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March 2018 Heat flows on hyperbolic spaces
Marius Lemm, Vladimir Markovic
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J. Differential Geom. 108(3): 495-529 (March 2018). DOI: 10.4310/jdg/1519959624


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.

Funding Statement

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


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Marius Lemm. Vladimir Markovic. "Heat flows on hyperbolic spaces." J. Differential Geom. 108 (3) 495 - 529, March 2018.


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

zbMATH: 06846984
MathSciNet: MR3770849
Digital Object Identifier: 10.4310/jdg/1519959624

Rights: Copyright © 2018 Lehigh University


Vol.108 • No. 3 • March 2018
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