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 Sn−1,n≥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 Sn−1,n≥3, can be extended to a harmonic quasi-isometry of the n-dimensional hyperbolic space.