Journal of Differential Geometry

Heat flows on hyperbolic spaces

Marius Lemm and Vladimir Markovic

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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.


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

Article information

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

Received: 14 November 2015
First available in Project Euclid: 2 March 2018

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

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