Isoperimetric control of the Steklov spectrum

Details Isoperimetric control of the Steklov spectrum Isoperimetric control of the Steklov spectrum

2011-03-15

arxiv.org/abs/1103.2863v1

J. Funct. Anal. 261 (2011), no. 5, 1384-1399

Mathematics

Spectral Theory

Scientific paper

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.

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