By Mahowald M., Priddy S. (eds.)

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By definition X- (0) = 0 . )>dVol 3Xpr- = 2 Re J F r* 5 \ — = 2Re J ( 2~ ~ I

GTn) = e" i n § is a unitary character. ) is in H (F . , + || grad f || 2 ] dVol d§ < » , p € BF"^. ' -functions on hypersurfaces. H1 is clearly a complete Hilbert space. 4 U 12 3 extends to a unitary isomorphism of W ' (K IT) onto H (F ^ X [0,2*17)) . Furthermore U grad F = grad UF . Proof: If F € C * O H 3 / T ) then: (UF)(p,S) = F(T n p) 6 £ in? ) = « in? ) . Taking limits we obtain this formula for all oa 12 3 F € W ' (H /T) . ) Let F € C* (H3/T) ; the limit of 2 L -gradient for almost every 2TT J 0 Thus the restriction of for UF implies that P € SF" r* .

And note that "p** p** The continuity of U f(p) implies that P* J* (M^ *) r U*f(p)Y-VdS + + P* J* (ST U*f(p)Y-N*lS = 0 . n+1 F r # )- P* Spectral Theory of 3-Manifolds 23 We obtain: Jo 3