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The complex Monge–Ampère equation, Zoll metrics and algebraization

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Abstract

Let M be a real analytic Riemannian manifold. An adapted complex structure on TM is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of TM. We prove here that the only real analytic Zoll metric on the n-sphere with an entire adapted complex structure on TM is the round sphere. Using similar ideas, we answer a special case of an algebraization question raised by the first author, characterizing some Stein manifolds as affine algebraic in terms of plurisubharmonic exhaustion functions satisfying the homogeneous complex Monge–Ampère equation. The result presented here is an extension to higher dimensions of an observation attributed to W. Stoll for the case of Riemann surfaces.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    Daniel M. Burns Jr.

  2. Department of Mathematics, University of Toronto, Toronto, Canada

    Kin Kwan Leung

Authors
  1. Daniel M. Burns Jr.
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  2. Kin Kwan Leung
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Correspondence to Daniel M. Burns Jr..

Additional information

Communicated by Ngaiming Mok.

Supported in part by NSF grant DMS-1105586.

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Burns, D.M., Leung, K.K. The complex Monge–Ampère equation, Zoll metrics and algebraization. Math. Ann. 371, 1–40 (2018). https://doi.org/10.1007/s00208-017-1547-x

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  • DOI: https://doi.org/10.1007/s00208-017-1547-x

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