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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A gap theorem for the complex geometry of convex domains


Author: Andrew Zimmer
Journal: Trans. Amer. Math. Soc. 370 (2018), 7489-7509
MSC (2010): Primary 32T15; Secondary 53C24, 32F45, 32Q15
DOI: https://doi.org/10.1090/tran/7284
Published electronically: June 26, 2018
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Abstract: In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex.

One consequence of our general result is the following: for any dimension there exists some $ \epsilon > 0$ so that if the squeezing function on a smoothly bounded convex domain is greater than $ 1-\epsilon $ outside a compact set, then the domain is strongly pseudoconvex (and hence the squeezing function limits to one on the boundary). Another consequence is the following: for any dimension $ d$ there exists some $ \epsilon > 0$ so that if the holomorphic sectional curvature of the Bergman metric on a smoothly bounded convex domain is within $ \epsilon $ of $ -4/(d+1)$ outside a compact set, then the domain is strongly pseudoconvex (and hence the holomorphic sectional curvature limits to $ -4/(d+1)$ on the boundary).


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Additional Information

Andrew Zimmer
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Address at time of publication: Department of Mathematics, College of William and Mary, 5734 S. University Avenue, Room 208C, Chicago, Illinois 60637
Email: amzimmer@wm.edu

DOI: https://doi.org/10.1090/tran/7284
Keywords: Strongly pseudoconvex domains, squeezing function, Bergman metric, holomorphic sectional curvature
Received by editor(s): October 13, 2016
Received by editor(s) in revised form: April 16, 2017, and May 15, 2017
Published electronically: June 26, 2018
Additional Notes: The author was supported in part by NSF Grant #1400919.
Article copyright: © Copyright 2018 American Mathematical Society

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