2025/06/11 by Denis Vinokurov, Vinokurov, Denis · 3 citations
Mathematics · #53C43 (Secondary) #58J50 (Primary) 58E20 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Graph theory and applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.DG #math.SP #msc:53C43 #msc:58E20 #msc:58J50
paper · pdf · doi:10.48550/arxiv.2506.09328
Theorem 1.1 strengthened via the addition of Section 5.2; former Section 1.4.1 removed as no longer needed
openalex publication_date 2025/06/11 · openalex created_date 2025/10/10 · arxiv created 2026/07/30 · arxiv updated 2026/07/31 · openalex updated_date 2026/08/01
We study the problem of maximizing the k-th eigenvalue functional over the class of absolutely continuous measures on a closed Riemannian manifold of dimension m≥ 3. Extending the work of Karpukhin and Stern on the first eigenvalue, we prove that, for every k≥ 1, the supremum is attained by a measure induced by a harmonic map into a finite-dimensional sphere. The map is smooth outside a closed singular set of Hausdorff dimension at most m-7, and is therefore smooth when 3 ≤ m ≤ 6. We further prove that this dimension bound is optimal: for every m ≥ 7 and every integer 0≤ d ≤ m-7, there exists a maximizing harmonic map on the m-dimensional round sphere whose singular set has Hausdorff dimension d.