Localized gluing of Riemannian metrics in interpolating their scalar curvature

TL;DR

This paper introduces a method to smoothly interpolate and glue Riemannian metrics, preserving scalar curvature properties, and extends existing gluing techniques to more general settings including asymptotically Euclidean and Schwarzschild metrics.

Contribution

It provides a new gluing technique for Riemannian metrics that interpolates scalar curvature and extends Corvino's gluing method to non-constant scalar curvature metrics.

Findings

Successfully interpolates scalar curvature between two metrics

Gluing preserves scalar curvature inequalities

Extends gluing techniques to asymptotically Euclidean and Schwarzschild metrics

Abstract

We show that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature. The resulting smooth metric is the same as the starting ones outside the gluing region and has scalar curvature interpolating between the original ones. One can then glue metrics while maintaining inequalities satisfied by the scalar curvature. We also glue asymptotically Euclidean metrics to Schwarzschild ones and the same for asymptotically Delaunay metrics, keeping bounds on the scalar curvature, if any. This extend the Corvino gluing near infinity to non-constant scalar curvature metrics.