Related papers: On normalizations of Thurston measure on the space…

Given a complex structure, we investigate diverging sequences of projective structures on the fixed complex structure in terms of Thurston's parametrization. In particular, we will give a geometric proof to the theorem by Kapovich stating…

Let $S$ be a closed orientable surface with genus $g\geq 2$. For a sequence $\s_i$ in the Teichm\"uller space of $S$, which converges to a projective measured lamination $[\lam]$ in the Thurston boundary, we obtain a relation between $\lam$…

Measure homology was introduced by Thurston in order to compute the simplicial volume of hyperbolic manifolds. Berlanga endowed measure homology with a structure of graded locally convex (possibly non-Hausdorff) topological vector space. In…

We show that every $\mathbb R$-linear surjective isometry between the cotangent spaces to the Teichm\"uller space equipped with the Thurston norm is induced by some isometry between the underlying hyperbolic surfaces, which is an analogue…

Let $\Sigma_g$ be a closed oriented surface of genus g and let $H_\mathbb{Q}$ denote $H_1(\Sigma_g;\mathbb{Q})$ which we understand to be the standard symplectic vector space over $\mathbb{Q}$ of dimension $2g$. We introduce a canonical…

In his influential work, Thurston introduced a norm on the second homology group of compact orientable 3-manifolds M, which by duality also determines a dual norm on the second cohomology group. A natural question, initiated by Thurston, is…

We undertake a systematic study of the infinitesimal geometry of the Thurston metric, showing that the topology, convex geometry and metric geometry of the tangent and cotangent spheres based at any marked hyperbolic surface representing a…

In [Mor], we have introduced a notion of flat laminations on surfaces endowed with a flat structure, similar to geodesic laminations on hyperbolic surfaces. Here is a sequel to this article that aims at defining transversal measures on flat…

We view all smooth metrics $g$ on a closed surface $\Sigma$ through their Nash isometric embeddings $f_g: (\Sigma,g) \rightarrow (\mathbb{S}^{\tilde{n}}, \tilde{g})$ into a standard sphere of large, but fixed, dimension $\tilde{n}$. We…

For the symmetric space sigma model in the internal metric formalism we explicitly construct the lagrangian in terms of the axions and the dilatons of the solvable Lie algebra gauge and then we exactly derive the axion-dilaton field…

The present paper is composed of two parts. In the first one we define two pseudo-metrics $L_F$ and $K_F$ on the Teichmu\"uller space of semi-translation surfaces $\mathcal{TQ}_g(\underline k,\epsilon)$, which are the symmetric counterparts…

We describe a construction of ordered algebraic structures (ordered abelian semigroups, ordered commutative semirings, etc.) and describe applications to codimension-1 laminations. For a suitable ordered semi- algebraic structure $\mathbb…

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all…

The aim of this paper is to relate Thurston's metric on Teichm\"uller space to several ideas initiated by T. Sorvali on isomorphisms between Fuchsian groups. In particular, this will give a new formula for Thurston's asymmetric metric for…

The spaces of Riemannian metrics on a closed manifold $M$ are studied. On the space ${\mathcal M}$ of all Riemannian metrics on $M$ the various weak Riemannian structures are defined and the corresponding connections are studied. The space…

We consider Riemannian metrics compatible with the natural symplectic structure on T^2 x M, where T^2 is a symplectic 2-Torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its…

In these memos, we define a pregeometry $\mathcal{T}_{\mathbb{S}} ^{alg}$ and a geometry $\mathcal{G}_{\mathbb{S}} ^{alg}$ which integrate symplectic manifolds with $E_{\infty}$-ring sheaves, enabling the construction of…

Let $G=\mathop{A\ast B}\limits_C$ be an amalgamated product of finite rank free groups $A$, $B$ and $C$. We introduce atomic measures and corresponding asymptotic densities on a set of normal forms of elements in $G$. We also define two…

About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of {\em hyperbolic geometry}, as natural as Euclid's regular polyhedra. For a closed…

Witten's Gauged Linear $\sigma$-Model (GLSM) unifies the Gromov-Witten theory and the Landau-Ginzburg theory, and provides a global perspective on mirror symmetry. In this article, we summarize a mathematically rigorous construction of the…