Equator conjecture: Hofer diameter of the space of equators on S^2

Determine whether the space of equators of the two-sphere S^2—i.e., the set of Hamiltonian images of a fixed equator L0 endowed with the Lagrangian Hofer distance induced from the Hofer norm on Ham(S^2)—has infinite diameter.

Background

The paper studies Hofer-Lipschitz quasimorphisms arising from link spectral invariants and applies them to questions in Hofer geometry. A central question is whether the space of Hamiltonian isotopies of an equator in S2 has infinite Hofer diameter, often referred to as the equator conjecture.

Using a vanishing result for a specific quasimorphism on the stabilizer of the equator, the authors obtain an alternative: either the quasimorphism vanishes identically or the equator conjecture holds. The conjecture itself, however, remains unsettled and is explicitly identified as an open question.

References

However, it remains unknown whether the space of equators inside the sphere has infinite diameter. This appears as an open question in , and the affirmative answer is often referred to as the `equator conjecture'.

— On link quasimorphisms on the sphere and the equator conjecture  (2509.14996 - Serraille et al., 18 Sep 2025) in Section 1.2 (Lagrangian Hofer distance and the equator conjecture)