On the Relation between Tautly Imbedded Space Modulo an Analytic Subset $S$ and Hyperbolically Imbedded Space Modulo $S$

Let X be a complex manifold, M a relatively compact domain of X and S an analytic subset of X. We denote the open unit disk in the complex plane C by A, the polydisk Ax xA in C by A and the Kobayashi pseudodistance of M by du (see [Ko] for its definition and basic properties). The space of holomorphic maps from a manifold TV to a manifold M with compact-open topology will be denoted by Hoi (N9M). Following Kiernan-Kobayashi [K-K] and [L], we use the following terminologies. M is tautly imbedded modulo S in X if for each positive integer k and each sequence {/)•} in Hol(A, M) we have one of the following: (a) {/}} has a subsequence which converges in Hol(A,l); (b) for each compact set K c A and each compact set L a X\S, there exists an integer N such that fj(K)nL = (j) for j^N. M is hyperbolically imbedded modulo S in X if, for every pair of distinct points /?, q of M, closure of M, not both contained in S, there exist neighborhoods Vp and Vq of p and q respectively in X such that dM(Vpr\M, F^nM)>0. In [K-K], it was proved that if M is tautly imbedded modulo S in X, then M is hyperbolically imbedded modulo S in X and brought up the inverse problem. But we believe there is no results except for the case S = 4> (see Kiernan [Ki2] in case $=$). In this note, we deal with the inverse problem