Harmonic maps and the Schoen conjecture

Abstract

Abstract. We show that every quasisymmetric homeomorphism of the circle ∂H 2 admits a harmonic quasiconformal extension to the hyperbolic plane H 2 . This proves the Schoen Conjecture.

Quasiconformal and HQC mappings between Lyapunov Jordan domains

“…Very recently, concerning the initial Schoen Conjecture (and more generally the Schoen-Li-Wang conjecture) Markovic made a major breakthrough. In [21], Markovic used the result of Li and Tam that every diffeomorphism of S 2 admits a harmonic quasiisometric extension to show that every quasisymmetric homeomorphism of the circle ∂H 2 admits a harmonic quasiconformal extension to the hyperbolic plane H 2 . This proves the initial Schoen Conjecture.…”

Harmonic quasi-isometric maps between rank one symmetric spaces

“…Partial results towards the existence statement in the Schoen conjecture were obtained in [35], [14], [28], [22], [4]. A major breakthrough was then achieved by Markovic who proved successively the Li-Wang conjecture for the case X = Y = H 3 R in [24], for the case X = Y = H 2 R in [23] thus solving the initial Schoen conjecture, and very recently with Lemm for the case X = Y = H p R with p ≥ 3 in [18]. As a corollary of Theorem 1.1, we complete the proof of the Li-Wan conjecture.…”

“…and, adapting an idea of Markovic in [23], to focus on the set The contradiction will come from the fact that when both R and ρ R go to infinity, the two angles θ 1 := θ(v f (z), v h (z)) and θ 2 := θ(v h (z), v R ) converge to 0 uniformly for z in W R , while one can find z = z R in W R such that the other angle θ 0 = θ(v f (z), v R ) stays away from 0. Here is a rough sketch of the arguments used to estimate these three angles.…”

Maximal surfaces in Anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms

“…Classical quasiconformal extensions include, for instance, the Beurling-Ahlfors extension and the Douady-Earle extension. More recently, Markovic [Mar17] proved the existence of quasiconformal harmonic extensions, where the harmonicity is referred to the complete hyperbolic metric of H 2 . Moreover, the maximal dilatation of the classical extensions has been widely studied.…”