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EIGENVALUE ESTIMATES FOR DIRAC OPERATORS WITH PARALLEL CHARACTERISTIC TORSION
 

Summary: EIGENVALUE ESTIMATES FOR DIRAC OPERATORS WITH
PARALLEL CHARACTERISTIC TORSION
ILKA AGRICOLA, THOMAS FRIEDRICH, AND MARIO KASSUBA
Abstract. Assume that the compact Riemannian spin manifold (Mn
, g) admits a G-
structure with characteristic connection and parallel characteristic torsion (T = 0),
and consider the Dirac operator D1/3
corresponding to the torsion T/3. This operator
plays an eminent role in the investigation of such manifolds and includes as special
cases Kostant's "cubic Dirac operator" and the Dolbeault operator. In this article,
we describe a general method of computation for lower bounds of the eigenvalues of
D1/3
by a clever deformation of the spinorial connection. In order to get explicit
bounds, each geometric structure needs to be investigated separately; we do this in
full generality in dimension 4 and for Sasaki manifolds in dimension 5.
1. Introduction
Lower bounds for the first eigenvalue of the Riemannian Dirac operator Dg on a compact
Riemannian spin manifold depending on the scalar curvature are well known since more
than two decades (see [15], [24]). In past years, another operator of Dirac type turned
out to play a crucial role in the investigation of non-integrable geometric structures as

  

Source: Agricola, Ilka - Institut für Mathematik, Humboldt-Universität zu Berlin

 

Collections: Mathematics