Notation All manifolds are assumed to be smooth and connected. The notation we use is generally standard. C k+a , 0 a 1, is the space of function whos e /cth deriva- tives satisfy a Holder condition with exponent a, C" is the space of real analytic functions, an d th e Sobole v space HQ(SI) i s the completio n o f smoot h function s with compac t suppor t i n fl i n th e nor m (/a|Vw| 2 )1/2. Ou r Laplacian , A , o n real-valued function s ha s the sign so thatAw = +u" o n R 1 . The Ricci curvature of th e standard uni t spher e S" with metric g in R n+1 i s (n - l)g , an d it s scalar curvature is n(n 1) . If a is a one-form o n a Riemannian manifold , the n a* i s the dua l vecto r field . I n classical tenso r notatio n w e use a semicolon t o denote covariant differentiation. u„ is the volume of the unit sphere S" in R" +1 . vu
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