Stronger uniqueness under minimal asymptotic closeness

Determine whether the uniqueness of the Calabi–Yau metric in a fixed cohomology class on X = M \ D holds under the minimal assumption that a second Calabi–Yau metric \tilde{ω} satisfies |\tilde{ω} − ω|_{ω} → 0 as the ω-distance r → ∞; specifically, show that this vanishing asymptotic difference implies \tilde{ω} = ω without requiring any polynomial decay rate.

Background

The main uniqueness theorem in the paper requires polynomial closeness between two Calabi–Yau metrics in the same class. The authors note that removing this decay-rate assumption is a natural strengthening but faces technical obstacles, notably the inability to derive the refined asymptotic decomposition of the potential needed for integration by parts arguments without polynomial control.

They explicitly pose the problem of whether mere vanishing of the pointwise difference (without a rate) suffices to deduce equality of the metrics, highlighting a potential obstruction from metrics that might approach the Calabi model space only at logarithmic rates.