Discrete Spectrum and Weyl’s Asymptotic Formula for Incomplete Manifolds

Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete m-dimensional manifolds with singularities on sets of codimension at least 2. With certain restrictions on the metric, we establish that the spectrum is discrete and satisfies Weyl’s asymptotic formula. 1. Discreteness of the Spectrum When one studies the Morse index of minimal surfaces in Euclidean 3-space R or of mean curvature 1 surfaces in hyperbolic 3-space H, the problem reduces to the study of the number of eigenvalues less than 2 of the spectrum of the LaplaceBeltrami operator on Met1 surfaces [FC], [UY], [LR]. (Met1 surfaces are incomplete 2-dimensional manifolds with constant curvature 1 and isolated conical singularities.) Met1 surfaces are known to have pure point spectrum and satisfy Weyl’s asymptotic formula. Here we will show that the spectrum is discrete and that Weyl’s asymptotic formula holds for more general incomplete manifolds. We allow the dimension to be arbitrary; we do not make any specific assumptions about the curvature; and we allow more general singularities, of at least codimension 2 (in a sense to be made precise below). This more general setting allows us to consider singularities such as a product of an m− n dimensional metric cone with a portion of R (m ≥ n+ 2), one of our desired examples. In this example, the incomplete metric is singular only in the direction of the metric cone and not on the portion of R itself, so generally the incomplete manifolds and their metrics g̃ that we consider will not be conformally equivalent to open sets of compact Riemann manifolds, unlike the case of Met1 surfaces. With this in mind, we now define the types of incomplete manifolds and metrics g̃ that we will study here. Let (M, g) be a compact manifold of dimension m with smooth Riemannian metric g. Let N be a compact submanifold of dimension n with codimension m − n ≥ 2. Suppose further that in a neighborhood of N the metric g can be diagonalized; that 1991 Mathematics Subject Classification. Primary 58J50; Secondary 35J05, 35P10, 35P20, 53C20. 1 2 JUN MASAMUNE AND WAYNE ROSSMAN is, there exist local coordinates (x1, ..., xm−n, y1, ..., yn), where (0, ..., 0, y1, ..., yn) are coordinates forN , so that (dx1, ..., dxm−n, dy1, ..., dyn) is globally defined in some open neighborhood of N and so that the metric g is written