# A Comprehensive Introduction to Differential Geometry, Vol. by Michael Spivak By Michael Spivak

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G(p,q)=G(q,p) forall p, qeM Namely, for ul,u2 E Coo(M), one has with p~q. 6) /M L (/M 6(I9,q)ut (p) d#[g](p)) u2(q)d#[g](q) *'~/M (/M G(p,q)ul (p) d/~[g](p)) Lu2(q)d#[g](q) = /M (/M G(p'q)Lu2(q)d#[g](q)) uI(p) d#[g](p) = Cn [ JM u2(P)Ul (P) d#[gl(p) , which gives L(/MG(p,q)ul(p)d/~[g](p) ) With = Cnul(q) . 7) leads to L (/M G(p, q)Lu(p) d/~[g](p)) = cnLu(q) . 2 The conformal Laplacian 21 Since L is invertible, it follows M G(P, q)Lu(p) d#[g](p) = ¢nu(q) for all u E C ~ ( M ) . 6). 9 Let M be closed and let g be a Riemannian metric on M which is conformally equivalent to a Riemannian metric with positive scalar curvature.

Kleinian groups and moduli spaces one gets 6 z ( d / ® d w 1) = 0 . Because of tiz (cos 2 (w 2) dl ® dw 1) = cos 2 (w 2) 6z (d/® dw 1) 3 1 -2 Eei (cos 2 (w2)) (d/(¢i) dw I + dwl(ei) dl) i=1 = cos ~ (w 2) 6z ( d / ® d w ' ) , it follows that 5z (cos ~ (w 2) dl Q d w 1) = 0 and hence %(x,0)'~,0 = O. 12). (2) ), log 2 A ~'-'x,o'o~,,o, '°~,o)go,,e ) 4 )~ log 2 A f~3 (~,,o) &3~,o cos 2 ( ~ 2 ) gz (dl®dl, dl®dw 1) d#[gz] , where M(A,O):= { x e R 3 : 1 < Ixl < ~} and (~A,O :---- 00,,0 o 71-A,0 . Together with gz (dl ® dl, dl ® dw 1) = O, this yields (0h, 00) = o .

On the other hand, the definition of our canonical metric can be generalized to non-flat conformal structures at least in low dimensions. 3. K l e i n i a n groups and moduli spaces As mentioned in the last section, any flat conformal manifold (M, C) whose conformal structure C is scalar positive is a Kleinian manifold. In the first section of this chapter we use this to describe the canonical metric can(C) more explicitly, namely in terms of the underlying Kleinian group. In particular, this description proves that can(C) indeed is a Riemannian metric provided that (M, C) is not conformally diffeomorphic to the standard sphere.