The odd-dimensional long neck problem via spectral flow

TL;DR

This paper addresses the odd-dimensional long neck problem on spin manifolds by establishing a scalar-mean curvature comparison theorem using spectral flow, extending prior work and providing a quantitative version of Llarull's theorem.

Contribution

It introduces a new scalar-mean curvature comparison theorem for odd-dimensional spin manifolds and applies spectral flow of Callias operators to solve Gromov's long neck problem.

Findings

Complete solution to Gromov's long neck problem for spin manifolds

Quantitative version of Llarull's theorem on non-compact spin manifolds

Extension of previous curvature comparison results

Abstract

In this paper, we establish a scalar-mean curvature comparison theorem for the long neck problem on odd-dimensional spin manifolds. This extends previous work of Cecchini and Zeidler, and gives a complete answer to Gromov's long neck problem in terms of spin manifolds. As a related question, we prove a quantitative version of Llarull's theorem on non-compact spin manifolds. Our results are derived by studying the spectral flow of a family of Callias operators.