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.