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There are no harmonic spinors on a hypersurface $M$ of $S^4$ if the scalar curvature is positive by the Lichnerowicz theorem.  Therefore it is useful to look at the formula for the scalar curvature.  It is (see ):
$n(n-1)r = n(n-1) + n^2 H^2 - S$
generally for $S^{n+1}$ where $H$ is the mean curvature and $S$ is the squared length of the second fundamental form.  This will be negative obviously when $S$ is large compared to the mean curvature term.  The issue of when the scalar curvature will be positive is delicate but we can get some sense by considering the minimal surfaces with $H=0$ to consider what happens: a small number of possibilities arise all of which have positive scalar curvature.  Thus a simple conjecture that there should be rare or impossible to have negative scalar curvature; then we can conclude that massive fermions cannot exist in the physical universe.