Related papers: Minimal surfaces in germs of hyperbolic 3--manifol…

We show that the algebraic intersection form of hyperbolic surfaces of genus $g$ has a minimum in the moduli space and that the minimum grows in the order $(\log g)^{-2}$ in terms of the genus. We also describe the asymptotic behavior of…

We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already…

Let $M$ be the moduli space of rank 3 parabolic vector bundles over a Riemann surface with several punctures. By the Mehta-Seshadri correspondence, this is the space of rank 3 unitary representations of the fundamental group of the…

A Teichm\"uller space $Teich$ is a quotient of the space of all complex structures on a given manifold $M$ by the connected components of the group of diffeomorphisms. The mapping class group $\Gamma$ of $M$ is the group of connected…

We count the connected components in the moduli space of PU(p,q)-representations of the fundamental group for a closed oriented surface. The components are labelled by pairs of integers which arise as topological invariants of the flat…

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space and anti-de Sitter 3-space is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group…

For any maximal surface group representation into $\mathrm{SO}_0(2,n+1)$, we introduce a non-degenerate scalar product on the the first cohomology group of the surface with values in the associated flat bundle. In particular, it gives rise…

By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with $n$ cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension $n-3$. The Hermitian form comes from the area of the…

For any almost-simple group $G$ over an algebraically closed field $k$ of characteristic zero, we describe the automorphism group of the moduli space of semistable $G$-bundles over a connected smooth projective curve $C$ of genus at least…

Let G be a split semisimple algebraic group with trivial center. Let S be a compact oriented surface, with or without boundary. We define {\it positive} representations of the fundamental group of S to G(R), construct explicitly all…

Fixing a closed hyperbolic surface S, we define a moduli space AI(S) of unmarked hyperbolic 3-manifolds homotopy equivalent to S. This 3-dimensional analogue of the moduli space M(S) of unmarked hyperbolic surfaces homeomorphic to S has…

We extend Hitchin's results on "The self-duality equations on a Riemann surface" (Proc. LMS (3), vol. 55, 1987) to orbifold Riemann surfaces. We prove existence results for orbifold solutions of the Yang-Mills-Higgs equations and construct…

It is one of the wonderful ``coincidences'' of the theory of finite groups that the simple group G of order 25920 arises as both a symplectic group in characteristic 3 and a unitary group in characteristic 2. These two realizations of G…

The Hessian of a general cubic surface is a nodal quartic surface, hence its desingularisation is a K3 surface. We determine the transcendental lattice of the Hessian K3 surface for various cubic surfaces (with nodes and/or Eckardt points…

We describe the moduli space of Higgs pairs on an irreducible nodal curve of arithmetic genus one and its geometric structures in terms of the Hitchin map and a flat degeneration of the moduli space of Higgs bundles on an elliptic curve.

A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which is characterized by the condition that the second fundamental form is proportional to the metric. It is also a generalization of the notion of a…

We show that each connected component of the moduli space of smooth real binary quintics is isomorphic to an open subset of an arithmetic quotient of the real hyperbolic plane. Moreover, our main result says that the induced metric on this…

In this article, we revisit classical length identities enjoyed by simple closed curves on hyperbolic surfaces. We state and prove the rigidity of such identities over Teichm\"uller spaces. Due to this rigidity, certain collections of…

In this manuscript, we aim to classify and characterize the moduli space of homogeneous spin connections and homogeneous SU(2) connections on three-dimensional Riemannian homogeneous spaces. An analysis of the topology of the associated…

We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing…