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

2020/10/23 by Bruno Colbois, Colbois, Bruno, Katie Gittins +1
Computer Science · Mathematics · #35P15 #58C40 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2010.12248

openalex publication_date 2020/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain upper bounds for the Steklov eigenvalues \σ_k(M) of a smooth,\ncompact, connected, n-dimensional submanifold M of Euclidean space with\nboundary \Σ that involve the intersection indices of M and of \Σ.\nOne of our main results is an explicit upper bound in terms of the intersection\nindex of \Σ, the volume of \Σ and the volume of M as well as\ndimensional constants. By also taking the injectivity radius of \Σ into\naccount, we obtain an upper bound that has the optimal exponent of k with\nrespect to the asymptotics of the Steklov eigenvalues as k \→ \∞.\n

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