Eigenvalue estimates of Reilly type in product manifolds and eigenvalue comparison for strip domains

In the first part we derive sharp upper bounds of Reilly type for three kinds of eigenvalues in product manifolds Rk×Mn+1−k for any complete Riemannian manifold M. The eigenvalues include the first Laplacian eigenvalue on mean convex closed hypersurfaces, the first Steklov eigenvalue on domains with mean convex boundary, and the first Hodge Laplacian eigenvalue on closed hypersurfaces with certain convexity condition. In the second part, we prove a comparison result between the first Steklov eigenvalue of a strip domain in space forms and that of the corresponding warped product manifold.