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Suppose $M$ is a compact oriented riemannian $n$-manifold and $x:M\rightarrow \mathbf{R}^{n+p}$ is an isometric immersion.  The Takahashi’s theorem (TakahashiMinimalImmersions) says that $x$ is minimal if and only if the components of $x$ are all eigenfunctions of the Laplacian with nonzero eigenvalue $\lambda$, i.e. $\Delta x = \lambda x$.
We can try to apply this theorem in our case where we are interested in the question of whether a minimal submanifold of $S^4(1/h)$ can carry harmonic spinors (or massless fermions).  So for $M$ a compact three-manifold, let the first positive eigenvalue be $\lambda$ and we can immerse $M$ into a sphere of radius $\sqrt{3/\lambda}$.  By this criteria of minimal immersion, the first eigenvalue of rhe Laplacian will be fixed by knowledge of the radius $1/h$ of the ambient sphere: $\lambda = 3h^2$.  Therefore we know that $M$ must have fixed first eigenvalue.
Therefore we know the exact first eigenvalue of the Laplacian of the three-manifold $M$.