2011/03/15 by Bruno Colbois, Ahmad El Soufi, Alexandre Girouard · 1 citation
Mathematics · #math.SP #math.DG #math.MG #msc:58J50 #msc:58E11 #msc:35P15
published as J. Funct. Anal. 261 (2011), no. 5, 1384-1399
arxiv created 2011/03/15 · arxiv updated 2012/02/24
Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, the normalized Steklov eigenvalues are uniformly bounded above in terms of the genus. We also obtain a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its bounding hypersurface.