[Paper Review] Boundary Conditions for Scalar Curvature

Christian Bär, Bernhard Hanke|arXiv (Cornell University)|Jan 1, 2021

Advanced Operator Algebra Research27 references7 citations

TL;DR

This paper establishes an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area using the Atiyah-Patodi-Singer index formula, and proves a general deformation principle for boundary conditions that induces weak homotopy equivalences in spaces of such metrics. It further constructs compact manifolds where these metric spaces have nontrivial higher homotopy groups, refining known results on singularities and boundary deformations.

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 à la Miao and the deformation of boundary conditions à 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.

Motivation & Objective

• To identify topological obstructions to the existence of positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area.
• To characterize the extremal case where such obstructions vanish.
• To establish a general deformation principle for boundary conditions under lower scalar curvature bounds.
• To show that relaxing boundary conditions induces weak homotopy equivalences in spaces of positive scalar curvature metrics.
• To construct compact manifolds whose spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.

Proposed method

• Uses the Atiyah-Patodi-Singer index formula as a foundational tool to derive topological obstructions.
• Applies the index formula to spin manifolds with boundary, focusing on metrics with mean convex boundaries and positive scalar curvature.
• Develops a general deformation principle that allows continuous deformation of boundary conditions while preserving lower scalar curvature bounds.
• Demonstrates that such deformations induce weak homotopy equivalences between spaces of positive scalar curvature metrics.
• Applies the deformation principle to refine existing results on smoothing codimension-one singularities and deforming boundary conditions.
• Constructs explicit compact manifolds to realize nontrivial higher homotopy groups in the space of positive scalar curvature metrics with mean convex boundaries.

Experimental results

Research questions

• RQ1 What topological obstructions prevent the existence of positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area?
• RQ2 Under what conditions does the extremal case of the obstruction occur?
• RQ3 How do relaxations of boundary conditions affect the homotopy type of spaces of positive scalar curvature metrics?
• RQ4 Can the deformation principle be used to refine known constructions such as Miao’s smoothing of singularities or Brendle-Marques-Neves’ boundary deformations?
• RQ5 Do there exist compact manifolds for which the space of positive scalar curvature metrics with mean convex boundaries has nontrivial higher homotopy groups?

Key findings

• An obstruction to positive scalar curvature metrics with mean convex boundaries is established on spin manifolds of infinite K-area via the Atiyah-Patodi-Singer index formula.
• The extremal case—where the obstruction vanishes—is fully characterized in terms of the index-theoretic conditions.
• A general deformation principle is proven, showing that relaxing boundary conditions induces weak homotopy equivalences in the spaces of such metrics.
• The deformation principle allows refinement of prior results, including Miao’s smoothing of codimension-one singularities and Brendle-Marques-Neves’ boundary condition deformations.
• Explicit compact manifolds are constructed where the space of positive scalar curvature metrics with mean convex boundaries has nontrivial higher homotopy groups.