Teichm\"uller spaces as degenerated symplectic leaves in Dubrovin--Ugaglia Poisson manifolds

TL;DR

This paper explores the structure of Teichmüller spaces as special degenerated symplectic leaves within a Poisson algebra framework, linking geometric and algebraic aspects of Riemann surfaces with holes and orbifold points.

Contribution

It demonstrates that Teichmüller spaces for certain Riemann surfaces can be realized as real slices of degenerated symplectic leaves in a specific Poisson algebra.

Findings

Teichmüller spaces are realized as real slices of degenerated symplectic leaves.

The Goldman bracket between geodesic length functions is analyzed.

Connection established between geometric structures and algebraic Poisson manifolds.

Abstract

In this paper we study the Goldman bracket between geodesic length functions both on a Riemann surface $\Sigma_{g,s,0}$ of genus $g$ with $s=1,2$ holes and on a Riemann sphere $\Sigma_{0,1,n}$ with one hole and $n$ orbifold points of order two. We show that the corresponding Teichm\"uller spaces $\mathcal T_{g,s,0}$ and $\mathcal T_{0,1,n}$ are realised as real slices of degenerated symplectic leaves in the Dubrovin--Ugaglia Poisson algebra of upper--triangular matrices $S$ with 1 on the diagonal.