Circle bundles with PSC over large manifolds

TL;DR

This paper constructs numerous high-dimensional manifolds with circle bundle structures that admit positive scalar curvature metrics, addressing Gromov's question about enlargeable manifolds and their scalar curvature properties.

Contribution

It provides the first infinite family of such manifolds in all dimensions, using symplectic geometry techniques to achieve positive scalar curvature on their total spaces.

Findings

Existence of circle bundles over enlargeable manifolds with PSC metrics

Construction of large manifolds with controlled macroscopic dimension

Resolution of Gromov's question in all dimensions

Abstract

We construct infinitely many examples of macroscopically large manifolds of dimension $m \geq 4$ equipped with circle bundles whose total spaces admit metrics of positive scalar curvature and have macroscopic dimension at most $\lceil m/2 \rceil + 1$. In particular, we answer a question of Gromov on the existence of circle bundles over enlargeable manifolds whose total spaces admit metrics of positive scalar curvature, in all dimensions. Our constructions are based on techniques from symplectic geometry.