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

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 $f_i$ : $S_i$ goes to $T_i$ 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 to give a sufficient condition for gap rigidity to hold for (OMEGA,D)in the Zariski topology.

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Published 1 January 2004