Local theory of wave equations with timelike curves of conic singularities

TL;DR

This paper develops a comprehensive mathematical framework for analyzing wave equations on Lorentzian manifolds with timelike conic singularities, addressing existence, uniqueness, and regularity of solutions in complex geometric and analytic settings.

Contribution

It introduces a general theory accommodating various singularity types and tensorial equations without symmetry assumptions, extending previous frameworks to more general Lorentzian geometries.

Findings

Established existence and uniqueness of solutions.

Extended regularity results for wave equations.

Unified treatment of geometric and analytic singularities.

Abstract

We develop a general theory for the existence, uniqueness, and higher regularity of solutions to wave-type equations on Lorentzian manifolds with timelike curves of cone-type singularities. These singularities may be of geometric type (cone points with time-dependent cross sectional metric), of analytic type (such as asymptotically inverse square singularities or first order asymptotically scaling-critical singular terms), or any combination thereof. We can treat tensorial equations without any symmetry assumptions; we only require a condition of mode stability type for the stationary model operators defined at each point along the curve of cone points. In symmetric ultrastatic settings, we recover the solvability theory given by the functional calculus for the Friedrichs extension.