NONUNIQUENESS OF GEODESICS IN INFINITE-DIMENSIONAL TEICHMULLER-SPACES (II)

The non-uniqueness of geodesics joining two given points in universal Teichmuller space is proved in the previous paper [7]. The purpose of the present paper is to discuss the non-uniqueness of geodesics in any infinite-dimensional Teichmuller space. It is proved that if mu1 and mu2 are two extremal Beltrami differentials belonging to a point [mu] of a Teichmuller space and mu1 - mu2 does not belong to N-class, the paths [tmu1] and [tmu2] (0 less-than-or-equal-to t less-than-or-equal-to 1) are different geodesics joining [0] and [mu]. Making use of this theorem, the result of [7] is generalized to hold for any infinite-dimensional Teichmuller space. This is a complete answer to a problem posed by F.P. Gardiner [1].