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Of the the _2 w„. }(in uB-) on a nbhd. of the and G -* G as n-*°°. Let V¥ and V¥ denote the gran dients of ¥ and ¥ with respect to these metrics, and n V¥ n V¥ X and X the normalized fields and . t. G and G. Of course X ~n and X are defined only on M - y w , and M - y w. respecively. of nbhd. of Let {M } _ 2 -* M _ to a {M} Dn be a "small disc" about w for which the Morse lemma holds for ¥ and ¥ about w Q . Thus there exists 2 a map Q from a nbhd. of 0 in JR to a nbhd. of w Q so _ 9 9 that ^ Q Q ( Z ) = c Q - z - z = foQ(z) .

1 Let f : U -* H connected subset f(z) = and U be a C isometry on an open of the hyperbolic plane. Then Az + B +D i Cz A,B,C,D are real AD - BC = 1. Proof. The class of maps Az— —+ —B , AD - BC = 1 z -• - and all real are the group of isometries of the Poincare metric. Thus we must show that a local isometry is also a global isometry. It is clear that we can take f to be orientation pre- serving. Then an easy calculation shows that f must be PLATEAU'S PROBLEM 41 holomorphic and satisfy the non-linear condition i^..

To see this let on M cp. be the natural coordinate charts induced by the inclusions of M B. ocp. = T . }. ¥ n ( p ) =Znj(q>j1«Pj(p))Y( oo this . t h e n ^ n ( p ) = ^ . ( ^ ^ . (X u m f c p T V , As H, so that Vn : M n -• IR by Define If B. , ¥ -* V in ¥ : M -* IR such that on each J J oo converges JK J J C to = Znj(cp~1 (u)y(cp~1 (u) ) which (after viewing ¥ = V(cp~1(u)) as defined on . M ) proves the result.

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