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Bounding the A-hat genus using scalar curvature lower bounds and isoperimetric constants

Qiaochu Ma, Jinmin Wang, Guoliang Yu, Bo Zhu

Abstract

In this paper, we prove an upper bound on the $\widehat{A}$ genus of a smooth, closed, spin Riemannian manifold using its scalar curvature lower bound, Neumann isoperimetric constant, and volume. The proof of this result relies on spectral analysis of the Dirac operator. We also construct an example to show that the Neumann isoperimetric constant in this bound is necessary. Our result partially answers a question of Gromov on bounding characteristic numbers using scalar curvature lower bound.

Bounding the A-hat genus using scalar curvature lower bounds and isoperimetric constants

Abstract

In this paper, we prove an upper bound on the genus of a smooth, closed, spin Riemannian manifold using its scalar curvature lower bound, Neumann isoperimetric constant, and volume. The proof of this result relies on spectral analysis of the Dirac operator. We also construct an example to show that the Neumann isoperimetric constant in this bound is necessary. Our result partially answers a question of Gromov on bounding characteristic numbers using scalar curvature lower bound.

Paper Structure

This paper contains 11 sections, 8 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Main result
  4. Acknowledgment
  5. Preliminaries
  6. Lichnerowicz formula and Atiyah-Singer index theorem
  7. Kato's inequality
  8. The Neumann isoperimetric constant
  9. Proof of the main theorem
  10. Examples of Riemannian manifolds with large A-hat genus, small scalar curvature, and bounded volume
  11. An open problem

Key Result

Theorem 1.1

If a closed spin Riemannian manifold $M$ with dimension $m$ has scalar curvature $\mathrm{Sc}_g\geqslant k_0$ for some constant $k_0$, then there exists a constant $C_m>0$ depending only on $m$ such that

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • ...and 2 more