 

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, GromovLawson construction, weak homotopy equivalence. 
Subject: 
53C21, 55P10. 


Abstract
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 codimension 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 2connected, and the corresponding space of metrics of positive scalar curvature on the standard disk D^{n}. 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 S^{n}. Finally, we prove analogous results for the more general space where the boundary metric is left unfixed.


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


Author information
Mark Walsh:
Mathematics and Statistics
Maynooth University
Maynooth, County Kildare, Ireland
Mark.Walsh@mu.ie

