Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces

@article{Mirzakhani2006SimpleGA,
  title={Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces},
  author={Maryam Mirzakhani},
  journal={Inventiones mathematicae},
  year={2006},
  volume={167},
  pages={179-222}
}

Figures and Tables from this paper

Moduli of bordered Riemann Surfaces - Complex structure and K\
We introduce natural complex structures on the Teichmül-ler space of bordered Riemann surfaces. The general case follows from the situation of one closed geodesic γ on a Riemann surface X of genus pExpand
Families of Riemann Surfaces and Weil-Petersson Geometry
This book is the companion to the CBMS lectures of Scott Wolpert at Central Connecticut State University. The lectures span across areas of research progress on deformations of hyperbolic surfacesExpand
MODULI SPACES OF SURFACES
  • Yi Huang
  • Mathematics
  • Bulletin of the Australian Mathematical Society
  • 2015
Moduli space theory and Teichmüller space theory are concerned with Riemann surfaces. The moduli space M(R) of a topological surface R (with negative Euler characteristic) is the space of allExpand
The moduli space of maps with crosscaps: the relative signs of the natural automorphisms
Just as a symmetric surface with separating fixed locus halves into two oriented bordered surfaces, an arbitrary symmetric surface halves into two oriented symmetric half-surfaces, i.e. surfaces withExpand
JT gravity as a matrix integral
We present exact results for partition functions of Jackiw-Teitelboim (JT) gravity on two-dimensional surfaces of arbitrary genus with an arbitrary number of boundaries. The boundaries are of theExpand
Convex hulls in hyperbolic 3-space and generalized orthospectral identities
We begin this dissertation by studying the relationship between the Poincaré metric of a simply connected domain Ω ⊂ C and the geometry of Dome(Ω), the boundary of the convex hull of its complement.Expand
New identities for small hyperbolic surfaces
Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holedExpand
Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves
We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers ofExpand
A pr 2 01 8 UNIFORM BOUNDS FOR WEIL-PETERSSON CURVATURES
We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. AmongExpand
Geodesics with one self-intersection, and other stories
Abstract In this paper, we show that for any hyperbolic surface S , the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic γ with a singleExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 31 REFERENCES
The Fenchel-Nielsen deformation
The uniformization theorem provides that a Riemann surface S of negative Euler characteristic has a metric of constant curvature -1. A hyperbolic structure can be understood in terms of itsExpand
The Weil-Petersson volume of the moduli space of punc-tured spheres
A V-shaped clamp has opposed first and second ends. The first end can accommodate the head of a conventional reduction wrench. The second end supports a threaded rod that approaches the first endExpand
Areas of two-dimensional moduli spaces
Wolpert’s formula expresses the Weil-Petersson 2-form in terms of the Fenchel-Nielsen coordinates in case of a closed or punctured surface. The area-form in Fenchel-Nielsen coordinates is invariantExpand
Geodesics with bounded intersection number on surfaces are sparsely distributed
LET M be a surface of negative Euler characteristic, possibly with boundary, which is either compact or obtained from a compact surface by removing a finite set of points. Let D be the Poincar~ disc.Expand
THE ORTHOGONAL SPECTRUM OF A HYPERBOLIC MANIFOLD
Introduction. In this paper, we investigate the geometry of embedded hypersurfaces in hyperbolic manifolds of dimension greater than one. Our focus is on hypersurfaces that are either totallyExpand
Generalizations of McShane's identity to hyperbolic cone-surfaces
We generalize McShane’s identity for the length series of simple closed geodesics on a cusped hyperbolic surface [19] to a general identity for hyperbolic cone-surfaces (with all cone angles ≤ π),Expand
Weil-Petersson volumes and intersection theory on the moduli space of curves
In this paper, we establish a relationship between the Weil-Petersson volume Vgin(b) of the moduli space Mg,n(b) of hyperbolic Riemann surfaces with geodesic boundary components of lengths b\,...,bn,Expand
INTERSECTION THEORY ON MODULI SPACES OF HOLOMORPHIC BUNDLES OF ARBITRARY RANK ON A RIEMANN SURFACE
Let n and d be coprime positive integers, and define M(n,d) to be the moduli space of (semi)stable holomorphic vector bundles of rank n, degree d and fixed determinant on a compact Riemann surface E.Expand
Symplectic geometry and the Verlinde Formulas
The purpose of this paper is to give a proof of the Verlinde formulas by applying the Riemann-Roch-Kawasaki theorem to the moduli space of flat G-bundles on a Riemann surface Σ with marked points,Expand
Necessary and Sufficient Conditions for Mcshane’s Identity and Variations
Greg McShane introduced a remarkable identity for lengths of simple closed geodesics on the once punctured torus with a complete, finite volume hyperbolic structure. Bowditch later generalized thisExpand
...
1
2
3
4
...