Upper bounds for Steklov eigenvalues of submanifolds in Euclidean space via the intersection index

Colbois, Bruno; Gittins, Katie

Abstract

We obtain upper bounds for the Steklov eigenvalues σk(M)of a smooth, compact, n-dimensional submanifold M of Euclidean space with boundary Σ that involve the intersection indices of M and of Σ. One of our main results is an explicit upper bound in terms of the intersection index of Σ, the volume of Σand the volume of M as well as dimensional constants. By also taking the injectivity radius of Σinto account, we obtain an upper bound that has the optimal exponent of k with respect to the asymptotics of the Steklov eigenvalues as k→∞

Citation

Colbois, B., & Gittins, K. (2021). Upper bounds for Steklov eigenvalues of submanifolds in Euclidean space via the intersection index. Differential Geometry and its Applications, 78, Article 101777. https://doi.org/10.1016/j.difgeo.2021.101777

• Journal Article Type: Article
• Acceptance Date: May 17, 2021
• Online Publication Date: Jul 29, 2021
• Publication Date: 2021-10
• Deposit Date: Jul 30, 2021
• Publicly Available Date: Jul 30, 2021
• Journal: Differential Geometry and its Applications
• Print ISSN: 0926-2245
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 78
• Article Number: 101777
• DOI: https://doi.org/10.1016/j.difgeo.2021.101777