Asian Journal of Mathematics

On the Validity of Failure of Gap Rigidity for Certain Pairs of Bounded Symmetric Domains

Philippe Eyssidieux and Ngaiming Mok

Abstract

The purpose of the article is two-fold. First of all, we will show that in general gap rigidity already fails in the complex topology. More precisely, we show that gap rigidity fails for $\Delta^2, \Delta \ti 0$ by constructing a sequence of ramified coverings fi : Si goes to Ti between hyperbolic compact Riemann surfaces such that, with respect to norms defined by the Poincare metrics. Since any bounded symmetric domain of rank greater than or equal to 2 contains a totally geodesic bidisk, this implies that gap rigidity fails in general on any bounded symmetric domain of rank greater than or equal to 2. Our counter examples make it all the more interesting to find sufficient conditions for pairs(OMEGA,D)for which gap rigidity holds. This will be addressed in the second part of the article, where for irreducible, we generalize the results for holomorphic curves in [Mok2002]to give a sufficient condition for gap rigidity to hold for (OMEGA,D)in the Zariski topology.

Article information

Source
Asian J. Math., Volume 8, Number 4 (2004), 773-794.

Dates
First available in Project Euclid: 13 June 2005

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1118669700

Mathematical Reviews number (MathSciNet)
MR2127948

Zentralblatt MATH identifier
1085.32010

Citation

Eyssidieux, Philippe; Mok, Ngaiming. On the Validity of Failure of Gap Rigidity for Certain Pairs of Bounded Symmetric Domains. Asian J. Math. 8 (2004), no. 4, 773--794. https://projecteuclid.org/euclid.ajm/1118669700


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