A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature

TL;DR

This paper explores various conjectures and questions regarding how positive scalar curvature bounds influence the geometric size and shape of Riemannian manifolds, aiming to deepen understanding of scalar curvature implications.

Contribution

It compiles and discusses a series of open problems and conjectures about the geometric consequences of positive scalar curvature bounds.

Findings

Proposes new conjectures relating scalar curvature to manifold geometry

Raises questions about size and shape constraints under scalar curvature bounds

Provides a collection of open problems for future research

Abstract

We collect a few guesses on possible implications of a lower bound on the scalar curvature of a Riemannian manifold on the size and shape of this manifold.