Teichm\"uller theory for conic surfaces

TL;DR

This paper develops a deformation theory for conic constant curvature metrics on closed surfaces, defining a Teichmüller space for such metrics with cone angles less than 2π, using advanced elliptic operator techniques.

Contribution

It introduces a systematic deformation framework for conic metrics on surfaces, generalizing classical Teichmüller theory with new elliptic operator methods.

Findings

Defined and studied the Teichmüller space of conic metrics.

Extended Tromba's approach to conic surfaces.

Utilized elliptic conic operators for analysis.

Abstract

In this paper we develop a systematic deformation theory for conic constant curvature metrics on a closed surface when all cone angles are less than $2\pi$; in particular, we define and study the Teichm\"uller space $\mathcal{T}^{\mathrm{conic}}_{\gamma,k}$ of conic constant curvature metrics on a surface of genus $\gamma$ with $k$ conic points. The methods here are adopted from higher dimensional global analysis, generalizing Tromba's approach to the study of the standard Teichm\"uller space $\mathcal{T}_\gamma$. The main new ingredient is the theory of elliptic conic operators.