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Eigenvalue value estimates and stability of positive quaternion-Kähler manifolds

Yasushi Homma, Uwe Semmelmann

Abstract

In this article we study the stability problem for positive quaternion-Kähler manifolds. We give a description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and a special class of symmetric 2-tensors. We also give improved eigenvalue estimates for the Hodge-Laplacian on 2-forms. On the parallel subbundle Sym^2 E of the 2-form bundle we prove a sharp lower bound for the first non-zero eigenvalue.

Eigenvalue value estimates and stability of positive quaternion-Kähler manifolds

Abstract

In this article we study the stability problem for positive quaternion-Kähler manifolds. We give a description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and a special class of symmetric 2-tensors. We also give improved eigenvalue estimates for the Hodge-Laplacian on 2-forms. On the parallel subbundle Sym^2 E of the 2-form bundle we prove a sharp lower bound for the first non-zero eigenvalue.

Paper Structure

This paper contains 15 sections, 26 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quaternion-Kähler manifolds
  4. Generalized gradients and Weitzenböck formulas
  5. Eigenvalue estimates and Weitzenböck formulas on T M
  6. Weitzenböck formulas on HE
  7. Generalised gradients on HE compared with differential and divergence
  8. Eigenvalue estimates on $2$-forms
  9. Eigenvalue estimates on $\mathrm{Sym}^2E$
  10. Eigenvalue estimates on Sym^2 H
  11. Eigenvalue estimates on Sym^2H Λ^2_0E
  12. Eigenvalue estimates on symmetric 2-tensors
  13. Eigenvalue estimates on Λ^2_0E
  14. Eigenvalue estimates on Sym^2HSym^2E
  15. Infinitesimal Einstein deformations and destabilising directions

Key Result

Proposition 2.1

On sections of the bundle $\mathrm{Sym}^kH \otimes V_\rho$ the following formula holds $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (38)

  • Remark 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Corollary 3.3
  • Remark 3.4
  • Proposition 3.5
  • Remark 3.6
  • ...and 28 more