Harmonic maps into conic surfaces with cone angles less than $2\pi$

TL;DR

This paper establishes the existence and uniqueness of harmonic maps from closed surfaces to conic surfaces with cone angles less than 2π, including minimization properties for certain punctured surfaces.

Contribution

It proves existence and uniqueness of harmonic maps into conic surfaces with angles less than 2π, extending the theory to surfaces with conic singularities.

Findings

Harmonic maps exist and are unique in specified homotopy classes

Minimization in relative homotopy classes is achieved for cone angles ≤ π

Results apply to punctured Riemann surfaces with conic metrics

Abstract

We prove the existence and uniqueness of harmonic maps in degree one homotopy classes of closed, orientable surfaces of positive genus, when the target has conic points with cone angles less than $2\pi$. For a cone point $p$ of cone angle less than or equal $\pi$ we show that one can minimize, uniquely, in the relative homotopy class of a homeomorphism sending a fixed point $q$ in the domain to $p$. The latter can be interpreted as minimizing maps from punctured Riemann surfaces.