THE CARATHI~ODORY METRIC ON ABELIAN TEICHMIJLLER DISKS t By IRWIN KRA Let T(p,n) be the Teichmfiller space of Riemann surfaces of type (p,n) with 2p + n > 2. An important theorem of Royden [8] asserts that the Teichm/iller metric on T(p, n) agrees with the (hyperbolic) Kobayashi metric on this space. This paper is a contribution towards the determination of the Carath6odory metric on T(p,n). It is well known that the Carath6odory metric is complete (Earle [3]). However, very little else is known about this invariantly defined metric. We will prove the following Theorem 2'. Let D C T(p, n) be a Teichmiiller disk determined by a quadratic differential all of whose zeros are of even order. Then the restriction to D of the Carath~odory metric on T(p, n) agrees with the non-Euclidean metric on D, which is the same as the restriction to D of the Teichmiiller metric of T(p, n ). Our proof consists of first reducing the problem to compact surfaces, without punctures, of genus p _->2, and then studying certain bounded holomorphic functions obtained by the canonical period map II of T(p,0) into the Siegal half plane ~e, consisting of p • p symmetric matrices of positive definite imaginary part. A Teichm/iller disk D C T(p, n) will be called an abelian Teichmiiller disk if it is determined by a quadratic differential r which is the square of an abelian differential of the first kind. This concept is well defined on T(p,n) -- it does not depend on the choice of a base point for T(p, n) -- see w Our result has several consequences. Corollary 1. The metric induced on an abelian disk D C T(p,0) by the canonical imbedding H of T(p,0) into ~p from the Kobayashi metric on 5(p is the Teichmiiller metric. Corollary 2. The map lq:T(p,0)--->~p restricted to any abelian disc D is isometric (and hence injective ). The above result is slightly surprising since the Teichmiiiler space T(p, 0) is an infinite sheeted covering of the 3"orelli space ~-p, and the map I1 induces a generically two-to-one map lrI from Jp into :~p so that the following diagram ' Research partially supported by NSF Grant MCS 7801248. JOURNAL D'ANALYSE MATHI~MATIOUE, VoL 40 (1981)