For every closed manifold of dimension at least 3 and every k, the maximal k-th eigenvalue functional is attained by a measure induced by a locally stable harmonic map into a sphere.
On the singular set of stable- stationary harmonic maps
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Maximizing higher eigenvalues in dimensions three and above
For every closed manifold of dimension at least 3 and every k, the maximal k-th eigenvalue functional is attained by a measure induced by a locally stable harmonic map into a sphere.