HOROSPHERES IN HYPERBOLIC GEOMETRY 1. Introduction ...

14 E. GALLEGO, A. REVENTÓS, G. SOLANES, AND E. TEUFEL
as follows:
L ∗ = − 1 2 (W(c 1, c 1 + c 2 ) − L 1 − TC 1 − L 2 − TC 2 ) (21)
TC ∗ = 1 2 (W(c 1, c 1 + c 2 ) + L 1 + TC 1 + L 2 + TC 2 ) , (22)
with
∫
W(c 1 , c 1 + c 2 ) = TC ∗ − L ∗ = e ( ) w 1∗
(ẇ 1∗ ) 2 + 2ẅ 1∗ + κ 2 θ dσ1 1
θ 1
and the mixed with function
w 1∗ (σ 1 ) = − ln(1 + λ(σ 1 )).
Proof. By the assumptions on c 1 , c 2 and by Proposition 4.2 we
have for all three curves c 1 , c 2 , c ∗ that ǫ 1 = ǫ 2 = ǫ ∗ = −1 and
(κ g ) 1 , (κ g ) 2 , κ ∗ g > −1. Therfore (6) writes dσ = (κ g+1) ds, hence
∫ ∫ ∫
L(θ) = dσ = (κ g + 1) ds = κ g ds + L(c) . (23)
This and (17) gives
From (4) and (8) we get
θ
c
TC ∗ + L ∗ = TC 1 + L 1 + TC 2 + L 2 . (24)
ds =
L(c) = − 1 2
This applied to c ∗ yields
L(c ∗ ) = − 1 2
∫
1 − 〈¨θ, ¨θ〉
2
∫
θ
dσ,
κ 2 θ dσ + 1 2 L(θ).
c
θ ∗ κ 2 θ ∗ dσ∗ + 1 2 L(θ∗ ). (25)