## A gap theorem for the complex geometry of convex domains

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- by Andrew Zimmer PDF
- Trans. Amer. Math. Soc.
**370**(2018), 7489-7509 Request permission

## 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
- MR Author ID: 831053
- Email: amzimmer@wm.edu
- 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.
- © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3841856