishing a correspondence between mini-holomorphic bundles and HE-monopoles.
method Defining mini-holomorphic bundles, Dirac-type singularities, and admissible BHE-metrics.
result A special Hermitian metric (admissible BHE-metric) exists on slope stable mini-holomorphic bundles.
Constructs an asymptotic metric for moduli space of centred hyperbolic monopoles.
problem Analyzing the moduli space of centred hyperbolic monopoles.
method Point particle approximation and geodesic motion analysis.
result Obtains a hyperbolic analogue of negative mass Taub-NUT metric.
Minimal surfaces in symmetric spaces linked to Hitchin representations studied.
problem Properties of minimal surfaces in symmetric spaces associated with Hitchin representations.
method Investigation of properties of immersed minimal surfaces in symmetric spaces associated with specific subloci of Hitchin component.
result Minimal surfaces have strong rigidity properties and are strictly negatively curved.
Study shows slices to sums of adjoint orbits are related to specific manifolds and Hilbert schemes.
problem Understanding the geometry of slices to sums of adjoint orbits.
method Analyzing the isomorphisms and deformations of specific manifolds and Hilbert schemes.
result Slices to sums of adjoint orbits are isomorphic to specific manifolds and Hilbert schemes of points.
By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFT-type correspondence between classical 2+1 dimensional vacuum general relativity theory on S x R and SO(3) Hitchin theory (regarded as a classical conformal field theory) on the spacelike past boundary S, a compact, oriented Riemann surface…
New metrics for Anosov representations defined from Thurston's asymmetric metrics.
problem Defining metrics for Anosov representations.
method Generalizing Thurston's asymmetric metric to Anosov representations.
result Provides a (possibly asymmetric) Finsler distance in some cases.
This is the first in a series of papers in which we develop a twistor-based method of constructing hyperkaehler metrics from holomorphic functions and elliptic curves. As an application, we revisit the Atiyah-Hitchin manifold and derive in an explicit holomorphic coordinate basis closed-form formulas for, among other t…
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…
Study proves correspondence for special bundles on complex surfaces.
problem Proving correspondence for parabolic bundles on complex surfaces.
method Kobayashi-Hitchin correspondence for parabolic bundles over compact non-Kähler surfaces.
result Proved Kobayashi-Hitchin correspondence for specified bundles.
The paper establishes a correspondence for quiver bundles over generalized Kähler manifolds.
problem Establishing a correspondence between stability and Hermitian-Einstein metrics for quiver bundles.
method Analyzing I±-holomorphic quiver bundles over compact generalized Kähler manifolds. result A quiver bundle is (α,σ,τ)-polystable if and only if it admits an (α,σ,τ)-Hermitian-Einstein metric. 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.
The paper proves inequalities and consequences in Hilbert metrics and Hitchin representations.
problem Volume entropy rigidity and length spectrum comparison in specific geometric settings.
method Sharp distance inequalities and volume growth analysis.
result Volume entropy rigidity for Hilbert geometries and length spectrum comparison for Hitchin representations.
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. Extends Ooguri-Vafa symplectic form to framed Higgs bundles.
problem Local model for Hitchin moduli spaces near discriminant locus.
method Identifies Ooguri-Vafa form with Atiyah-Bott form on framed connections.
result Identifies Ooguri-Vafa metric with Hitchin's L2-metric on Hitchin section. Affine actions fail for Hitchin linear parts, except flat pseudo-Riemannian cases.
problem Properly discontinuous affine actions of surface groups with Hitchin linear part.
method Analysis of representations and pseudo-Riemannian metrics.
result Hitchin linear parts in SO(n,n−1) lead to non-properly discontinuous affine actions. Study on Blaschke locus with covariance metric properties.
problem Riemannian structure of Blaschke locus.
method Analysis of Blaschke locus as a manifold, study of covariance metric properties.
result Geodesics in Blaschke locus have infinite length with respect to the covariance metric.