Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the stability of Einstein metrics carrying a special twisted spinor

Diego Artacho

TL;DR

The paper proves linear semi-stability for a broad class of Einstein metrics with non-positive scalar curvature that admit a parallel twisted pure spin^r spinor. By constructing an isometric map from symmetric 2-tensors to twisted spinor-valued 1-forms and relating the linearised Einstein operator to a Dirac-type square, the authors derive a Bochner-type formula using twisted spinor identities to control curvature terms. Under mild restrictions on the dimensions and the twist parameter, they establish a nonpositive second variation, extending Dai–Wang–Wei’s results and applying to negative quaternion-Kähler manifolds of dimension >8. The approach leverages spinorial techniques to address stability questions for Einstein metrics in the non-positive curvature regime.

Abstract

We prove linear semi-stability for a large class of Einstein metrics of non-positive scalar curvature. More precisely, we show that any Einstein $n$-manifold with non-positive scalar curvature carrying a parallel twisted pure spin$^r$ spinor is linearly semi-stable, under mild restrictions on $n$ and $r$. We thus extend the parallel spin and spin$^c$ stability results of Dai--Wang--Wei. As an application, our result implies linear semi-stability for all negative quaternion-K{ä}hler manifolds of dimension greater than $8$.

On the stability of Einstein metrics carrying a special twisted spinor

TL;DR

The paper proves linear semi-stability for a broad class of Einstein metrics with non-positive scalar curvature that admit a parallel twisted pure spin^r spinor. By constructing an isometric map from symmetric 2-tensors to twisted spinor-valued 1-forms and relating the linearised Einstein operator to a Dirac-type square, the authors derive a Bochner-type formula using twisted spinor identities to control curvature terms. Under mild restrictions on the dimensions and the twist parameter, they establish a nonpositive second variation, extending Dai–Wang–Wei’s results and applying to negative quaternion-Kähler manifolds of dimension >8. The approach leverages spinorial techniques to address stability questions for Einstein metrics in the non-positive curvature regime.

Abstract

We prove linear semi-stability for a large class of Einstein metrics of non-positive scalar curvature. More precisely, we show that any Einstein -manifold with non-positive scalar curvature carrying a parallel twisted pure spin spinor is linearly semi-stable, under mild restrictions on and . We thus extend the parallel spin and spin stability results of Dai--Wang--Wei. As an application, our result implies linear semi-stability for all negative quaternion-K{ä}hler manifolds of dimension greater than .

Paper Structure

This paper contains 4 sections, 4 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Bochner-type formula
  4. Stability of metrics with parallel twisted pure spinors

Key Result

Theorem 1

Let $(M^n,g)$ be a Riemannian manifold carrying a parallel twisted pure spin$^r$ spinor, for some $r \ge 3$, $r\neq 4$, $n\neq 8$, and $n+4r-16 \neq 0$. Then $g$ is Einstein and, if $\mathrm{scal}_g \le 0$, it is linearly semi-stable. ∎

Theorems & Definitions (9)

  • Theorem 1
  • Definition 2: HS19
  • Lemma 3
  • proof
  • Corollary 4
  • proof
  • Remark 5
  • Lemma 6: HS19
  • proof : Proof of Theorem \ref{['thm:main']}