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This is a Quantitative Study study.

October 2, 2025Open Access

Maximizing higher eigenvalues in dimensions three and above

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This study explores maximizing eigenvalue functionals on Riemannian manifolds, indicating conditions for singularities and smooth harmonic maps.

Maximizing measures for eigenvalue functionals are smooth harmonic maps for dimensions 3 to 6, indicating structural properties of measures.
For higher dimensions (m ≥ 7), harmonic maps may have singularities with a Hausdorff dimension upper bound of m-7, revealing complex behavior.
The findings generalize previous work on the first eigenvalue, enhancing understanding of eigenvalue behavior in Riemannian geometry.
The presence of preset singular set dimensions showcases elasticity in harmonic map behavior relative to manifold dimensions.

Denis Vinokurov (2025) studied this question.

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This study explores maximizing eigenvalue functionals on Riemannian manifolds, indicating conditions for singularities and smooth harmonic maps.