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## Mass gap problem and the Yamabe problem

The general question of existence or nonexistence of fermions without masses is a difficult question that depends on the background metric.  The question is simpler in an S4 universe of static scaled 4-sphere background where physical universes arise as hypersurfaces.  By the Lichnerowicz formula the square of the Dirac operator is a positive second-order differential operator with a potential term a multiple of the scalar curvature.  Therefore when scalar curvature is positive then we cannot have zero mode eigenspinors of Dirac or massless fermions.  If we make the additional identification of mass with mean curvature, the question boils down to: which minimal hypersurfaces of a 4-sphere can be conformally deformed to a metric with positive scalar curvature.  For minimal surfaces, mean curvature $H=0$ and the formula for scalar curvature reduces to $R = n(n-1) - S^2$ where $S$ is the squared-length of the second fundamental form.  So the embedding metric might produce negative scalar curvature for minimal surfaces for example those for which a frame exists diagonalizing the second fundamental form with entries $\lambda, \lambda, -2\lambda$ with high $\lambda$.  In this case we cannot apply the Lichnerowicz formula directly but we can seek conformal deformations to positive curvature and still conclude nonexistence of harmonic spinors.  The famous Yamabe problem is that of seeking conformal transformations to constant scalar curvature metrics.  The issue of existence of positive scalar curvature metrics is delicate because of topological restrictions but fortunately for us Yau, Schoen, Gromov, Lawson and others have given a great deal of information about this problem.  The mass gap problem is tightly linked to the Yamabe problem for minimal hypersurfaces of a 4-sphere.

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