Conjectures on Convergence and Scalar Curvature

TL;DR

This paper surveys progress and challenges related to conjectures on the compactness and geometric stability of sequences of Riemannian manifolds with nonnegative scalar curvature, highlighting recent developments and open questions.

Contribution

It provides a comprehensive overview of the current state of research on scalar curvature convergence conjectures and related examples, consolidating collective insights from a large collaborative effort.

Findings

Survey of progress on scalar curvature convergence conjectures

Identification of key examples and counterexamples

Highlighting open problems and future directions

Abstract

Here we survey the compactness and geometric stability conjectures formulated by the participants at the 2018 IAS Emerging Topics Workshop on {\em Scalar Curvature and Convergence}. We have tried to survey all the progress towards these conjectures as well as related examples, although it is impossible to cover everything. We focus primarily on sequences of compact Riemannian manifolds with nonnegative scalar curvature and their limit spaces. Christina Sormani is grateful to have had the opportunity to write up our ideas and has done her best to credit everyone involved within the paper even though she is the only author listed above. In truth we are a team of over thirty people working together and apart on these deep questions and we welcome everyone who is interested in these conjectures to join us.