Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 6

Special Issue in honor of Professor Blaine Lawson’s 80th birthday

Guest Editors: Shiu-Yuen Cheng, Paulo Lima-Filho, and Stephen Shing-Toung Yau

Positive scalar curvature on manifolds with boundary and their doubles

Pages: 2919 – 2950

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n6.a12

Authors

Jonathan Rosenberg (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)

Shmuel Weinberger (Department of Mathematics, University of Chicago, Illinois, U.S.A.)

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 $M = \operatorname{Dbl} (X, \partial X)$.

Keywords

positive scalar curvature, mean curvature, surgery, bordism, $K$-theory, index

2010 Mathematics Subject Classification

Primary 53C21. Secondary 19L41, 53C27, 55N22, 58J22.

Received 4 January 2022

Received revised 25 May 2022

Accepted 2 February 2023

Published 30 January 2024