Geometry of ends, spectrum of Laplacian, scattering, and inverse problem

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(1) I obtained a sharp criterion of the curvature which shows infinitely many discrete spectrum of the Laplace-Beltrami operator.(2) I obtained a sharp result which clarifies the relation between the measure growth rate and the absence of embedded eigenvalues of drift Laplacians.(3) I proved that the limiting absorption principle holds on Riemannian manifolds having ends with various measure growth rates.(4) I provided the explicitly calculated radii of geodesic balls of a complete noncompact Riemannian submanifold which must exit from the cylindrically bounded domain U, in case its mean curvature is sufficiently small.