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Griffiths Extremality, Interpolation of Norms, and Kähler Quantization

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Abstract

Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge–Ampère type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kähler geometry, related to the construction of flat maps for the Mabuchi metric.

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Acknowledgements

We would like to thank László Lempert for numerous suggestions improving the presentation of the paper. We thank the referees for careful reading and helpful comments. The first named author has been partially supported by NSF Grants DMS-1610202 and DMS-1846942(CAREER). The second named author has been partially supported by NSF Grant DMS-1764167.

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  1. University of Maryland, College Park, USA

    Tamás Darvas

  2. Purdue University, West Lafayette, USA

    Kuang-Ru Wu

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  1. Tamás Darvas
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Correspondence to Tamás Darvas.

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Darvas, T., Wu, KR. Griffiths Extremality, Interpolation of Norms, and Kähler Quantization. J Geom Anal 32, 203 (2022). https://doi.org/10.1007/s12220-022-00940-0

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