Résumé : We will describe the construction of complete non-compact Ricci-flat manifolds of dimension 7 and 8 with holonomy $G_{2}$ and Spin(7) respectively. The examples we consider all have non-maximal volume growth and an asymptotic geometry, so-called ALC geometry, that generalises to higher dimension the asymptotic geometry of 4-dimensional ALF hyperkähler metrics. The interest in these metrics is motivated by the study of codimension 1 collapse of compact manifolds with exceptional holonomy. The constructions we will describe are based on the study of adiabatic limits of ALC metrics on principal Seifert circle fibrations over asymptotically conical orbifolds, cohomogeneity one techniques and the desingularisation of ALC spaces with isolated conical singularities. The talk is partially based on joint work with Mark Haskins and Johannes Nordstrm.

Keywords : $G_{2}$ manifold; Einstein metric; spin (7) manifold