Local models for handle-crushing harmonic maps

Determine whether the harmonic maps from punctured Riemann surfaces to the hyperbolic plane constructed with prescribed Scherk asymptotics provide local models for handle-crushing harmonic maps from higher-genus surfaces to lower-genus surfaces.

Background

The paper constructs harmonic maps from a compact Riemann surface minus finitely many punctures to the hyperbolic plane, with each puncture having prescribed Scherk-map asymptotics. These maps have trivial induced homomorphisms on fundamental groups. The authors explicitly leave unresolved whether this class of maps can serve as local models for handle-crushing phenomena, in which harmonic maps collapse handles and map surfaces of higher genus to surfaces of lower genus.

References

It remains a question that whether these maps give local models of handle-crushing harmonic maps from higher genus surfaces to lower genus surfaces.

— Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics  (2609.18297 - Li et al., 16 Sep 2026) in Section 1, Introduction, immediately after Theorem 1 (Main theorem)