Schoen–Yau conjecture

In mathematics, the Schoen–Yau conjecture is a disproved conjecture in hyperbolic geometry, named after the mathematicians Richard Schoen and Shing-Tung Yau.

It was inspired by a theorem of Erhard Heinz (1952). One method of disproof is the use of Scherk surfaces, as used by Harold Rosenberg and Pascal Collin (2006).

Let ${\displaystyle \mathbb {C} }$ be the complex plane considered as a Riemannian manifold with its usual (flat) Riemannian metric. Let ${\displaystyle \mathbb {H} }$ denote the hyperbolic plane, i.e. the unit disc

endowed with the hyperbolic metric

E. Heinz proved in 1952 that there can exist no harmonic diffeomorphism

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