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Scalar and mean curvature comparison via the Dirac operator

Simone Cecchini, Rudolf Zeidler

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 $\hat{\mathrm{A}}$-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.

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 -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.

Paper Structure

This paper contains 14 sections, 40 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Length of the neck
  3. Estimates of bands
  4. Extremality and rigidity results
  5. Higher index theory and future directions
  6. Relative Dirac bundles
  7. Index theory for relative Dirac bundles
  8. Spectral estimates
  9. A long neck principle with mean curvature
  10. Relative K- and A-hat-area
  11. Estimates of bands
  12. The general warped product rigidity theorem
  13. Rigidity of bands
  14. Scalar-mean extremality and rigidity of warped products

Key Result

Theorem 1.4

Let $(M,g)$ be a compact connected Riemannian spin manifold with non-empty boundary, $n=\dim M \geq 2$ even, and let $\Phi\colon M\to \mathrm{S}^n$ be a smooth area non-increasing map. Assume that $\mathrm{scal}_g \geq n(n-1)$. Moreover, suppose there exists $l\in \left(0,\pi/n\right)$ such that $\m

Figures (2)

  • Figure 1: The long neck problem
  • Figure 2: Construction of the map $\Psi$

Theorems & Definitions (99)

  • Conjecture 1.1: gromovFourLecturesScalar2019v3
  • Conjecture 1.2: Gromov:MetricInequalitiesScalar
  • Conjecture 1.3: Gromov:MetricInequalitiesScalar
  • Theorem 1.4: see \ref{['sec:LongNeck']}
  • Corollary 1.5
  • Definition 1.6
  • Theorem 1.7: see \ref{['S:K-area']}
  • Remark 1.8
  • Theorem 1.9: cf. \ref{['cor:symmetric-band-estimate', 'cor:half-band-estimate']}
  • Corollary 1.10: cf. \ref{['cor:total-band-estimate']}
  • ...and 89 more