A perturbation of spacetime Laplacian equation

TL;DR

This paper investigates a perturbed spacetime Laplacian equation involving a gradient term, analyzing its properties within an initial data set to understand the effects of perturbations on solutions.

Contribution

It introduces a new perturbation of the spacetime Laplacian equation and studies its behavior in the context of initial data sets with tensor and function perturbations.

Findings

Derived properties of the perturbed equation

Analyzed the influence of the perturbation on solutions

Provided insights into the stability of solutions under perturbation

Abstract

We study a perturbation \begin{equation} \Delta u + P | \nabla u| = h | \nabla u|, \end{equation} of spacetime Laplacian equation in an initial data set $(M, g, p)$ where $P$ is the trace of the symmetric 2-tensor $p$ and $h$ is a smooth function.