Limits of harmonic maps and crowned hyperbolic surfaces

TL;DR

This paper investigates the limits of harmonic diffeomorphisms from degenerating Riemann surfaces to a fixed hyperbolic target, revealing new geometric structures and providing a novel proof of harmonic map existence to crowned hyperbolic surfaces.

Contribution

It establishes the existence of limiting harmonic maps from conformal limits of degenerating surfaces to crowned hyperbolic targets, extending previous work with new geometric insights.

Findings

Limiting harmonic maps from degenerating surfaces to crowned hyperbolic surfaces exist.

Conformal limits involve attaching half-planes and cylinders to the critical graph.

New proof of harmonic map existence from punctured surfaces to crowned hyperbolic targets.

Abstract

We consider harmonic diffeomorphisms to a fixed hyperbolic target $Y$, from a family of domain Riemann surfaces degenerating along a Teichm\"{u}ller ray. We use the work of Minsky to show that there is a limiting harmonic map from the conformal limit of the Teichm\"{u}ller ray, to a crowned hyperbolic surface. The target surface is the metric completion of the complement of a geodesic lamination on $Y$. The conformal limit is obtained by attaching half-planes and cylinders to the critical graph of the holomorphic quadratic differential determining the ray. As an application, we provide a new proof of the existence of harmonic maps from any punctured Riemann surface to a given crowned hyperbolic target of the same topological type.