Compact Riemannian manifolds with exceptional holonomy (Q2771290)

The present article is a comprehensive survey on compact manifolds with exceptional holonomy, and is particularly concerned with their construction. The possible holonomy groups of a Riemannian metric were already classified in 1955 by Berger, the exceptional cases being the groups \(G_2\) and Spin(7), but it was not until 1994-5 that compact 7- and 8-dimensional manifolds with exceptional holonomy were constructed for the first time by the author, generalizing the Kummer construction of metrics with holonomy SU(2) on the \(K3\) surface.NEWLINENEWLINENEWLINEIf \(T^7\) denotes the 7-dimensional torus and \(\Gamma\) is a finite group of isometries preserving a given flat \(G_2\)-structure on \(T^7\), the quotient \(T^7/ \Gamma\) is a singular compact manifold. Let \(M\) denote its resolution. In order to construct a metric with holonomy \(G_2\) on \(M\), one writes down a 1-parameter family of \(G_2\)-structures on \(M\) with small torsion first, and then shows that, for sufficiently small parameter values, the elements of this family can be deformed to a \(G_2\)-structure with zero torsion. NEWLINENEWLINENEWLINEA similar construction is employed to obtain 8-dimensional Spin(7)-manifolds, and, by considering different groups \(\Gamma\), one obtains many topologically distinct 7- and 8-dimensional manifolds with exceptional holonomy. The crucial point in the construction consists in rewriting the deformation problem as a nonlinear elliptic partial differential equation, which is solved by iteration and inductive estimates, using elliptic regularity and Sobolev embedding techniques. The paper contains an introduction to holonomy groups of Riemannian manifolds and explains the main issues involved in the construction of compact \(G_2\)- and Spin(7)-manifolds. A full exposition of the subject can be found in the meanwhile published book of the same author on compact manifolds with special holonomy. Since the methods employed in the book are, however, of much more generality, the uninitiated reader might find it helpful to have a look at the survey paper first, in order to get an overview of the main ideas.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00021].