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The Yamabe problem for dimension 3 proved by Richard Schoen always gives us a constant scalar curvature metric by conformal transformation to positive scalar curvature for those hypersurfaces that carry positive scalar curvature but is it not possible to get a minimal surface with topology might make psc impossible, such as $S^1 \times S^1 \times S^1$.  One useful tool to restrict possible cases is to observe that the second fundamental form is going to be identical to a stress-energy or energy-momentum tensor.  To the extent that massless objects cannot have extremely large stress-energy tensor, we can reduce the possibility of existence of massless fermions.