All manifolds are assumed to be smooth and connected. The notation we use is
generally standard. C
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
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|
Ou r Laplacian , A , o n
real-valued function s ha s the sign so thatAw = +u" o n R
. The Ricci curvature
of th e standard uni t spher e S" with metric g in R
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"
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