A Bogovski\v{i}-type operator for Corvino-Schoen hyperbolic gluing

TL;DR

This paper develops a specialized solution operator for the linearized constant scalar curvature equation in hyperbolic space, enabling advanced gluing techniques that extend existing methods to hyperbolic geometries.

Contribution

It introduces a new solution operator with favorable support and regularity properties, facilitating Corvino-Schoen hyperbolic gluing and extending recent gluing methods to hyperbolic space.

Findings

Constructed a solution operator with good support propagation

The operator gains two derivatives relative to standard norms

Extended gluing techniques to hyperbolic geometries

Abstract

We construct a solution operator for the linearized constant scalar curvature equation at hyperbolic space in space dimension larger than or equal to two. The solution operator has good support propagation properties and gains two derivatives relative to standard norms. It can be used for Corvino-Schoen-type hyperbolic gluing, partly extending the recently introduced Mao-Oh-Tao gluing method to the hyperbolic setting.