Proves a simplified version of Hitchin's theorem.
problem Constructing hyper-Kähler structures.
method Concrete variant of Hitchin's theorem.
result Applies to real manifolds without constructing hyper-Kähler structures.
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48 results for “Hitchin's theorem”
This paper shows connections between two complex mathematical theories are equivalent.
problem Establishing equivalence between two complex mathematical theories.
method Using geometric quantisation and conformal field theory, the paper establishes equivalence between the Hitchin connection and the Knizhnik-Zamolodchikov connection.
result The Hitchin and Knizhnik-Zamolodchikov connections are projectively equivalent in genus zero.
Entropy rigidity theorem for cusped Hitchin representations.
problem Entropy rigidity for Hitchin representations of cusped groups.
method Introduction of (1,1,2)-hypertransverse groups and transverse representations.
result Hausdorff dimension of conical limit set agrees with simple root entropy.
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.
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.
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.
The paper proves quasi-Einstein structures on manifolds admit Killing vector fields and provides new examples.
problem Classifying quasi-Einstein structures and understanding their properties.
method Analyzing quasi-Einstein equations and exploring their connections to Hitchin's equations.
result A class of quasi-Einstein structures on closed manifolds must admit a Killing vector field.
Study shows how to count and equidistribute cusped Hitchin representations with entropy gaps.
problem Counting and equidistribution of cusped Hitchin representations.
method Renewal theorem of Kesseböhmer and Kombrink applied to count and equidistribute.
result Entropy gaps at infinity allow for counting and equidistribution results.
This note is based on a talk given at the 2019 ISAAC Congress in Aveiro, Portugal. We give an expository account of joint work with Daniele Alessandrini and Gye-Seon Lee on Hitchin components for orbifold groups (arXiv:1811.05366), recasting part of it in the language of analytic orbi-curves. This reduces the computati…
Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.
problem Rigidity of geodesic planes in Hitchin manifolds.
method Constructing a specific surface group and analyzing its action on the Hitchin manifold.
result Existence of floating geodesic planes in Hitchin manifolds with fractal closures.
On Higgs bundlesmath.AG
Study on Higgs bundles over Riemann surfaces.
problem Investigate the moduli space of Higgs bundles.
method Explicit description of moduli space, study of Narasimhan-Seshadri theorem, nonabelian Hodge correspondence, Hitchin equations.
result Derive an explicit description of the moduli space and study its properties.
Generic Hitchin representations avoid hyperplanes in Lie algebras.
problem Properties of Hitchin representations in Lie algebras.
method Defined and used hyperplanes in Lie algebras to show .
result Generic -Hitchin representations avoid hyperplanes in the Lie algebra of .
We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with tr…
Study confirms Chern's conjecture on compact Hessian manifolds and classifies their topologies.
problem Global topological constraints and structural properties of compact Hessian manifolds.
method Novel fibration and splitting theorems, Chern's conjecture, Hitchin systems, Cheng-Yau solution.
result Topological classification of complete Hessian surfaces and closed orientable Hessian 3-manifolds.
Study k-positive surface group representations and their degenerations.
problem Understanding the behavior of surface group representations under degenerations.
method Introduced k-positive representations and studied their degenerations using a limit theorem for positively ratioed representations.
result Degenerations of k-positive representations can lead to limits that are at least (k-3)-positive and irreducible limits are (k-1)-positive.
We give an independent proof of a theorem of Danciger of Zhang: surface groups with Hitchin linear part cannot act properly on the affine space
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…
Solves generalized Kazdan-Warner equations on foliated manifolds.
problem Existence and uniqueness of solutions to generalized Kazdan-Warner equations on foliated manifolds.
method Extends theorem to compact foliated manifolds, provides examples of PDEs.
result Solves the transverse Hitchin equation and its generalizations.
We prove a gluing theorem for solutions of Hitchin's self-duality equations with logarithmic singularities on a rank-2 vector bundle over a noded Riemann surface representing a boundary point of Teichmüller moduli space.
We investigate the -monopole invariants of symplectic -manifolds and Kähler surfaces with real structures. We prove the nonvanishing theorem for real symplectic -manifolds which is an analogue of Taubes' nonvanishing theorem of the Seiberg-Witten invariants for symplectic -manifolds. Further…
We establish global existence of smooth solutions to heat flow for Yang-Mills-Higgs functional on Kahler fibrations. As an application, we give a new proof of the key inequality for Mundet's Hitchin-Kobayashi correspondence theorem using the heat flow technique.
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 proves unboundedness of a functional on G2 forms and describes manifold limits.
problem Proving unboundedness of a functional on G2 forms and describing manifold limits.
method Scaling arguments, geometric estimates, collapsing theorem for orbifolds.
result Explicit descriptions of large volume limits of two G2 manifolds.
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.
Minimal surfaces linked to Higgs bundles in pseudo-hyperbolic spaces.
problem Proving Labourie's theorem and extending it to new cases.
method Establishing infinitesimal rigidity of minimal surfaces and using it to prove the theorem.
result New proof of Labourie's theorem and extension to Collier's components.
Study on 4D solitons with curvature constraints.
problem Characterizing gradient shrinking Ricci solitons with positive modified sectional curvature.
method Sharp pinching conditions, weighted integral gap results, Hitchin-Thorpe inequality.
result Locally Kähler property under specific curvature conditions.
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.
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.
This research constructs hypercomplex structures from twistor spaces.
problem Understanding hyperkähler metrics and structures.
method Utilizing twistor spaces and Kodaira-Spencer deformation theory.
result Facilitates construction of hypercomplex structures on parameter spaces.
Real spectrum compactification of Hitchin components, Weyl chamber valued lengths, and dual spacesmath.GT
This paper connects real closed fields to Hitchin representations and their properties.
problem Understanding representations of surface groups over real closed fields.
method Tarski-Seidenberg transfer principle and multiplicative Bonahon-Dreyer coordinates.
result Hitchin representations correspond to -positive representations over real closed fields.
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.
Study asymptotics of hyperkähler geometry on singular fibers of Hitchin moduli space.
problem Asymptotic hyperkähler geometry of -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 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 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.
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 .
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 or to have compatible harmonic metrics.
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 harmonic 1-form can be deformed into a sequence of solutions.
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…
We survey some recent developments in the asymptotic geometry of the Hitchin moduli space, starting with an introduction to the Hitchin moduli space and hyperkähler geometry.
We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus, which generalizes the fa…
Kobayashi-Hitchin Correspondence for Saturated Reflexive Parabolic Sheaves on Kähler manifoldsmath.DG
Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.
problem Understanding the Kobayashi-Hitchin correspondence for specific sheaves.
method Using Hermitian-Yang-Mills flow on Kähler manifolds with simple normal crossing divisors.
result Established the correspondence for saturated reflexive parabolic sheaves.
We consider Hitchin's hyperkähler metric on the -Hitchin moduli space moduli space over a compact Riemann surface. We prove that the difference between the metric and a simpler "semiflat" hyperkähler metric is exponentially-decaying along generic rays in the Hitchin moduli s…
We show that a Hitchin representation is determined by the spectral radii of the images of simple, non-separating closed curves. As a consequence, we classify isometries of the intersection function on Hitchin components of dimension 3 and on the self-dual Hitchin components in all dimensions. As an important tool in t…
This is our second paper in a series to study gravitational instantons, i.e. complete hyperkäler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: 1.The asymptotic rate of gravitational instantons to the standard models can be improved automatically. 2.Any ALF-D_k gravitational instant…
This paper proves Hitchin moduli spaces are ALG gravitational instantons.
problem Proving Hitchin moduli spaces are ALG gravitational instantons.
method Computing Torelli parameters for each Hitchin moduli space corresponding to different parabolic data.
result All Hitchin moduli spaces studied are ALG- gravitational instantons.
We construct Dirac operators on foliations by applying the Bismut-Lebeau analytic localization technique to the Connes fibration over a foliation. The Laplacian of the resulting Dirac operators has better lower bound than that obtained by using the usual adiabatic limit arguments on the original foliation. As a consequ…
This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is…