A Note about Isotopy and Concordance of Positive Scalar Curvature Metrics on Compact Manifolds with Boundary

TL;DR

This paper explores isotopy and concordance of positive scalar curvature metrics on compact manifolds with boundary, introducing new variants of minimal concordance and analyzing their properties, especially for surfaces.

Contribution

It introduces two variants of minimal concordance for metrics with boundary and provides a comprehensive analysis for surfaces, expanding understanding of spectral stability conditions.

Findings

Defined new notions of minimal concordance

Developed tools for analyzing metrics with boundary

Achieved a complete picture for surfaces

Abstract

We study notions of isotopy and concordance for Riemannian metrics on manifolds with boundary and, in particular, we introduce two variants of the concept of minimal concordance, the weaker one naturally arising when considering certain spaces of metrics defined by a suitable spectral ''stability'' condition. We develop some basic tools and obtain a rather complete picture in the case of surfaces.