Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces

Details Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces

2010-04-27

arxiv.org/abs/1004.4788v1

Proc. Lond. Math. Soc. 102, No. 6, 1127-1160 (2011)

Mathematics

Differential Geometry

46 pages

Scientific paper

10.1112/plms/pdq048

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.

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