ISBN 978-3-86853-794-9

978-3-86853-794-9, Reihe Mathematik

Sebastian Stock
Evolution of Geometries with Torsion

150 Seiten, Dissertation Universität Köln (2011), Softcover, A5

Zusammenfassung / Abstract

We reduce the embedding problem for hypo SU(2) and SU(3)-structures to the embedding problem for hypo G2-structures into parallel Spin(7)-manifolds. The latter will be described in terms of gauge deformations. This description involves the intrinsic torsion of the initial G2-structure and allows us to prove that the evolution equations, for all of the above embedding problems, do not admit non-trivial longtime solutions. For G2-structures we introduce a new flow, which generalizes Hitchin’s flow equations. This intrinsic torsion flow admits unique solutions in the real analytic category.

We extend the Kähler-Ricci flow to SU(n)-structures and characterize under which conditions this flow converges to a parallel SU(n)-structure. This approach also yields an extension of the Ricci flow to G2 and Spin7-structures. For SU(3)- structures in dimension seven we derive the analogue of the Gray-Hervella classification. Based on this classification, we define a type of G2-structure which can be regarded as the seven dimensional analogue of Kähler SU(3)-structures. This type of G2-structures allow a fibrewise Ricci flow that converges to a Ricci flat G2 -structure.