Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces

We investigate cohomogeneity-one metrics whose principal orbit is an Aloff-Wallach space SU(3)/U(1). In particular, we are interested in metrics whose holonomy is contained in Spin(7). Complete metrics of this kind which are not product metrics have exactly one singular orbit. We prove classification results for metrics on tubular neighborhoods of various singular orbits. Since the equation for the holonomy reduction has only few explicit solutions, we make use of power series techniques. In order to prove the convergence and the smoothness near the singular orbit, we apply methods developed by Eschenburg and Wang. As a by-product of these methods, we find many new examples of Einstein metrics of cohomogeneity one.Comment: 46 page

G2-orbifolds with ADE-singularities

Mannigfaltigkeiten mit Holonomie G2 sind sowohl in der reinen Mathematik als auch der mathematischen Physik ein Gegenstand aktiver Forschung. Das Ziel dieser Arbeit ist es, zusätzlich zu den bekannten glatten Beispielen Orbifaltigkeiten mit einer G2-Struktur und Singularitäten von einem speziellen Typ zu finden. Hierzu modifizieren wir die bekannten Konstruktionsmethoden für G2-Mannigfaltigkeiten. Neben G2-Orbifaltigkeiten mit einer Vielzahl von unterschiedlichen Singularitäten finden wir auch eine glatte G2-Mannigfaltigkeit, deren Betti-Zahlen keinem bisher bekannten Beispiel entsprechen

Special cohomogeneity one metrics with Q^111 or M^110 as principal orbit

We classify all cohomogeneity one manifolds with principal orbit Q^111=SU(2)^3/U(1)^2 or M^110=(SU(3) x SU(2))/(SU(2) x U(1)) whose holonomy is contained in Spin(7). Various metrics with different kinds of singular orbits can be constructed by our methods. It turns out that the holonomy of our metrics is automatically SU(4) and that they are asymptotically conical. Moreover, we investigate the smoothness of the metrics at the singular orbit.Comment: 37 pages, no figure

Spaces admitting homogeneous G2-structures

AbstractWe classify all seven-dimensional manifolds which admit a homogeneous cosymplectic G2-structure. The motivation for this classification is that each of these spaces is a possible principal orbit of a parallel Spin(7)-manifold of cohomogeneity one