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Positive scalar curvature on manifolds with boundary and their doubles

Jonathan Rosenberg, Shmuel Weinberger

Abstract

This paper is about positive scalar curvature on a compact manifold $X$ with non-empty boundary $\partial X$. In some cases, we completely answer the question of when $X$ has a positive scalar curvature metric which is a product metric near $\partial X$, or when $X$ has a positive scalar curvature metric with positive mean curvature on the boundary, and more generally, we study the relationship between boundary conditions on $\partial X$ for positive scalar curvature metrics on $X$ and the positive scalar curvature problem for the double $M=\operatorname{Dbl}(X,\partial X)$.

Positive scalar curvature on manifolds with boundary and their doubles

Abstract

This paper is about positive scalar curvature on a compact manifold with non-empty boundary . In some cases, we completely answer the question of when has a positive scalar curvature metric which is a product metric near , or when has a positive scalar curvature metric with positive mean curvature on the boundary, and more generally, we study the relationship between boundary conditions on for positive scalar curvature metrics on and the positive scalar curvature problem for the double .
Paper Structure (7 sections, 20 theorems, 4 equations, 9 figures)

This paper contains 7 sections, 20 theorems, 4 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The Relatively $1$-Connected Case
  3. The Simply Connected Case
  4. Obstruction Theory for the Relative Problem
  5. More Complicated Cases with Non-trivial Fundamental Groups
  6. The Gromov-Lawson Doubling Theorem
  7. Open Problems

Key Result

Theorem 1.1

Let $X$ be a compact manifold with boundary, of dimension $n$, and let $M=\operatorname{Dbl}(X,\partial X)$ denote the double of $X$ along the boundary $\partial X$. If $X$ admits a metric of positive scalar curvature with positive mean curvature $H>0$ along the boundary $\partial X$(with respect to

Figures (9)

  • Figure 1: A $3$-holed sphere
  • Figure 2: Computing the $\alpha_\Gamma$-invariant
  • Figure 3: The spin bordism $W$ from $(Y, \partial Y)$ to $(X, \partial X)$
  • Figure 4: The graph of groups for computing $\pi_1(M)$
  • Figure 5: The Atiyah-Hirzebruch spectral sequence for computing $ko_n(B\pi_1(M))$. The corresponding spectral sequence for $KO_n(B\pi_1(M))$ is similar except that it extends into the fourth quadrant, and the map from this spectral sequence to that one preserves differentials affecting the line $p+q=n$. Typical differentials are shown in color.
  • ...and 4 more figures

Theorems & Definitions (45)

  • Theorem 1.1: MR569070
  • Theorem 1.2: MR759265
  • Theorem 1.3: MR759265
  • Remark 1.4
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • ...and 35 more