  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.
