Two-dimension vanishing, splitting and positive scalar curvature

TL;DR

This paper establishes new bounds on the asymptotic and topological dimensions of manifolds with positive scalar curvature under curvature and volume conditions, extending Gromov's conjectures.

Contribution

It proves sharp upper bounds on the asymptotic cone dimension and Betti numbers for manifolds with positive scalar curvature and nonnegative Ricci or sectional curvature.

Findings

Asymptotic cone dimension is at most n-2 under given conditions.

First Betti number upper bound is n-2 for compact and n-3 for non-compact manifolds.

Provides fibration and rigidity theorems for manifolds achieving bounds.

Abstract

We prove several analogs of Gromov's macroscopic dimension conjecture with extra curvature assumptions. More explicitly, we show that for an open Riemannian $n$-manifold $(M,g)$ of nonnegative Ricci (resp. sectional) curvature, if it has uniformly positive scalar curvature and it is uniformly volume noncollapsed, then the essential (resp. Hausdorff) dimension of an asymptotic cone, as a notion of largeness, has a sharp upper bound $n-2$, which is $2$ less than the upper bound for an open Riemannian manifold with only nonnegative Ricci curvature. As a consequence, the dimension of space of linear growth harmonic functions of $M$ has upper bound $n-1$ which is also $2$ less than the sharp bound $n+1$ when $M$ only has nonnegative Ricci curvature. We also prove the first Betti number upper bound is $n-2$ if $M$ is compact, and $n-3$ if $M$ is non-compact. When $M$ is compact we show a fibration theorem over torus, and a rigidity theorem for the fiber when the first Betti number upper bound is achieved.