Oh’s volume-minimization conjecture for the Clifford torus in CP^n

Determine whether the monotone Clifford torus T^n = μ^{-1}(1/(n+1), …, 1/(n+1)) in CP^n minimises the unsigned Riemannian volume among all Lagrangian submanifolds Hamiltonian isotopic to T^n, thereby establishing or refuting Oh’s conjecture.

Background

The paper connects Lagrangian intersection counts with volume via Crofton-type integral geometry formulas. In CPn, bounding the mean surplusection over images of RPn by projective unitary isometries corresponds to lower bounds on volume. Within this context, Oh conjectured that the monotone Clifford torus minimises volume in its Hamiltonian isotopy class, which would imply strong constraints on intersection behaviour and surplusection distribution.

Despite progress on intersection bounds and improved volume lower bounds (e.g., the Goldstein bound), the exact volume-minimising property remains unresolved, making it a central open problem tied to surplusection phenomena.

References

Bounding the volume of \phi(K) from below is a notoriously thorny problem in general: Oh [p. 192] conjectured that the monotone Clifford torus minimises volume amongst Lagrangians in its Hamiltonian isotopy class, but this conjecture remains open thirty years later.

— Lagrangian Surplusection Phenomena (2408.14883 - Rizell et al., 27 Aug 2024) in Section 2.1 (Crofton formula)