Schoen–Yau's harmonic-map parabolicity conjecture

Schoen and Yau posed the nonexistence conjecture in 1985. It asserted that a hyperbolic open Riemann surface admits no proper harmonic map to $\mathbb{R}^2$, with a related assertion for properly projected minimal surfaces.

Known results

• Heinz (1952): no harmonic diffeomorphism maps the unit disk onto $\mathbb{R}^2$.
• Božin (1999): an explicit proper harmonic map $\mathbb{D}\to\mathbb{R}^2$ refutes the harmonic-map assertion.
• Alarcón and López (2010): every open Riemann surface admits a conformal minimal immersion with proper projection to $\mathbb{R}^2$.
• A 2013 survey records the minimal-surface assertion as false for every open Riemann surface.

2009–2010 counterexamples

The 2009 paper constructs proper harmonic maps from suitable hyperbolic domains, including the unit disk, to $\mathbb{R}^2$. The 2010 result further gives minimal immersions with proper planar projection, so both stated assertions are reported false; the cited sources provide the mathematical claims but this card does not independently verify their proofs.

Current status (as of September 2026): Published sources report counterexamples to both the harmonic-map conjecture and its minimal-surface consequence; no part of the stated conjecture remains open, subject to independent verification of the cited proofs.