Heat flows on hyperbolic spaces

TL;DR

This paper introduces new methods for analyzing heat flow convergence on hyperbolic spaces, proving that quasiconformal maps on spheres extend to harmonic quasi-isometries in hyperbolic space, confirming a longstanding conjecture.

Contribution

It develops novel techniques for heat flow convergence and proves the Schoen-Li-Wang conjecture for all dimensions n ≥ 3.

Findings

Quasiconformal maps extend to harmonic quasi-isometries in hyperbolic space.

Heat flow starting from these extensions converges to harmonic maps.

Confirms the Schoen-Li-Wang conjecture for n ≥ 3.

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.