Rigid comparison geometry for Riemannian bands and open incomplete manifolds

TL;DR

This paper develops comparison theorems relating curvature bounds to geometric widths in Riemannian manifolds, including incomplete and open cases, with applications to classical theorems and new rigidity results using spacetime harmonic functions.

Contribution

It introduces sharp comparison theorems for manifolds with curvature bounds, including incomplete cases, and characterizes model geometries with rigidity statements, addressing conjectures and classical results.

Findings

Sharp inequalities relating curvature bounds to manifold width.

Rigidity characterizations of model geometries including lens spaces.

New proofs of classical comparison theorems using spacetime harmonic functions.

Abstract

Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci curvature, and contains a variety of theorems which provide sharp relationships between this bound and notions of {\em{width}}. Some inequalities leverage geometric quantities such as boundary mean curvature, while others involve topological conditions in the form of linking requirements or homological constraints. In several of these results open and incomplete manifolds are studied, one of which partially addresses a conjecture of Gromov in this setting. The majority of results are accompanied by rigidity statements which isolate various model geometries -- both complete and incomplete -- including a new characterization of round lens spaces, and other models that have not appeared elsewhere. As a byproduct, we additionally give new and quantitative proofs of several classical comparison statements such as Bonnet-Myers' and Frankel's Theorem, as well as a version of Llarull's Theorem and a notable fact concerning asymptotically flat manifolds. The results that we present vary significantly in character, however a common theme is present in that the lead role in each proof is played by \emph{spacetime harmonic functions}, which are solutions to a certain elliptic equation originally designed to study mass in mathematical general relativity.