Harmonic projections in negative curvature II: large convex sets

An important result in the theory of harmonic maps is due to Benoist--Hulin: given a quasi-isometry $f:X\to Y$ between pinched Hadamard manifolds, there exists a unique harmonic map at a finite distance from $f$. Here we show existence of harmonic maps under a weaker condition on $f$, that we call non-collapsing -- we require that the following two conditions hold uniformly in $x\in X$: (1) average distance from $f(x)$ to $f(y)$ for $y$ on the sphere of radius $R$ centered at $x$ grows linearly with $R$ (2) the pre-image under $f$ of small cones with apex $f(x)$ have low harmonic measures on spheres centered at $x$. Using these ideas, we also continue the previous work of the author on existence of harmonic maps that are at a finite distance from projections to certain convex sets. We show this existence in a pinched negative curvature setting, when the convex set is large enough. For hyperbolic spaces, this includes the convex hulls of open sets in the sphere at infinity with sufficiently regular boundary.