Volume collapse under Ricci lower bound and eigenvalue pinching

Determine whether the volume Vol(M, ω) of compact Kähler manifolds (M, ω) of complex dimension n that satisfy Ric(ω) > ω and λ_{n+3}(Δ_ω) ≤ 1 + δ can collapse, i.e., whether such manifolds can have arbitrarily small volume under these conditions.

Background

In addressing the proposed Kähler analogue of the Petersen–Aubry eigenvalue pinching rigidity, the authors highlight a key obstacle: understanding volume behavior under the conditions Ric(ω) > ω and λ_{n+3} ≤ 1 + δ. Volume non-collapsing is crucial for applying Gromov–Hausdorff convergence and limit space structure theorems, which underpinned the authors’ proofs in the Kähler–Einstein and almost Kähler–Einstein settings.

The uncertainty about possible volume collapse under these assumptions suggests a foundational gap in the current methodology for proving the conjectured rigidity without the Kähler–Einstein condition. Resolving this would clarify whether eigenvalue pinching alone prevents collapse and enable broader application of limit space techniques.