Comparison and rigidity for scalar curvature

Introduction

Questions involving the scalar curvature bridge many areas inside mathematics including geometric analysis, differential geometry, and algebraic topology, and they are naturally related to the mathematical description of general relativity.

Methods to Probe Scalar Curvature

There are two main flavours of methods to probe the geometry of scalar curvature:

1. One goes back to Lichnerowicz and uses various versions of index theory for the Dirac equation on spinors.
2. The other is broadly based on minimal hypersurfaces and was initiated by Schoen and Yau.

On both types of methods, there has been tremendous progress over recent years sparked by novel quantitative comparison and rigidity questions due to Gromov and by ongoing attempts to arrive at a deeper geometric understanding of lower scalar curvature bounds.

Unified Standpoint

In this proposal, we view established landmark results, such as the non-existence of positive scalar curvature on the torus, together with the more recent quantitative problems from a conceptually unified standpoint. Here, a comparison principle for scalar and mean curvature along maps between Riemannian manifolds plays the central role.

Development of New Tools

Guided by this point of view, we aim to develop fundamentally new tools to study scalar curvature that bridge long-standing gaps between the existing techniques. This includes:

• A far-reaching generalization of the Dirac operator approach expanding upon techniques pioneered by the PI.
• Novel applications of Bochner-type methods.

Study of Comparison Problems

We will also study analogous comparison problems on domains with singular boundaries motivated by:

• A first synthetic characterization of lower scalar curvature bounds in terms of polyhedral domains.
• The general quest for extending the study of scalar curvature beyond smooth manifolds.

Rigidity Questions

At the same time, we will treat subtle almost rigidity questions corresponding to the rigidity aspect of our comparison principle.