Related papers: A projection from filling currents to Teichm\"ulle…

A Teichm\"uller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichm\"uller curves of positive genus. Our methods are based on…

Given a closed, oriented surface X of genus g>1, and a semisimple Lie group G, let R_G be the moduli space of reductive representations of the fundamental group of X in G. We determine the number of connected components of R_PGL(n,R), for…

Let $S$ be a compact, connected, oriented surface, possibly with boundary, of negative Euler characteristic. In this article we extend Lindenstrauss-Mirzakhani's and Hamenst\"adt's classification of locally finite mapping class group…

We prove, for any n, that there is a closed connected orientable surface S so that the hyperbolic space H^n almost-isometrically embeds into the Teichm\"uller space of S, with quasi-convex image lying in the thick part. As a consequence,…

Given an embedding of a smooth projective curve $X$ of genus $g\geq1$ into $\mathbb{P}^N$, we study the locus of linear subspaces of $\mathbb{P}^N$ of codimension 2 such that projection from said subspace, composed with the embedding, gives…

We provide a complete classification of Teichm\"uller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichm\"uller curves intersect the boundary of the…

Let $M$ be a closed hyperbolic $3$-manifold. A homotopy class $[S]$ of surfaces in $M$ is filling if any representative cuts $M$ into components contractible in $M$. We prove that there exist $\epsilon_0, g_0>0$ such that every homotopy…

An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichm\"{u}ller and moduli spaces. The tensor is evaluated on the gradients of geodesic-lengths for disjoint geodesics. A precise lower…

We consider circle patterns on surfaces with complex projective structures. We investigate two symplectic forms pulled back to the deformation space of circle patterns. The first one is Goldman's symplectic form on the space of complex…

In this paper, we construct spines, i.e., $\Mod_g$-equivariant deformation retracts, of the Teichm\"uller space $\T_g$ of compact Riemann surfaces of genus $g$. Specifically, we define a $\Mod_g$-stable subspace $S$ of positive codimension…

We classify all closed 1-connected manifolds $M$ which look like projective planes, i.e. with integral homology $H_*(M)=Z^3$. Furthermore, we give an explicit construction of these manifolds as Thom spaces of open disk bundles.

Thurston boundary of the universal Teichm\"uller space $T(\mathbb{D})$ is the space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of $\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of generalized Teichm\"uller type…

We construct a new Riemannian metric on Goldman space $\mathcal{B}(S)$, the space of the equivalence classes of convex projective structures on the surface $S$, and then prove the new metric, as well as the metric of Darvishzadeh and…

We study the curvature of the moduli space M_g of curves of genus g with the Siegel metric induced by the period map. We give an explicit formula for the holomorphic sectional curvature of M_g along a Schiffer variation at a point P on the…

This paper has 3 principal goals: (1) to survey what is know about mapping class and Torelli groups of simply connected compact Kaehler manifolds, (2) supplement these results, and (3) present a list of questions and open problems to…

We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.

This is an introduction to the algebraic aspect of Teichm\"uller dynamics, with a focus on its interplay with the geometry of moduli spaces of curves as well as recent advances in the field.

The space of Hitchin representations of the fundamental group of a closed surface $S$ into $\text{SL}_n\mathbb{R}$ embeds naturally in the space of projective oriented geodesic currents on $S$. We find that currents in the boundary have…

The primary goal of this paper is to find a homotopy theoretic approximation to moduli spaces of holomorphic maps Riemann surfaces into complex projective space. There is a similar treatment of a partial compactification of these moduli…

A projective structure on a compact Riemann surface X of genus g is given by an atlas with transition functions in PGL(2,C). Equivalently, a projective structure is given by a projective sl(2,C)-bundle over X equipped with a section s and a…