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## The standard mass gap problem versus S4 mass gap problem

The standard mass gap problem is to show whether there is a mass gap for a Yang-Mills theory in flat Minkowski space.  I do not claim to solve this problem.  I solve a mass gap problem for the model of the universe that is a scaled 4-sphere of numerical radius $1/h$.  I have provided empirical evidence that the actual universe itself must be a scaled 4-sphere.  With this model, a physical universe can only occur as a smooth hypersurface of $S^4(1/h)$.  The S4 mass gap problem is to show that no smooth hypersurface $M$ can carry a nontrivial harmonic spinor.  Nontrivial harmonic spinors are the same as massless fermions:  the time-dependent Dirac equation can be solved just as in the case of wave equations multiplying $e^{i \omega t}$ with the eigenspinors.

Now it is well-known that no spin manifold with positive scalar curvature can carry harmonic spinors by the Lichnerowicz formula for $D^2 = \nabla^*\nabla + S/4$ where $S$ is the scalar curvature which applies to the ambient $S^4$ as well as minimal submanifolds such as the equator and the Clifford tori.

Suppose $M$ is a physical hypersurface of $S^4$ that only contains massless fermions and no massive fermions.  Mass corresponds to geometry of $M$ by increasing its mean curvature.  If we accepts the correspondence, then a physical universe with only massless fermions will be a minimal hypersurface which have rigidity in $S^4$: the possible scalar curvatures are discrete and positive by the Chern conjecture and its resolution.  Since all the minimal hypersurfaces have positive scalar curvature, no massless fermions are possible.  We can then conclude that there is a mass gap for these hypersurfaces.

The correspondence between mass and mean curvature is elucidated by the work of LawnRothIsometricImmersionS4Spinors who show how two ‘antiparticle’ massive fermion solutions of the Dirac operator:
$D\phi_1 = 3/2(H+k)\phi_1$ and $D\phi_2 = -3/2(H+k)\phi_2$

where $H$ is a function can be used to embed the manifold into a sphere whose radius is a simple function of $k$ for which $H$ appears as the mean curvature of the embedding.  Their work shows that the correspondence between mass and mean curvature can be made precise.