Boundary conditions for scalar curvature

Abstract

Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites a la Miao and the deformation of boundary conditions a la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.

Minor typos fixed. Otherwise identical to the published version up to layout

References in corpus (2)

• Generalized Cylinders in Semi-Riemannian and Spin Geometry
• Scalar Curvature of Manifolds with Boundaries: Natural Questions and Artificial Constructions

Cited by in corpus (7)

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• A note on the long neck principle and spectral width inequality of geodesic collar neighborhoods
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• A Note about Isotopy and Concordance of Positive Scalar Curvature Metrics on Compact Manifolds with Boundary
• Spaces of Positive Scalar Curvature metrics on totally nonspin Manifolds with spin boundary
• $\mathrm{K}$-cowaist on complete foliated manifolds