Prescribing scalar curvatures: on the negative Yamabe case
Abstract
The problem of prescribing conformally the scalar curvature on a closed Riemannian manifold of negative Yamabe invariant is always solvable, when the function K to be prescribed is strictly negative, while sufficient and necessary conditions are known for K≤ 0. For sign changing K Rauzy showed solvability, if K is not too positive. We revisit this problem in a different variational context, thereby recovering and quantifying the principle existence result of Rauzy and show under additional assumptions, that for a sign changing K solutions to the conformally prescribed scalar curvature problem, while existing, are not unique.
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