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Positive scalar curvature and an equivariant Callias-type index theorem for proper actions

Hao Guo, Peter Hochs, Varghese Mathai

Abstract

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M/G$ is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in $M$. We also deduce some other applications of this index theorem. If $G$ is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.

Positive scalar curvature and an equivariant Callias-type index theorem for proper actions

Abstract

For a proper action by a locally compact group on a manifold with a -equivariant Spin-structure, we obtain obstructions to the existence of complete -invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in . We also deduce some other applications of this index theorem. If is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete -invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.

Paper Structure

This paper contains 26 sections, 33 theorems, 109 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Results on positive scalar curvature
  3. Obstructions
  4. Existence for connected Lie groups
  5. An index theorem
  6. The $G$-Callias-type index
  7. Hypersurfaces and the index theorem
  8. Properties of the $G$-Callias-type index
  9. Sobolev modules
  10. Vanishing
  11. Homotopy invariance
  12. A relative index theorem
  13. Proof of the $G$-Callias-type index theorem
  14. An index on the cylinder
  15. Attaching a half-cylinder
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $H\subset M$ be a $G$-invariant, cocompact hypersurface with trivial normal bundle, that partitions $M$ into two open sets. If $M$ admits a complete, $G$-invariant Riemannian metric with nonnegative scalar curvature, and positive scalar curvature in a neighbourhood of $H$, then for a $\mathop{\mathrm{Spin}}\nolimits$-Dirac operator $D^H$ on $H$.

Figures (5)

  • Figure 1: The cylinder $N \times \mathbb{R}$
  • Figure 2: The manifold $M$
  • Figure 3: The manifold $M_C$
  • Figure 4: The manifold $M_C^-$
  • Figure 5: The manifold $M_- \cup_N M_-^-$

Theorems & Definitions (73)

  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5: $G$-Callias-type index theorem
  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Corollary 2.4
  • Definition 2.5
  • Example 2.6
  • ...and 63 more