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

Alessandro Carlotto, Chao Li

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.

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

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.
Paper Structure (4 sections, 13 theorems, 7 equations)

This paper contains 4 sections, 13 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Setup and key definitions
  3. Isotopy in $\boldsymbol{\mathscr R_{R>0, H=0}}$ vs. minimal concordance
  4. Spaces of metrics defined by a spectral stability condition

Key Result

Lemma 3.1

Let us consider on the manifold $M=X\times J$ a smooth metric of the form where $u\in C^{\infty}(M)$ and the map $J\ni t \mapsto h_t(x)\in \mathscr R(X)$ is also smooth. Then the following equations hold: Note that for the first two equations we have considered $X\times\{t\}$ as boundary of $X\times [0,t]$, i.e., we have worked with respect to the unit normal $u^{-1}\partial_t$; $R_{h_t}$ denote

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • Proposition 3.2
  • Corollary 3.3
  • Corollary 3.4
  • ...and 10 more