Four Lectures on Scalar Curvature

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. Unlike manifolds with controlled sectional and Ricci curvatures, those with their scalar curvatures bounded from below are not configured in specific rigid forms but display an uncertain variety of flexible shapes similar to what one sees in geometric topology. Yet, there are definite limits to this flexibility, where determination of such limits crucially depends, at least in the known cases, on two seemingly unrelated analytic means: index theory of Dirac operators and the geometric measure theory.1 The emergent picture of spaces with Sc.curv ≥ 0, where topology and geometry are intimately intertwined, is reminiscent of the symplectic geometry, 2 but the former has not reached yet the maturity of the latter. The mystery of scalar curvature remains unsolved. What follows is an extended account of my lectures, delivered during the Spring 2019 at IHES. In Section 1, we give an outline of results, techniques, and problems in scalar curvature. In Section 2, we spend a few dozen pages on background Riemannian geometry, with another dozen in Section 3.3.3 on Clifford algebras and Dirac operators. In Section 3, we overview main topics in geometry and topology of manifolds with their scalar curvatures bounded from below, state theorems, explain the ideas of their proofs, and formulate a variety of problems and conjectures. In Sections 4 and 5, we reformulate, in a more precise and general form, what was stated in the earlier sections and expose technical aspects of the proofs. In Section 6, we describe connective links between different facets of scalar curvature presented in the earlier sections with an emphasis on open problems.