It is known that a maximal collection of essential annuli in H 2 could contain exactly 1, or 2, or at most 3 annuli, and a maximal collection of essential annuli in H n with n ≥ 3 can contains at most 4n - 5 annuli, and the bound is best possible. In the present paper, we show that a maximal collection of essential annuli in H n with n ≥ 3 contains at least two annuli, and for each m, 2 ≤ m ≤ 4n - 5, there exists a maximal collection of essential annuli in H n which contains exactly m annuli.
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Yin et al. (2009) studied this question.