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

TL;DR

This paper establishes an upper bound on the -hat genus of spin manifolds based on scalar curvature, isoperimetric constants, and volume, advancing understanding of geometric invariants influenced by curvature constraints.

Contribution

It introduces a novel bound on the -hat genus involving scalar curvature and isoperimetric constants, and demonstrates the necessity of the isoperimetric term through examples.

Findings

Bound on -hat genus using scalar curvature and isoperimetric constants

Spectral analysis of Dirac operator as key technique

Counterexample showing necessity of isoperimetric constant

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.