arXiv research

Study improves Simpson's estimate for cyclic harmonic bundles.

Develops Lie-theoretic perspective on Hitchin's equations for cyclic G-Higgs bundles.

Unified rigidity theorem for cyclic and alternating surfaces.

Solves Dirichlet problem for generalized Hitchin's equation on cyclic Higgs bundles.

Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.

Complete solutions found for Toda equations on non-compact surfaces.

The paper studies entropy of harmonic metrics on cyclic Higgs bundles.

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 paper studies entropy and free energy for harmonic metrics on cyclic Higgs bundles.

The paper extends the uniqueness of complete harmonic metrics to subharmonic weights and proves their existence on the unit disc.

Develops complex harmonic maps for Teichmüller theory, proving new theorems.

Solves generalized Kazdan-Warner equations on foliated manifolds.

Let $S$ be a closed surface of genus at least $2$. For each maximal representation $ρ: π_1(S)\rightarrow\mathsf{Sp}(4,\mathbb{R})$ 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…

The paper classifies Toda equations for noncompact symmetric spaces and their solutions.

The study finds nondegenerate harmonic 1-forms using symmetry conditions.

Geometric methods for surface group representations in higher rank SL(2m+1,R).

Develop a variational framework for statistical inference on cyclic interactions.

This paper generalizes a result about bounded differentials to higher-order differentials and studies their geometric implications.

Axis bundles in free-by-cyclic groups have non-generic monodromies.

Study uses Vinberg pairs for Higgs bundles, revealing their role.

Conditions for equivariant bundles on 4-manifolds with cyclic actions.

Criterion for flat circle bundles using intrinsically harmonic forms.

The aim of this paper is to calculate the eta invariants and the dimensions of the spaces of harmonic spinors of an infinite family of closed flat manifolds. It consits of some flat manifolds M with cyclic holonomy groups.

The paper examines the limit of harmonic flow on flat vector bundles.

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 study (i) asymptotic behaviour of wild harmonic bundles, (ii) the relation between semisimple meromorphic flat connections and wild harmonic bundles, (iii) the relation between wild harmonic bundles and polarized wild pure twistor $D$-modules. As an application, we show the hard Lefschetz theorem for algebraic semis…

Study on existence of harmonic metrics for non-Hermitian Yang-Mills bundles.

We give a construction of cyclic cocycles representing the equivariant characteristic classes of equivariant bundles. Our formulas generalize Connes' Godbillon-Vey cyclic cocycle. An essential tool of our construction is Connes-Moscovici's theory of cyclic cohomology of Hopf algebras.

This paper generalizes a topological invariant to cyclic Higgs bundles.

Study on harmonic metrics for rank 3 Higgs bundles in Hitchin section.

We prove the Kobayashi-Hitchin correspondence between good wild harmonic bundles and polystable good filtered $λ$-flat bundles satisfying a vanishing condition. We also study the correspondence for good wild harmonic bundles with the homogeneity with respect to a group action, which is expected to provide another way t…

Study harmonic metrics on Higgs bundles on non-compact Riemann surfaces.

Study harmonicity of normal almost contact structures on Riemannian manifolds.

Two constructions of Chern character for equivariant vector bundles in noncommutative geometry.

Study Higgs sections and flat sections for nonlinear harmonic bundles.

Harmonic metrics are established for SO0(n,n)-Higgs bundles on non-compact hyperbolic surfaces.

We define the notion of a loop Hodge structure -- an infinite dimensional generalization of a Hodge structure -- and prove that a suitable variation of this object over a complex manifold is equivalent to the datum of a harmonic bundle. Hence one can study harmonic bundles using classical tools of Hodge theory, especia…

The paper develops harmonic theory on vector bundles with singular metrics and extends results from complex geometry.

Study shows unique harmonic metrics for certain Higgs bundles over non-compact surfaces.

Study harmonicity on tangent bundles with a specific metric.

We introduce a Hopf algebroid associated to a proper Lie group action on a smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is equal to the de Rham cohomology of invariant differential forms. When the action is cocompact, we develop a generalized Hodge theory for the de Rham cohomology of inv…

We study harmonic sections of a Riemannian vector bundle whose total space is equipped with a 2-parameter family of metrics which includes both the Sasaki and Cheeger-Gromoll metrics. This enables the theory of harmonic unit sections to be extended to bundles with non-zero Euler class.

Study harmonicity of metrics in generalized Kantowski-Sachs spacetime.

This survey studies equivariant harmonic maps arising from Higgs bundles. We explain the non-abelian Hodge correspondence and focus on the role of equivariant harmonic maps in the correspondence. With the preparation, we review current progress towards some open problems in the study of equivariant harmonic maps.

Let $E$ be a holomorphic vector bundle. Let $θ$ be a Higgs field, that is a holomorphic section of $End(E)\otimesΩ^{1,0}_X$ satisfying $θ^2=0$. Let $h$ be a pluriharmonic metric of the Higgs bundle $(E,θ)$. The tuple $(E,θ,h)$ is called a harmonic bundle. Let $X$ be a complex manifold, and $D$ be a normal crossing divi…

Study characterizes martingales on fiber bundles for harmonic map analysis.

From the homotopy groups of two cubic spherical 3-manifolds we construct the isomorphic groups of deck transformations acting on the 3-sphere. These groups become the cyclic group of order eight and the quaternion group respectively. By reduction of representations from the orthogonal group to the identity representati…

We show a relationship between Chern-Simons 1- and 3-forms and harmonic forms on a principal bundle. Doing so requires one to consider an adiabatic limit. For the 3-form case, assume that G is simple and the corresponding Chern-Weil 4-form is exact. Then, the Chern-Simons 3-form on the princpal bundle G-bundle, minus a…