On Certain Geometric Operators Between Sobolev Spaces of Sections of Tensor Bundles on Compact Manifolds Equipped with Rough Metrics

Authors

DOI:

https://doi.org/10.37256/cm.3220221278

Keywords:

compact manifolds, Sobolev spaces, tensor bundles, convergence, Laplacian, vector Laplacian, covariant derivative, metric distortion tensor

Abstract

The study of Einstein constraint equations in general relativity naturally leads to considering Riemannian manifolds equipped with nonsmooth metrics. There are several important differential operators on Riemannian manifolds whose definitions depend on the metric: gradient, divergence, Laplacian, covariant derivative, conformal Killing operator, and vector Laplacian, among others. In this article, we study the approximation of such operators, defined using a rough metric, by the corresponding operators defined using a smooth metric. This paves the road to understanding to what extent the nice properties such operators possess, when defined with smooth metric, will transfer over to the corresponding operators defined using a nonsmooth metric. These properties are often assumed to hold when working with rough metrics, but to date the supporting literature is slim.

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Published

2022-04-19

How to Cite

1.
Behzadan A, Holst M. On Certain Geometric Operators Between Sobolev Spaces of Sections of Tensor Bundles on Compact Manifolds Equipped with Rough Metrics. Contemporary Mathematics [Internet]. 2022 Apr. 19 [cited 2022 Jun. 28];3(2):89-140. Available from: https://ojs.wiserpub.com/index.php/CM/article/view/1278