Teichm\"uller theory for conic surfaces

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.