Remark 3.2 . [03GU]
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Remark 3.2.
In [TY90] the volume growth and curvature decay rates of the -dimensional Calabi metric are estimated as and , respectively. When , this suggests that is borderline not in . However, while the volume estimate is sharp for all , the curvature estimate is sharp only for . For , the leading term in the asymptotic expansion of the curvature vanishes because the Calabi-Yau metric on an elliptic curve is flat, and the true curvature decay rate of for is . This was also pointed out by R. Kobayashi in [Kob90] but is perhaps most easily seen in the Gibbons-Hawking picture.