2020/04/28 by В. И. Медведев, Vladimir Medvedev, Medvedev, Vladimir · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.DG #math.SP
paper · pdf · doi:10.48550/arxiv.2004.13776
46 pages, 5 figures. To appear in Canadian Journal of Mathematics
openalex publication_date 2020/04/28 · arxiv created 2021/04/25 · arxiv updated 2021/04/27 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Let Σ be a compact surface with boundary. For a given conformal class c on Σ the functional σk^*(Σ,c) is defined as the supremum of the k-th normalized Steklov eigenvalue over all metrics on c. We consider the behaviour of this functional on the moduli space of conformal classes on Σ. A precise formula for the limit of σk^*(Σ,cn) when the sequence \cn\ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander-Nadirashvili invariants of closed manifolds defined as infcσk^*(Σ,c), where the infimum is taken over all conformal classes c on Σ. We show that these quantities are equal to 2πk for any surface with boundary. As an application of our techniques we obtain new estimates on the k-th normalized Steklov eigenvalue of a non-orientable surface in terms of its genus and the number of boundary components.