Vanshing Theorems for Quaternionic Kaehler Manifolds

TL;DR

This paper explores the relationships between holonomy, representation theory, and differential operators on quaternionic Kähler manifolds, leading to eigenvalue estimates and vanishing theorems for harmonic forms.

Contribution

It provides simplified proofs of eigenvalue bounds and vanishing theorems for quaternionic Kähler manifolds using representation theory and Weitzenböck formulas.

Findings

Eigenvalue estimates for Dirac and Laplace operators

Vanishing of certain odd Betti numbers in negative scalar curvature cases

Identification of representations contributing to harmonic forms

Abstract

We discuss a peculiar interplay between the representation theory of the holonomy group of a Riemannian manifold, the Weitzenboeck formula for the Hodge-Laplace operator on forms and the Lichnerowicz formula for twisted Dirac operators. For quaternionic Kaehler manifolds this leads to simple proofs of eigenvalue estimates for Dirac and Laplace operators. Moreover it enables us to determine which representations can contribute to harmonic forms. As a corollary we prove the vanishing of certain odd Betti numbers on compact quaternionic Kaehler manifolds of negative scalar curvature. We simplify the proofs of several related results in the positive case.