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.
A Course in Metric Geometry
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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.