Discrete spectrum and Weyl's asymptotic formula for incomplete manifolds

TL;DR

This paper investigates the spectral properties of Laplacians on incomplete manifolds with singularities, establishing discreteness of spectrum and Weyl's asymptotic behavior under certain metric conditions.

Contribution

It extends spectral analysis to more general incomplete manifolds with higher codimension singularities, providing new conditions for discreteness and asymptotic formulas.

Findings

Spectrum is discrete under specified conditions.

Weyl's asymptotic formula holds for these manifolds.

Results generalize previous work on incomplete surfaces.

Abstract

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.