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Extremal metrics involving scalar curvature

Shota Hamanaka

TL;DR

The paper investigates extremal metrics with respect to scalar curvature by establishing sufficient conditions under which rigidity types, as formulated by Listing, fail. It develops a perturbative framework using a TT-tensor deformation $g_s=g- s\,\overset{\circ}{\mathrm{Ric}}_g$ (together with conformal directions) and derives precise sign criteria for $R_{g_s}$ relative to $R_g\cdot\|g\|_{1,g_s}$ or related norms; these criteria yield nonrigidity results (Theorems 1–4). The authors provide concrete non-Einstein examples satisfying the hypotheses, including products of Einstein spaces, left-invariant metrics on 3D unimodular Lie groups, and canonical variations of Riemannian submersions, illustrating broad applicability beyond Einstein metrics. They also discuss connections to Yamabe extremality, Dirac-type rigidity results, and potential extensions to singular or boundary-condition settings, outlining several open questions about weakening assumptions and the landscape of extremal scalar-curvature metrics.

Abstract

We investigate extremal metrics at which various types of rigidity theorems involving scalar curvatures hold. The rigidity we discuss here is related to the rigidity theorems presented by Mario Listing in his previous preprint. More specifically, we give some sufficient conditions for metrics not to be rigid in this sense. We also give several examples of Riemannian manifolds that satisfy such sufficient conditions.

Extremal metrics involving scalar curvature

TL;DR

The paper investigates extremal metrics with respect to scalar curvature by establishing sufficient conditions under which rigidity types, as formulated by Listing, fail. It develops a perturbative framework using a TT-tensor deformation (together with conformal directions) and derives precise sign criteria for relative to or related norms; these criteria yield nonrigidity results (Theorems 1–4). The authors provide concrete non-Einstein examples satisfying the hypotheses, including products of Einstein spaces, left-invariant metrics on 3D unimodular Lie groups, and canonical variations of Riemannian submersions, illustrating broad applicability beyond Einstein metrics. They also discuss connections to Yamabe extremality, Dirac-type rigidity results, and potential extensions to singular or boundary-condition settings, outlining several open questions about weakening assumptions and the landscape of extremal scalar-curvature metrics.

Abstract

We investigate extremal metrics at which various types of rigidity theorems involving scalar curvatures hold. The rigidity we discuss here is related to the rigidity theorems presented by Mario Listing in his previous preprint. More specifically, we give some sufficient conditions for metrics not to be rigid in this sense. We also give several examples of Riemannian manifolds that satisfy such sufficient conditions.

Paper Structure

This paper contains 14 sections, 21 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Previous rigidity results for metrics with positive scalar curvature
  3. Preliminaries
  4. Proofs of Main Theorems
  5. Examples
  6. On the product of two Einstein spaces
  7. Examples on Lie groups
  8. Examples on the total spaces of Riemannian submersions with totally geodesic fibers
  9. Conclusion
  10. Other rigidity results
  11. A sufficient condition for a metric to be a positive Yamabe metric
  12. For singular metrics
  13. About our assumptions
  14. Appendix

Key Result

Theorem 1.1

Let $(M_{0}^{n}, g_{0})~(n \ge 3)$ be an oriented spin closed Riemannian manifold with nonnegative curvature operator, positive Ricci curvature and non-vanishing Euler characteristic. Suppose that $(M^{n}, g)$ is an oriented closed Riemannian manifold and $f : M \rightarrow M_{0}$ is a spin map of n then $\alpha := \mathrm{area} (f)$ is a (positive) constant and $f : (M, \alpha \cdot g) \rightarro

Theorems & Definitions (39)

  • Theorem 1.1: listing2010scalar
  • Theorem 1.2: listing2010scalar
  • Theorem 1.3
  • Remark 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.2
  • Definition 1.1
  • Definition 1.2
  • ...and 29 more