Harmonic projections in negative curvature

TL;DR

This paper constructs harmonic maps close to nearest-point retractions to convex sets in negatively curved manifolds, providing new tools for understanding geometric structures in hyperbolic spaces.

Contribution

It introduces methods to approximate nearest-point retractions with harmonic maps in negatively curved manifolds, extending previous results to broader settings.

Findings

Harmonic maps can be constructed near nearest-point retractions in hyperbolic space.

The construction applies to convex hulls of sets at infinity with certain dimensional constraints.

Results hold for manifolds with isometry groups acting with cobounded orbits.

Abstract

In this paper we construct harmonic maps that are at a bounded distance from nearest-point retractions to convex sets, in negatively curved manifolds. Specifically, given a quasidisk $Q$ in hyperbolic space, we construct a harmonic map to the hyperbolic plane that corresponds to the nearest-point retraction to the convex hull of $Q$. If $M$ is a pinched Hadamard manifold so that its isometry group acts with cobounded orbits, and if $S$ is a set in the boundary at infinity of $M$, with the property that all elements of its orbit under the isometry group of $M$ have dimension less than $\frac{n-1}{2}$, we show that the nearest-point retraction to the convex hull of $S$ is a bounded distance away from some harmonic map.