Abstract
Tian and Yau constructed in J. Am. Math. Soc., 3(3):579–609, 1990, a complete Ricci-flat Kähler metric on the complement of an ample and smooth anticanonical divisor. For the explicitly constructed referential metric ω of Tian and Yau (J. Am. Math. Soc., 3(3):579–609, 1990) we prove a property that \({\|\partial\overline\partial u\|_\omega}\) has the same decay rate as Δ ω u provided u satisfies some decay conditions on higher Laplacians. As an application we describe the behaviour of this metric towards the boundary divisor and prove the best possible decay rate of the difference to ω.
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Part of the work on this paper was supported by the DFG priority program ‘Global Methods in Complex Geometry’.
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Koehler, B., Kühnel, M. On asymptotics of complete Ricci-flat Kähler metrics on open manifolds. manuscripta math. 132, 431–462 (2010). https://doi.org/10.1007/s00229-010-0354-4
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DOI: https://doi.org/10.1007/s00229-010-0354-4