New York Journal of Mathematics
Volume 26 (2020), 853-930


Mark Walsh

The space of positive scalar curvature metrics on a manifold with boundary

view    print

Published: September 9, 2020.
Keywords: space of Riemannian metrics of positive scalar curvature, manifold with boundary, surgery, bordism, spin, Gromov-Lawson construction, weak homotopy equivalence.
Subject: 53C21, 55P10.

We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary. These metrics extend a fixed boundary metric and take a product structure on a collar neighbourhood of the boundary. We show that the weak homotopy type of this space is preserved by certain surgeries on the boundary in co-dimension at least three. Thus, under reasonable circumstances there is a weak homotopy equivalence between the space of such metrics on a compact spin manifold W, of dimension n ≥ 6 and whose boundary inclusion is 2-connected, and the corresponding space of metrics of positive scalar curvature on the standard disk Dn. Indeed, for certain boundary metrics, this space is weakly homotopy equivalent to the space of all metrics of positive scalar curvature on the standard sphere Sn. Finally, we prove analogous results for the more general space where the boundary metric is left unfixed.


The author acknowledges support from Simons Foundation Collaboration Grant No. 280310.

Author information

Mark Walsh:
Mathematics and Statistics
Maynooth University
Maynooth, County Kildare, Ireland