On the spectral geometry of spaces with cone-like singularities

Abstract

I describe an extension of a portion of the theory of the Laplace operator on compact riemannian manifolds to certain spaces with singularities. Although this approach can be extended to include quite general spaces, this paper will confine itself to the case of manifolds with cone-like singularities. These singularities are geometrically the simplest possible, but they already serve to illustrate new phenomena that are typical of the more general situation. Moreover, by inductive arguments, the study of simplicial complexes whose simplices have constant curvature and totally geodesic faces (e.g., p.l. manifolds) can in large measure be reduced to the study of cone-like singularities.