Gauge mediation of supersymmetry breakdown has many attractive features and can be realized in phenomenologically interesting string-motivated models. We point out that in models with the Polonyi-like field stabilized at a low expectation value by quantum corrections, gravity seems to limit from above the admixture of gravity mediation to the dominant gauge mediation channel. However, we also point out that in a class of such models the low energy metastable supersymmetry breaking vaccum appears to be cosmologically disfavoured. These features should hold also in the case of typical stabilized models with anomalous U(1) groups.
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