Llarull type theorems for bands in Three and Four dimensions

TL;DR

This paper extends Llarull's theorem to three and four-dimensional bands using warped μ-bubble techniques, establishing new scalar curvature bounds in these dimensions.

Contribution

It introduces Llarull type theorems for 3- and 4-dimensional bands based on spectral scalar curvature bounds, employing warped μ-bubble methods.

Findings

Llarull type theorems established for 3- and 4-dimensional bands

Spectral scalar curvature bounds are used to derive geometric constraints

New techniques extend previous scalar curvature results to higher dimensions

Abstract

Llarull's theorem asserts that the scalar curvature and the metric on the $n$-sphere cannot be bounded below at the same time by those of the standard $n$-sphere. Using the warped $\mu$-bubble method, we develop Llarull type theorems for three and four-dimensional bands with spectral scalar curvature bounds.