arXiv research

We study simple root flows and Liouville currents for Hitchin representations. We show that the Liouville current is associated to the measure of maximal entropy for a simple root flow, derive a Liouville volume rigidity result, and construct a Liouville pressure metric on the Hitchin component.

Study of Hitchin map on specific Higgs bundles.

In this article we define new flows on the Hitchin components for PGL(V). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present…

Characterizes flag geometries for Hitchin representations in SL3(R).

Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.

New flows introduced for symplectic geometry.

The Hitchin flow constructs eight-dimensional Riemannian manifolds (M,g) with holonomy in Spin(7) starting with a cocalibrated G_2-structure on a seven-dimensional manifold. As Sp(2)\subseteq SU(4)\subseteq Spin(7), one may also obtain Calabi-Yau fourfolds or hyperKähler manifolds via the Hitchin flow. In this paper, w…

Investigates nearly Kähler and parallel G2 manifolds using Hitchin functionals.

In this short paper, we prove a Hitchin-Thorpe type inequality for closed 4-manifolds with non-positive Yamabe invariant, and admitting long time solutions of the normalized Ricci flow equation with bounded scalar curvature.

Infinite volume found in the thick part of $\mathrm{PSL}_n(\mathbb{R})$-Hitchin-Riemann moduli space.

Study shows infinite volumes of moduli spaces for certain groups.

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.

We investigate left-invariant Hitchin and hypo flows on $5$-, $6$- and $7$-dimensional Lie groups. They provide Riemannian cohomogeneity-one manifolds of one dimension higher with holonomy contained in $SU(3)$, $G_2$ and $Spin(7)$, respectively, which are in general geodesically incomplete. Generalizing results of Cont…

Study on flows of G2-structures on contact Calabi-Yau 7-manifolds.

In 1990, Hitchin's proved a component of the space of representations of a surface group in SL(n,R) is homeomorphic to a ball. For n=2,3 this component has been identified with the holonomies of geometric structures (hyperbolic for n=2, or real projective for n=3). In the preprint "Anosov flows, Surface groups and Curv…

Solutions of Hitchin's self-duality equations corresponds to special real sections in the Deligne-Hitchin moduli space -- twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would allow for complex analytic …

Study of parabolic Higgs bundles on curves with special fixed points.

The paper proves unboundedness of a functional on G2 forms and describes manifold limits.

The paper connects Ricci flow and harmonic spinors, proving new inequalities.

In this article, we produce infinite families of 4-manifolds with positive first betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the non-vanishing of the stable cohomotopy Seiberg-Witten invariants of their connected sums. Elementary building blocks used in the earlier …

Surveying Hitchin representations of Fuchsian groups.

The paper studies Lagrangian structures in Higgs bundle moduli spaces and their conformal limits.

Geodesic concavity and hypersymplectic structures in $G2$-structures space.

This paper is the first step in a systematic project to study examples of Kähler manifolds with positive holomorphic sectional curvature ($H > 0$). Previously Hitchin proved that any compact Kähler surface with $H>0$ must be rational and he constructed such examples on Hirzebruch surfaces $M_{2, k}=\mathbb{P}(H^{k}\opl…

This paper deals with moduli spaces of framed principal bundles with connections with irregular singularities over a compact Riemann surface. These spaces have been constructed by Boalch by means of an infinite-dimensional symplectic reduction. It is proved that the symplectic structure induced from the Atiyah--Bott fo…

In this paper we study non-singular solutions of Ricci flow on a closed manifold of dimension at least 4. Amongst others we prove that, if M is a closed 4-manifold on which the normalized Ricci flow exists for all time t>0 with uniformly bounded sectional curvature, then the Euler characteristic $χ(M)\ge 0$. Moreover, …

Defines and parametrizes $\mathfrak{sl}(2)$-type singular fibres in symplectic and odd orthogonal Hitchin systems.

Real projective surfaces with Hitchin holonomy can be related via grafting.

Study asymptotics of hyperkähler geometry on singular fibers of Hitchin moduli space.

Generic Hitchin representations generate dense subgroups.

New estimates for Hitchin's equations at high energy.

Study shows rapid decay of Hitchin metric from semi-flat metric on Higgs bundles.

Study of a basic Hitchin equation on Sasakian 3-folds, showing hyperKähler metric.

Study pressure metrics for cusped Hitchin representations.

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

The paper studies sequences of solutions to Hitchin-Simpson equations on Kähler manifolds.

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 consider Hitchin's hyperkähler metric $g_{L^2}$ on the $SU(n)$-Hitchin moduli space moduli space over a compact Riemann surface. We prove that the difference between the metric $g_{L^2}$ and a simpler "semiflat" hyperkähler metric $g_{\mathrm{sf}}$ 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 paper proves Hitchin moduli spaces are ALG gravitational instantons.

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…

In this paper we construct a Hitchin connection in a setting, which significantly generalizes the setting covered by the first author previously, which in turn was a generalisation of the moduli space case covered by Hitchin in his original work on the Hitchin connection. In fact, our construction provides a Hitchin co…

The abstract discusses rational approximations for Hitchin representations on surfaces.

Generic Hitchin representations avoid hyperplanes in Lie algebras.

Describes spectral data for singular fibres of a specific Hitchin system.

In this paper we calculate the curvature of the Hitchin connection. We further show that a slight (possibly trivial) modification of the Hitchin connection has curvature equal to an explict given multiple of the Weil-Petersen symplectic form on Teichmüller space.

We study the gradient flow lines of a Yang-Mills-type functional on the space of gauged holomorphic maps $\mathcal{H}(P,X)$, where $P$ is a principal bundle on a Riemann surface $Σ$ and $X$ is a Kähler Hamiltonian $G$-manifold. For compact $Σ$, possibly with boundary, we prove long time existence of the gradient flow. …