Chapter 2: Scalar Curvature and Generalized Callias Operators

In this survey, we give an overview of recent applications of Callias operators to the geometry of scalar curvature. A Callias operator is an operator of the form ℬ_ψ =𝒟 + 𝒢_ψ, where 𝒟 is a Dirac operator and 𝒢_ψ is an order zero term depending on a scalar-valued function ψ. The zero order term modifies the Schrödinger–Lichnerowicz formula by a differential expression in the function ψ that can be related to distance estimates. This fact allows to use the Dirac method to derive sharp quantitative estimates in the presence of lower scalar curvature bounds in the spirit of metric inequalities with scalar curvature as proposed by Gromov.