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## Resolution of S4 mass gap problem modulo ‘mass is mean curvature’

The problem is to show that a minimal hypersurface $\Sigma$ in $S^4(1/h)$ cannot have nontrivial harmonic spinors.  This can be done using the the solvability of the boundary value problem for Dirac operator on either piece $\Omega_1, \Omega_2$ of $S^4(1/h)$ cut by $\Sigma$.  The boundary condition is the Atiyah-Patodi-Singer conditions.  This is due to Hijazi et. al. who use elliptic estimates discovered by Farinelli and Schwarz.

THEOREM (Hijazi et. al. HijaziDiracHypersurfaces)

Let $\Omega$ be a compact Riemannian spin manifold with boundary $\partial \Omega = \Sigma$.  Let $\pi_+:\Gamma(S\Sigma)\rightarrow \Gamma(S\Sigma)$ denote the projection onto the space of nonnegative eigenspinors.  Denote by $N$ the normal of the hypersurface and $\gamma$ Clifford multiplication. The inhomogeneous boundary value problem for the Dirac operator
$\bar{D}\psi = \Psi$ on $\Omega$

$\pi_+\psi = \pi_+\phi$ on $\Sigma$

has a smooth solution for each $\Psi\in\Gamma(S\Omega)$ and $\phi\in\Gamma(S\Sigma)$ satisfying the following integrability condition

$\int_\Omega \langle \Psi,\Phi \rangle d\Omega + \int_\Sigma \langle \gamma(N)\phi, \Phi \rangle d\Sigma = 0$

for any harmonic spinor $\Phi$ on $\Omega$ such that $\pi_+\Phi = 0$.  The solution is unique up to an arbitrary spinor of this type.

In case $\Sigma$ has non-negative mean curvature and $\Omega$ has non-negative scalar curvature, the integrability conditions are satisfied and one can solve the boundary value problem.  In particular for minimal hypersurfaces of $S^4(1/h)$, we can always solve the boundary value problem from either component cut by $\Sigma$.

In order to show nonexistence of harmonic spinors on $\Sigma$ argue by contradiction.  Assuming $\phi$ a harmonic spinor on $\Sigma$  exists, solve the APS boundary value problem with $\Psi=0$ for both sides which is possible by minimal condition (for non-negative mean curvature on each side).  We have $\pi_+\phi = \phi$.  We can glue the solutions together to produce a harmonic spinor on $S^4(1/h)$ which then must be zero because the scalar curvature is positive by the Lichnerowicz theorem.  Therefore $\Sigma$ cannot have harmonic spinors.