Geometric regularization on Riemannian and Lorentzian manifolds

TL;DR

This paper develops a geometric regularization framework for distributional sections on Riemannian and Lorentzian manifolds, using elliptic operators and functional calculus, with applications to wave equations and spacelike foliations.

Contribution

It introduces a novel regularization method based on elliptic operators that preserves key distributional properties on manifolds, linking Riemannian and Lorentzian geometries.

Findings

Regularization preserves singular support and wavefront set.

Bounds on convergence rates of regularized wave operators.

Compatibility of regularization with spacelike foliation restrictions.

Abstract

We investigate regularizations of distributional sections of vector bundles by means of nets of smooth sections that preserve the main regularity properties of the original distributions (singular support, wavefront set, Sobolev regularity). The underlying regularization mechanism is based on functional calculus of elliptic operators with finite speed of propagation with respect to a complete Riemannian metric. As an application we consider the interplay between the wave equation on a Lorentzian manifold and corresponding Riemannian regularizations, and under additional regularity assumptions we derive bounds on the rate of convergence of their commutator. We also show that the restriction to underlying space-like foliations behaves well with respect to these regularizations.