Heat flows on hyperbolic spaces

Marius Lemm, Vladimir Markovic
2018 Journal of differential geometry  
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 S n−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 S n−1 , n ≥ 3, can be extended to a harmonic quasi-isometry of the n-dimensional hyperbolic space.
doi:10.4310/jdg/1519959624 fatcat:l6yvq7i6ffatngz62rxa5fctsq