Manifolds of Riemannian metrics with prescribed scalar curvature

Fischer, A. E. and J. E. Marsden

Bull. Amer. Math. Soc., 80, (1974), 479-484

Abstract:

Throughout, M will denote a C$\scriptstyle \infty$ compact connected oriented n -manifold, n $ \geq$ 2 . Let $ \rho$ : M$ \to$$ \mathbb {R}$ be a C$\scriptstyle \infty$ function, $ \mathcal {M}$ the space of C$\scriptstyle \infty$ riemannian metrics on M and

$\displaystyle \mathcal {M}$$\scriptstyle \rho$ = {g $\displaystyle \in$ $\displaystyle \mathcal {M}$ : R(g) = $\displaystyle \rho$}

where R(g) is the scalar curvature of g . A superscript s will denote objects in the corresponding Sobolev space, s > $ {\tfrac{1}{2}}$n + 1 (one can also treat Ws, p spaces in the same way), and we also allow s = $ \infty$ so $ \mathcal {M}$$\scriptstyle \infty$ = $ \mathcal {M}$ .

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