The problem is to show that a minimal hypersurface in cannot have nontrivial harmonic spinors. This can be done using the the solvability of the boundary value problem for Dirac operator on either piece of $S^4(1/h)$ cut by . 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 be a compact Riemannian spin manifold with boundary . Let denote the projection onto the space of nonnegative eigenspinors. Denote by the normal of the hypersurface and Clifford multiplication. The inhomogeneous boundary value problem for the Dirac operator

on $\Omega$

on $\Sigma$

has a smooth solution for each and satisfying the following integrability condition

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

In case 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 , we can always solve the boundary value problem from either component cut by .

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

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