CHOI - TREIBERGS

variable such that a{s) and b{t) are strictly increasing and a(+00) = +00, a(—oo) = —00, b(-foo) = +00, and b(—00) = -00. The conformaHty condition (4) can be reformulated as

{ 1 - f\a { s ) f ) a' { s ) Us^ - \~f {a{s))4'{t)Ut^ =h[ds^-^ [coslis fdt^l

Therefore the conformahty condition becomes

( •=°^^^^ ) -(1-/'(аИПа'(.Г By separating variables and using (2),

{ cosh s)a'{s) ,

where Л is a constant to be determined to make Ф a difFeomorphism. For the

hyperboloid , Л = 1, fc = 0 and f{u) = (1 4-w^)^/^. Thus the solution for the hyperboloid is a(5) = sinh5 and b{t) = t. In general, there is no exphcit formula for a{s). However, it is easy to see that the equation for a{s) has a solution, and to make Ф a difFeomorphism, Л must be chosen so that a(+oo) = +00, a(—00) = —00. Integrating (5),

[ ' ,ffl^Lu ds=X г -^= 2A(arctan e^ - ^). (6)

Jq f{a{s)Y +fc 7o cosh5 4

On the other hand, (2) can be rewritten as

1 /'

P + k [/4^(2fc_l)/2+fc2]l/2-

Substituting и = a{s) and w = f{u), the left hand side of (6) becomes

Jofi^oW + k Jo f{u?+k Jß [ш4+(2к-1Н+Р]1/2- ^'

Since и —^ oo as 5 —> oo, Lemma 2 implies that it; —► oo. Hence we have, by letting s and It; —> oo in Formulas (6) and (7),

2 f°^ dw

( 8 )

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