  In this paper we develop new methods for studying the convergence problem for
the heat flow on negatively curved spaces and prove that any quasiconformal map
of the sphere $\mathbb{S}^{n-1}$, $n\geq 3$, can be extended to the
$n$-dimensional hyperbolic space such that the heat flow starting with this
extension converges to a quasi-isometric harmonic map. This implies the
Schoen-Li-Wang conjecture that every quasiconformal map of $\mathbb{S}^{n-1}$,
$n\geq 3$, can be extended to a harmonic quasi-isometry of the $n$-dimensional
hyperbolic space.
