The Non-compact Case
A closely related question is the so-called "non-compact Yamabe problem", which asks: on a smooth, complete Riemannian manifold (M,g) which is not compact, does there exist a conformal metric of constant scalar curvature that is also complete? The answer is no, due to counterexamples given by Jin (1988).
Read more about this topic: Yamabe Problem
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“One thing in any case is certain: man is neither the oldest nor the most constant problem that has been posed for human knowledge.”
—Michel Foucault (1926–1984)
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