$\mathrm{K}$-cowaist on complete foliated manifolds

TL;DR

This paper extends Gromov's K-cowaist concept to complete foliated manifolds, deriving curvature estimates and conditions under which the leafwise scalar curvature is non-positive, especially when the manifold or foliation is spin.

Contribution

It introduces a generalized K-cowaist for complete foliated manifolds and establishes curvature bounds under certain topological conditions.

Findings

If the generalized K-cowaist is infinite and either the manifold or foliation is spin, then the infimum of leafwise scalar curvature is at most zero.

Provides estimates on leafwise scalar curvature using K-cowaist and A-hat-cowaist concepts.

Extends Gromov's curvature invariants to non-compact, foliated settings.

Abstract

Let $(M,F)$ be a connected (not necessarily compact) foliated manifold carrying a complete Riemannian metric $g^{TM}$. We generalize Gromov's $\mathrm{K}$-cowaist using the coverings of $M$, as well as defining a closely related concept called the $\widehat{\mathrm{A}}$-cowaist. Let $k^F$ be the associated leafwise scalar curvature of $g^F = g^{TM}|_F$. We obtain some estimates on $k^F$ using these two concepts. In particular, assuming that the generalized $\mathrm{K}$-cowaist is infinity and either $TM$ or $F$ is spin, we show that $\inf(k^F)\leq 0$.