Scalar positive immersions

TL;DR

This paper demonstrates that on certain manifolds, positive scalar curvature metrics can be realized through immersions into Euclidean space, extending classical results with a new extrinsic construction method.

Contribution

It introduces an extrinsic analog of Gromov-Lawson surgery, enabling the realization of positive scalar curvature metrics via immersions close to Whitney bounds.

Findings

Positive scalar curvature metrics can be immersed into Euclidean space.

The dimension of the immersion is close to Whitney's classical bounds.

A new extrinsic surgery technique is developed for constructing such metrics.

Abstract

As shown by Gromov-Lawson and Stolz the only obstruction to the existence of positive scalar curvature metrics on closed simply connected manifolds in dimensions at least five appears on spin manifolds and is given by the non-vanishing of the $\alpha$-genus of Hitchin. When unobstructed we shall realize a positive scalar curvature metric by an immersion into Euclidean space whose dimension is uniformly close to the classical Whitney upper bound for smooth immersions. Our main tool is an extrinsic counterpart of the well-known Gromov-Lawson surgery procedure for constructing positive scalar curvature metrics.