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Rationally convex domains and singular Lagrangian surfaces in \(\mathbb {C}^2\)

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Abstract

We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in \(\mathbb {C}^2\). We recall some classical obstructions and prove some deeper ones related to symplectic and contact topology. We explain the close connection to Lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces. Our proof also gives a complete characterization of Lagrangian surfaces with open Whitney umbrellas, answering a question first posed by Givental in 1986.

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Notes

  1. In the terminology of Cieliebak–Eliashberg, the closure of a smoothly bounded strictly pseudoconvex domain is called an \(i\) -convex domain, and a strictly plurisubharmonic function is called an i-convex function. In this paper, “domain” means open connected set.

  2. We reserve the term knot for links with a single connected component.

  3. The sign discrepancy arises because, for an oriented Lagrangian plane \(P \subset \mathbb {C}^2\), we have \(P \oplus iP = \mathbb {C}^2\), but the induced orientation on \(P \oplus iP\) as a direct sum is the opposite of the orientation on \(\mathbb {C}^2\) as a complex vector space.

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Acknowledgments

We would like to thank Yasha Eliashberg for suggesting this problem and for numerous informative discussions. We also thank Roger Casals and Emmy Murphy for enlightening conversations regarding Sect. 5.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Gubkina 8, 119991, Moscow, Russia

    Stefan Nemirovski

  2. Fakultät für Mathematik, Ruhr-Universität Bochum, 44780, Bochum, Germany

    Stefan Nemirovski

  3. Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, 94305, CA, USA

    Kyler Siegel

Authors
  1. Stefan Nemirovski
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  2. Kyler Siegel
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Correspondence to Kyler Siegel.

Additional information

The first author was partially supported by DFG project SFB/TR-12 and RFBR Grant 14-01-00709-a. The second author was partially supported by NSF Grant DGE-114747.

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Nemirovski, S., Siegel, K. Rationally convex domains and singular Lagrangian surfaces in \(\mathbb {C}^2\) . Invent. math. 203, 333–358 (2016). https://doi.org/10.1007/s00222-015-0598-4

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