Develops Lie-theoretic perspective on Hitchin's equations for cyclic G-Higgs bundles.
problem Analytic and geometric properties of harmonic maps.
method Lie-theoretic perspective on Hitchin's equations for cyclic G-Higgs bundles.
result Proves Dai-Li's conjecture on the monotonicity of the energy density and negative curvature conjecture for Coxeter cyclic G-Higgs bundles.
Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.
problem Addressing Hermitian-Einstein equation for cyclic Higgs bundles.
method Introducing generalizations using subharmonic functions and proving existence, uniqueness, and convergence of heat equations.
result Existence, uniqueness, and convergence of solutions for heat equations.
Study uses Vinberg pairs for Higgs bundles, revealing their role.
problem Understanding the role of Vinberg pairs in Higgs bundle theory.
method Exploring Vinberg pairs defined by cyclic gradings of a Lie algebra in Higgs bundle theory.
result Vinberg pairs play a significant role in Higgs bundle theory.
Solves Dirichlet problem for generalized Hitchin's equation on cyclic Higgs bundles.
problem Existence and uniqueness of solutions to the Dirichlet problem.
method Formulated using subharmonic functions; generalizes Hitchin's equation for diagonal harmonic metrics on cyclic Higgs bundles.
result Existence and uniqueness of solutions to the Dirichlet problem.
This paper generalizes a topological invariant to cyclic Higgs bundles.
problem Defining and studying a topological invariant for cyclic Higgs bundles.
method Using a complex semisimple Lie group and its Lie algebra, the authors construct special cyclic Higgs bundles and define a topological invariant.
result The authors generalize the definition and properties of the Toledo invariant to arbitrary (G0,g1⊕g1−m)-Higgs pairs. In this paper, we derive a maximum principle for a type of elliptic systems and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion f associated to cyclic Higgs bundles. Also, we obtain a lower and up…
We introduce a class of Higgs bundles called cyclic which lie in the Hitchin component of representations of a compact Riemann surface into the split real form of a simple Lie group. We then prove that such a Higgs bundle is equivalent to a certain class of solutions to the affine Toda equations. We further explain thi…
The study connects monopole chains to Higgs bundles and classifies symmetric chains.
problem Classifying symmetric monopole chains invariant under cyclic actions.
method Formulation of a correspondence between monopole chains and spectral data, using the Nahm transform.
result Classification of symmetric monopole chains of charge k.
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.
Classifies solutions of Toda equations near singularities.
problem Classifying solutions of Toda equations near singularities.
method Analyzes meromorphic and essential singularities of r-differentials. result Classifies all solutions on C for finite sums of exponentials of polynomials. Complete solutions found for Toda equations on non-compact surfaces.
problem Solving Toda equations on non-compact Riemann surfaces.
method Introduced complete solutions and proved existence and uniqueness using Toda equations and harmonic bundle techniques.
result Existence and uniqueness of complete solutions to Toda equations on non-compact Riemann surfaces.
The paper studies entropy of harmonic metrics on cyclic Higgs bundles.
problem Quantifying the degree of mutual misalignment of Hermitian metrics.
method Introduces entropy function and estimates its bounds for cyclic Higgs bundles.
result Entropy difference converges to a finite real number for β>−1. The paper studies entropy and free energy for harmonic metrics on cyclic Higgs bundles.
problem Quantifying the degree of mutual misalignment of metrics on Higgs bundles.
method Introduced entropy and free energy to quantify mutual misalignment; provided conditions for entropy and free energy to change.
result Extended work on boundedness of functions related to entropy and free energy on the unit disc.
We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real simple Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspo…
Study Higgs bundles and flat connections on quasi-regular Sasakian manifolds.
problem Investigate Higgs bundles and flat connections on compact Sasakian manifolds.
method Introduce quasi-regularity and regularity of vector bundles, relate to orbibundles, extend non-abelian Hodge correspondence.
result Extend non-abelian Hodge correspondence to quasi-regular Sasakian manifolds.
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.
New representation theory for surface groups to SO0(2,3).
problem Understanding representations of surface groups into special orthogonal groups.
method Non-maximal Anosov representations via Higgs bundles and non-Abelian Hodge correspondence.
result Generalization of Filip's result on weight 3 variation of Hodge structures.
Geometric methods for surface group representations in higher rank SL(2m+1,R).
problem Representations of surface groups in higher rank SL(2m+1,R).
method Para-complex and pseudo-Riemannian geometric techniques.
result One-to-one correspondence between Higgs bundles and isotropic P-alternating surfaces.
The paper classifies Toda equations for noncompact symmetric spaces and their solutions.
problem Classifying Toda equations for noncompact symmetric spaces.
method Interpreting Toda equations as equations for metrics on holomorphic principal bundles and using stability criteria.
result Existence of solutions to geometric Toda equations for totally noncompact pairs.
The paper studies holomorphic curves in a pseudo-Riemannian space and their moduli space.
problem Understanding the moduli space of holomorphic curves in a pseudo-Riemannian space.
method Using Frenet framing and G2′-Higgs bundles, the paper describes the moduli space of equivariant alternating holomorphic curves. result Equivariant alternating holomorphic curves are infinitesimally rigid.
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.
We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we compute its dimension explicitly. We then give applications to the study of…
On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's r…
Let S be a closed surface of genus at least 2. For each maximal representation ρ:π1(S)→Sp(4,R) in one of the 2g−3 exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric spa…
Identifies images of determinant morphism for specific co-Higgs bundles.
problem Determining images of determinant morphism for co-Higgs bundles.
method Identifying images of the determinant morphism of trace-free co-Higgs bundles modeled on rank 2 Schwarzenberger bundles.
result Identified images of the determinant morphism for specific co-Higgs bundles.
Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
problem Finding conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
method Restriction of Donaldson's functional to diagonal metrics on Higgs bundles with non-holomorphic Higgs fields.
result Provides necessary and sufficient conditions for the functional to attain a minimum.
The paper extends orthogonal decomposition results to hermitian Higgs bundles.
problem Classical propositions on holomorphic vector bundles do not always extend to Higgs bundles.
method The approach involves extending propositions on orthogonal decompositions and the second fundamental form to hermitian Higgs bundles.
result Extended propositions concerning orthogonal decompositions and the second fundamental form have applications in Higgs bundles.
Generalizes Higgs bundles theory using a vector bundle twist.
problem Extending Higgs bundles theory to incorporate vector bundle twists.
method Defined a Hitchin map and spectral correspondence, stated Hitchin-Kobayashi correspondence.
result Established a theory halfway between curve and higher-dimensional variety Higgs bundles.
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. In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional for Higgs pairs on a Hermitian vector bundle (E,H0) over a compact Kähler manifold (M,ω). We study the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity, and show that the limiting Higgs sheaf is isomorph…
New Poisson structures found on Higgs bundle moduli spaces.
problem Constructing Poisson structures on moduli spaces of Higgs bundles.
method Via Lie algebroids on stacky curves, focusing on parabolic Higgs bundles.
result Provides new examples of Poisson structures on moduli spaces.
We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle (E,H0) over a Riemann surface X. It is already known the gradient flow with initial data (A0,φ0) converges to a critical point (A∞,φ∞) of this functional. Using a modified Chern-Wei…
Classifies very stable Higgs bundles for complex groups.
problem Classifying Higgs bundles for arbitrary complex groups.
method Classification based on stability and Higgs field properties.
result Extends previous classification for GLn to arbitrary groups.
Study Higgs bundles and their reductions to prove stability and cohomology properties.
problem Analyzing Higgs bundles and their stability conditions.
method Introduced H-nflatness, proved stability conditions, and used cohomology rings.
result H-nflat Higgs bundles are either stable or reducible to a parabolic subgroup.
Diagonal metrics solve Hermitian-Einstein equations for decomposed Higgs bundles.
problem Existence of diagonal pluriharmonic metrics in G-Higgs bundles. method Analyzes Higgs bundles over compact Kähler manifolds, decomposes vector bundles, and uses torus action to relate stability and conditions.
result Necessary and sufficient conditions for the existence of diagonal metrics solving Hermitian-Einstein equations.
The paper extends the uniqueness of complete harmonic metrics to subharmonic weights and proves their existence on the unit disc.
problem Finding complete harmonic metrics on Riemann surfaces for subharmonic weights.
method Extending Li-Mochizuki's theorem to subharmonic weights and proving existence on the unit disc.
result Complete harmonic metrics exist on the unit disc for subharmonic weights.
Study very stable Higgs bundles on Riemann surfaces, linking to multiplicity and mirror symmetry.
problem Existence and properties of very stable Higgs bundles.
method Bialynicki-Birula theory, C∗-actions, Hecke transformations, Fourier-Mukai transforms. result Precise formula for multiplicity of very stable components of global nilpotent cone.
In this note, by using the Yang-Mills-Higgs flow, we show that semistable Higgs bundles with vanishing the first and second Chern numbers over compact Käher manifolds must admit a filtration whose quotients are Hermitian flat Higgs bundles.
We generalize the classical Beauville-Narasimhan-Ramanan correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs V-bundles. Using this correspondence along with Bott-Morse theoretic techniques we provide an exact component count for moduli spaces of maximal parabolic $\text{Sp}\left…
Defines and classifies toric co-Higgs bundles on projective toric varieties.
problem Classifying co-Higgs bundles on toric varieties.
method Using Klyachko's fan filtration and studying the co-Higgs bundle fiber at a closed point.
result Provides a Lie-theoretic classification of toric co-Higgs bundles.
We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact Kähler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they…
We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the ba…
Geometric structures on 5-manifolds from surface group representations of G2'.
problem Constructing geometric structures on 5-manifolds from G2'-surface group representations.
method Using Higgs bundles and partial flag manifolds of G2' to construct geometric structures.
result Developing maps of geometric structures are the domain of discontinuity.
In this article, we study the Higgs vector bundles (E,θ) over a compact Calabi-Yau manifolds X. We use Yang-Mills-Higgs flow to prove that if a semistable Higgs bundle with vanishing Chern classes over a compact connected Calabi-Yau manifold, then the Higgs field θ is trivial. In particular, the vector bundle E…
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. Study Poisson metrics on noncompact Kähler manifolds and their Higgs bundle applications.
problem Existence of Poisson metrics on flat vector bundles over noncompact Riemannian manifolds.
method Generalization of Corlette-Donaldson-Hitchin-Simpson's nonabelian Hodge correspondence to noncompact Kähler manifolds.
result Existence of Poisson metrics on Higgs bundles over noncompact Kähler manifolds.
We prove a Hitchin-Kobayashi correspondence for extensions of Higgs bundles. The results generalize known results for extensions of holomorphic bundles. Using Simpson's methods, we construct moduli spaces of stable objects. In an appendix we construct Bott-Chern forms for Higgs bundles
Study families of flat connections with nilpotent Higgs fields, showing similar monodromy to regular Higgs bundles.
problem Investigate Cimes-families of flat connections with nilpotent Higgs fields. method Analyze families of flat connections including real twistor lines and conformal limits, deducing monodromy similarities.
result Traces of holonomies are asymptotically exponential in rational powers of the parameter of the family.