## Journal of Differential Geometry

### Heat flows on hyperbolic spaces

#### Abstract

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.

#### Note

Vladimir Markovic is supported by the NSF grant number DMS-1500951.

#### Article information

Source
J. Differential Geom., Volume 108, Number 3 (2018), 495-529.

Dates
First available in Project Euclid: 2 March 2018

https://projecteuclid.org/euclid.jdg/1519959624

Digital Object Identifier
doi:10.4310/jdg/1519959624

Mathematical Reviews number (MathSciNet)
MR3770849

Zentralblatt MATH identifier
06846984

#### Citation

Lemm, Marius; Markovic, Vladimir. Heat flows on hyperbolic spaces. J. Differential Geom. 108 (2018), no. 3, 495--529. doi:10.4310/jdg/1519959624. https://projecteuclid.org/euclid.jdg/1519959624