Hofer–Zehnder capacity of closed constant-curvature surfaces

Determine the Hofer–Zehnder capacity of unit disk bundles over all closed surfaces of constant curvature, extending the known computations for the round sphere, round projective plane, and flat reversible Finsler torus.

Background

The paper summarizes known capacity computations for several constant-curvature surfaces: the round two-sphere, the round real projective plane, and the flat two-torus with a reversible Finsler metric. It then identifies the unresolved problem of extending this list to every closed surface of constant curvature. Such a result would broaden the known relationship between Hofer–Zehnder capacity and the systolic geometry of the base surface.

References

We don’t know how to extend this list to all closed surfaces of constant curvature.

— Hofer-Zehnder capacity as a geodesic selector  (2608.20177 - Bimmermann et al., 20 Aug 2026) in Section 1, page 3, paragraph beginning “Constant curvature”

Even for flat Klein bottles K, the capacity is unknown in general, while for closed orientable hyperbolic surfaces even its finiteness remains open [ABB+26].

— Hofer-Zehnder capacity as a geodesic selector  (2608.20177 - Bimmermann et al., 20 Aug 2026) in Section 1, page 3, paragraph beginning “Constant curvature”