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Boundary conditions for scalar curvature

Christian Baer, Bernhard Hanke

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.

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.

Paper Structure

This paper contains 13 sections, 26 theorems, 132 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Nonexistence of metrics with positive scalar curvature and mean convex boundary
  3. Harmonic spinors on manifolds with boundary
  4. Infinite K-area
  5. Nonexistence of metrics with positive scalar curvature and mean convex boundary
  6. Deformations of the metric
  7. Applications
  8. Subspaces of R>sigma(M) given by geometric conditions on the boundary
  9. Additional conditions on the boundary metric
  10. Additional conditions in the interior
  11. Spaces of metrics with mean-convex singularities
  12. Min-Oo's conjecture
  13. Homotopy groups of spaces of metrics with lower scalar curvature and mean curvature bounds

Key Result

Lemma \oldthetheorem

Let $p\in M$. Then all eigenvalues $\lambda$ of $\mathscr{K}^E_p$ satisfy

Figures (4)

  • Figure 1: Identify opposite sides of cube to obtain $3$-torus and remove the red solid torus
  • Figure 2: $M=S^1\times I$ is area-enlargeable
  • Figure 3: The functions $\varphi_1$, $\psi_1$, and $\chi_\delta$
  • Figure 4: The functions $S_1$ and $S_2$

Theorems & Definitions (66)

  • Lemma \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • ...and 56 more