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Bergman interpolation on finite Riemann surfaces. Part II: Poincaré-Hyperbolic Case

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Abstract

We formulate the Bergman-type interpolation problem on finite open Riemann surfaces covered by the unit disk. Our version of the interpolation problem generalizes Bergman-type interpolation problems previously studied by Seip, Berndtsson, Ortega Cerdà, and a number of other authors. We then prove sufficient conditions for a sequence to be interpolating. When the curvature of the weight in question is bounded in an appropriate sense, we show that the sufficient conditions are almost necessary, but not quite. The results extend work of Ortega Cerdà, who resolved the case in which the boundary of the surface is pure 1-dimensional. Our version of the interpolation problem effectively changes the geometry of the underlying space near the 0-dimensional boundary components, or punctures, thereby linking in a crucial way with the previous article (Varolin, J d’Anal Math, 2015, to appear) in this two-part series.

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Notes

  1. Strictly speaking, this transformation changes the weight function \(\varphi \). However, the new weight function satisfies the same hypotheses, and in particular the same curvature bounds, so the result holds for the original weight as well: see the analogous comment in [16].

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Acknowledgments

I am grateful to Henri Guenancia, Long Li, Jeff McNeal, Quim Ortega Cerdà and Alex Schuster for many stimulating conversations both past and present, and without which this work would not have come to be. I am also grateful to the anonymous referee for very useful and interesting remarks.

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  1. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794-3651, USA

    Dror Varolin

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  1. Dror Varolin
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Varolin, D. Bergman interpolation on finite Riemann surfaces. Part II: Poincaré-Hyperbolic Case. Math. Ann. 366, 1137–1193 (2016). https://doi.org/10.1007/s00208-015-1344-3

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