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Envelopes with Prescribed Singularities

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Abstract

We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact Kähler manifolds have the optimal \(C^{1,1}\) regularity on a Zariski open set. This also proves regularity of certain pluricomplex Green’s functions on Kähler manifolds. We then go on to prove the same regularity for envelopes when the manifold is assumed to have boundary. As an application, we answer affirmatively a question of Ross–Witt-Nyström concerning the Hele-Shaw flow on an arbitrary Riemann surface.

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Acknowledgements

We would like to thank Julius Ross for useful comments about the Hele-Shaw flow and for encouraging us to prove Theorem 1.2. I would also like to thank my advisor Valentino Tosatti for introducing me to this circle of questions, for greatly improving the format of this paper, and for his continued patience and guidance.

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  1. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208, USA

    Nicholas McCleerey

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Correspondence to Nicholas McCleerey.

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Partially supported by NSF RTG Grant DMS-1502632.

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McCleerey, N. Envelopes with Prescribed Singularities. J Geom Anal 30, 3716–3741 (2020). https://doi.org/10.1007/s12220-019-00215-1

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