Perspectives in Scalar Curvature

The following sections are included:

• VOLUME I
• Contents

We overview major topics on manifolds with positive scalar curvatures and also with specific, positive or negative, lower bounds on their scalar curvatures.

The first two chapters and the beginning sections of the third one include the necessary background material on the differential and on the metric geometry, Clifford algebras, spinors and differential operators. This is accompanied by a presentation of the basic concepts of the theory and an introduction to the techniques of minimal hypersurfaces, μ-bubbles and index theorems for twisted Dirac operators applied to the scalar curvature problems.

We reproduce the proofs of several old theorems on geometry and topology of manifolds with their scalar curvatures bounded from below, present a few new ones and formulate a variety of open problems…

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.

We discuss in this short survey the notion of d_p convergence and its application to studying limits of sequence of Riemannian manifolds ($M_i^n$, g_i) whose scalar curvatures and entropies are bounded from below by small constants. We will build examples showing how the more classical notions of Gromov–Hausdorff and Intrinsic Flat convergence must fail in the context of lower scalar bounds. More fundamentally, we will see how a notion of a metric space itself is the wrong notion for such limits, as distance functions may degenerate. We will see how to fix these problems by weakening the convergence criteria with respect to the d_p-structure. The d_∞-structure corresponds to the classical metric structure, but for any p < ∞ a d_p-structure still comes equipped with a well behaved analysis. Our main result is to show that if a manifold with lower scalar and entropy bounds R, μ ≥ −ϵ(n), then a ball must be d_p-close to a Euclidean ball for p = p(n, ϵ) < ∞. We also study more general limits, and have applications which include a priori estimates on such spaces.

Recently, it has been observed that the scalar curvature of Riemannian 3-manifolds exerts a nontrivial influence on the topology of the fibers of distinguished scalar- and circle-valued functions, generalizing some aspects of the relationship between scalar curvature and stable minimal surfaces in dimension three. In this article, we survey several recent applications of these observations to the geometry of three-manifolds with lower scalar curvature bounds and related problems in general relativity. Along the way, we indicate some natural open questions and directions for further study.

This survey introduces the hyperbolic unfolding correspondence that links the geometric analysis of minimal hypersurfaces with that of Gromov hyperbolic spaces. Problems caused from hypersurface singularities oftentimes become solvable on associated Gromov hyperbolic spaces. Applied to scalar curvature geometry, this yields an inductive smoothing scheme that eliminates such singularities. We explain the two main lines of this argument: a top-down singular analysis using iterated blow-ups of the given hypersurface and a bottom-up smoothing following the tree of blow-ups backward.

Dimension 4 provides a peculiarly idiosyncratic setting for the interplay between scalar curvature and differential topology. Here we will explain some of the peculiarities of the 4-dimensional realm via a careful discussion of the Yamabe invariant (or “sigma constant”). In the process, we will also prove some new results, and point out open problems that continue to represent key challenges in the subject.

The following section is included:

• Index