Scalar and mean curvature comparison via the Dirac operator

Abstract

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we thereby give refined answers to questions on metric inequalities recently proposed by Gromov. This includes optimal estimates for Riemannian bands and for the long neck problem. In the case of bands over manifolds of non-vanishing A-genus, we establish a rigidity result stating that any band attaining the predicted upper bound is isometric to a particular warped product over some spin manifold admitting a parallel spinor. Furthermore, we establish scalar-and mean curvature extremality results for certain log-concave warped products. The latter includes annuli in all simply-connected space forms. On a technical level, our proofs are based on new spectral estimates for the Dirac operator augmented by a Lipschitz potential together with local boundary conditions.

The positive mass theorem and distance estimates in the spin setting

“…This can be interpreted as a "long neck principle", originally proposed by Gromov for certain compact manifolds with boundary (compare [13, p. 87] and [4]), in the context of the positive mass theorem. Indeed, on a technical level, the present paper combines Witten's proof of the positive mass theorem with the technique systematically developed by the authors in [5] of using Callias operators on spin manifolds to obtain scalar-and mean curvature comparison results related to Gromov's metric inequalities programme. See also [6,8,14,22,28,30,31] for other related work in this area.…”

“…Note that the operator B ψ we study here is essentially just the operator / D + iψ together with its formal adjoint / D − iψ, both subject to chiral boundary conditions on the interior boundary. The reason for considering both at the same time is simply a matter of convenience because it makes certain computations more symmetric and fits more neatly into the formal setup we considered in [5].…”

On Gromov's dihedral extremality and rigidity conjectures

“…In the past several years, Gromov has formulated an extensive list of conjectures and open questions on scalar curvature [15,16,17,18]. This has given rise to new perspectives on scalar curvature and inspired a wave of recent activity in this area [8,9,13,17,18,19,20,25,28,31,32,34,35,36]. In particular, Gromov proposed two rigidity conjectures: the dihedral extremality conjecture (Conjecture 1.1) [16] and the dihedral rigidity conjecture (Conjecture 1.2) [15] about comparisons of scalar curvature, mean curvature and dihedral angles for compact manifolds with corners, which can be viewed as scalar curvature analogues of the Alexandrov's triangle comparisons for spaces whose sectional curvature is bounded below [1,2].…”

Density and positive mass theorems for incomplete manifolds

“…Also, their asymptotics are slightly stronger than ours, but we do not believe that this is essential. We would also like to point the reader to the very interesting paper [CZ20].…”