arXiv research

Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.

Riemann moduli spaces are quantum ergodic for certain dimensions.

In this paper we study the smooth moduli space of closed Riemann surfaces. This smooth moduli is an infinite cover of the usual moduli space $\mathscr{M}_g$ of closed Riemann surfaces, and is identified with the Schottky space of rank $g.$ The main theorem of the paper is: Closed Riemann surfaces are uniformizable by S…

Paper connects volumes of moduli spaces of super Riemann surfaces to integrals over stable Riemann surfaces.

Existence of symplectic structure shown on Seiberg-Witten moduli space product of Riemann surfaces.

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

Constructs moduli spaces for complex affine and dilation surfaces.

In this paper we continue our study on the moduli spaces of flat G-bundles, for any semi-simple Lie group G, over a Riemann surface by using heat kernel and Reidemeister torsion. Formulas for intersection numbers on the moduli spaces over a Riemann surface with several boundary components, over non-orientable Riemann s…

New invariant for Riemann surfaces connects to moduli space classes.

Survey of computations for Riemann surface moduli spaces, focusing on unstable homology.

The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.

Paper introduces hybrid curves moduli space and canonical measures.

Constructs a combinatorial model for bordered Riemann surfaces with a compactification.

The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of $λ$-connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli spa…

We prove a Gauss-Bonnet theorem for (finite coverings of) moduli spaces of Riemann surfaces endowed with the McMullen metric. The proof uses properties of an exhaustion of moduli spaces by compact submanifolds with corners and the Gauss-Bonnet formula of Allendoerfer and Weil for Riemannian polyhedra.

We compute the automorphism groups of the Dolbeault, de Rham and Betti moduli spaces for the multiplicative group ${\mathbb C}^*$ associated to a compact connected Riemann surface.

We survey our recent new results on the geometry of Teichmuller and moduli spaces of Riemann surfaces and Calabi-Yau manifolds.

The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact Kähler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, Kähler) structures $Ω_{Ψ_0}$ on the moduli space, parametrised by $Ψ_0$, a section o…

In this paper we modify the coordinate construction in our previous paper on the universal moduli space of pair consisting of a Riemann Surfaces and a stable holomorphic bundles on the Riemann Surface, so as to produce a new set of coordinates, which are in fact K\" ahler coordinates on this universal moduli space. Fur…

Study automorphism equivariant Hitchin index for Riemann surfaces.

Constructs a new mathematical structure for Riemann surfaces with projective structures.

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.

New bounds show triangulated surfaces are evenly distributed in moduli space.

Estimates the rational homological dimension of Riemann surfaces with boundary and marked points.

This survey covers earlier work of the author as well as recent work on Riemann's moduli space, its canonical cell decomposition and compactification, and the related operadic structure of arc complexes.

Riemann surfaces are two-dimensional manifolds with a conformal class of metrics. It is well known that the harmonic action functional and harmonic maps are tools to study the moduli space of Riemann surfaces. Super Riemann surfaces are an analogue of Riemann surfaces in the world of super geometry. After a short intro…

Survey on Higgs bundle moduli spaces and their connected components.

We consider the moduli space of flat $SO(2n+1)$-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring ge…

Constructs geometric models for moduli spaces of Higgs bundles over Riemann sphere.

These are notes on the theory of super Riemann surfaces and their moduli spaces, aiming to collect results that are useful for a better understanding of superstring perturbation theory in the RNS formalism.

Coarse density of subspaces in moduli space of Riemann surfaces.

Study SO(3) vortices over orbifold Riemann surfaces and monopoles.

The moduli space of holomorphic fiber bundles ${\cal M}_n(\Si)$ over a compact Riemann surface $\Si$ is considered. A formula for the regularised determinant and an other for the symplectic form at trivial bundle are proposed.

Mathematical analysis of Riemann surfaces and their moduli spaces using hybrid Laplacians.

The study proves semisimplicity of totally geodesic subvarieties in moduli spaces of Riemann surfaces.

Study continuity of phi-invariant for degenerating graphs.

In this paper, the moduli space of singular unitary Hermitian--Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bund…

This is a survey of our recent results on the geometry of moduli spaces and Teichmuller spaces of Riemann surfaces appeared in math.DG/0403068 and math.DG/0409220. We introduce new metrics on the moduli and the Teichmuller spaces of Riemann surfaces with very good properties, study their curvatures and boundary behavio…

On the thick part of the moduli space of Riemann surfaces, where there is a positive lower bound of the systole of the surface, we show that all Weil-Petersson Riemannian curvatures are bounded, independent of the genus of the surface.

Survey of methods for computing volumes of moduli spaces.

The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another complex analytic invariant called the harmonic volume. It is a natural extension o…

Computes the Euler characteristic and rational homology of moduli spaces of multi-monopoles.

As was shown by Harer the second homology of ${\mathbb M}_g$, the moduli space of compact Riemann surfaces of genus $g$, is of rank 1, provided $g \geq 3$. This means a nontrivial second de Rham cohomology class on ${\mathbb M}_g$ is unique up to constant factor. But several canonical 2-forms on the moduli space have b…

The Weil-Petersson metric's curvature vanishes for surfaces with short geodesics.

A new projective structure on Riemann surfaces differs from the uniformization theorem's structure.

We show that the prequantum line bundle on the moduli space of flat $SU(2)$ connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of fl…

In this article we show that any finite cover of the moduli space of closed Riemann surfaces of $g$ genus with $g\geq 2$ does not admit any complete finite-volume Hermitian metric of non-negative scalar curvature. Moreover, we also show that the total mass of the scalar curvature of any almost Hermitian metric, which i…

The paper constructs a canonical connection on bundles over Riemann surfaces and relates it to the theta divisor.