On Calabi-Yau cones, uniformly bounded cscK metrics are unique up to cone automorphisms, and uniformly bounded Kähler metrics with bounded scalar curvature are asymptotically conical with polynomial decay.
Lectures on Kähler Manifolds
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A Liouville Theorem and $C^{\alpha}$-Estimate for Calabi-Yau Cones
On Calabi-Yau cones, uniformly bounded cscK metrics are unique up to cone automorphisms, and uniformly bounded Kähler metrics with bounded scalar curvature are asymptotically conical with polynomial decay.