<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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 <italic>C</italic><sup>0</sup><italic>−</italic>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.</p></abstract>

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.

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

Dipartminto di Scienze di Base ed Applicate per l’Ingegneria

Università degli Studi di Roma La Sapienza

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

We set out to obtain estimates of the Laplacian Spectrum of Riemannian manifolds with non-empty boundary. This was achieved using standard doubled manifold...

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

Published Online: Sep 26, 2019

Page range: 179 - 211

Received: Aug 22, 2018

Accepted: Sep 12, 2018

DOI: https://doi.org/10.2478/auom-2019-0027

Keywords
Double of a manifold, Laplacian, Topological invariants, Regularized metric with control of curvature

© 2019 Luca Sabatini, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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

Published Online: Sep 26, 2019

Page range: 179 - 211

Received: Aug 22, 2018

Accepted: Sep 12, 2018

DOI: https://doi.org/10.2478/auom-2019-0027

KeywordsDouble of a manifold, Laplacian, Topological invariants, Regularized metric with control of curvature

© 2019 Luca Sabatini, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Double of a manifold, Laplacian, Topological invariants, Regularized metric with control of curvature