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We prove that in Riemannian manifolds the $k$-th Steklov eigenvalue on adomain and the square root of the $k$-th Laplacian eigenvalue on its boundarycan be mutually controlled in terms of the maximum principal curvature of theboundary under sectional curvature conditions. As an application, we derive aWeyl-type upper bound for Steklov eigenvalues. A Pohozaev-type identity forharmonic functions on the domain and the min-max variational characterizationof both eigenvalues are important ingredients.

Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds