Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary

We set out to obtain estimates of the Laplacian Spectrum of Riemannian manifolds with non-empty boundary. This was achieved using standard doubled manifold techniques. In simple terms, we pasted two copies of the same manifold along their common boundary thereby obtaining a Riemannian manifold with empty boundary and with a C⁰−metric. This made it possible to adapt some estimates of the spectrum dependent on the volume or genus of the manifold as calculated in recent years by several authors. In order to extend further estimates that depend on the curvature, it is necessary to regularize the metric of the doubled manifold so that the new metric is isometric to that of each copy and such that the curvature has a finite lower bound. Controlling the curvature in this way also makes estimates of topological invariants available.