Abstract. In an earlier paper, [9], we showed that the moduli space of deformations of a smooth, compact, orientable special Lagrangian submanifold L in a symplectic manifold X with a non-integrable almost complex structure is a smooth manifold of dimension H 1 (L), the space of harmonic 1-forms on L. We proved this first by showing that the linearized operator for the deformation map is surjective and then applying the Banach space implicit function theorem. In this paper, we obtain the same surjectivity result by using a different method, the Fredholm Alternative, which is a powerful tool for compact operators in linear functional analysis.

Abstract. In this paper, we show the existence of (co-oriented) contact structures on certain classes of G 2 -manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure (and so any manifold with G 2 -structure) admits an almost contact structure. We also construct explicit almost contact metric structures on manifolds with G 2 -structures.

No abstract

McLean proved that the moduli space of coassociative deformations of a compact coassociative 4-submanifold C in a G 2 -manifold (M, ϕ, g) is a smooth manifold of dimension equal to b 2 + (C). In this paper, we show that the moduli space of coassociative deformations of a noncompact, asymptotically cylindrical coassociative 4-fold C in an asymptotically cylindrical G 2 -manifold (M, ϕ, g) is also a smooth manifold. Its dimension is the dimension of the positive subspace of the image of H 2 cs (C, R) in H 2 (C, R).

As is well known, when D6 branes wrap a special lagrangian cycle on a non compact CY 3-fold in such a way that the internal string frame metric is Kahler there exists a dual description, which is given in terms of a purely geometrical eleven dimensional background with an internal metric of G 2 holonomy. It is also known that when D6 branes wrap a coassociative cycle of a non compact G 2 manifold in presence of a self-dual two form strength the internal part of the string frame metric is conformal to the G 2 metric and there exists a dual description, which is expressed in terms of a purely geometrical eleven dimensional background with an internal non compact metric of Spin (7) holonomy. In the present work it is shown that any G 2 metric participating in the first of these dualities necessarily participates in one of the second type. Additionally, several explicit Spin(7) holonomy metrics admitting a G 2 holonomy reduction along one isometry are constructed. These metrics can be described as R-fibrations over a 6-dimensional Kahler metric, thus realizing the pattern Spin(7) → G 2 → (Kahler) mentioned above. Several of these examples are further described as fibrations over the Eguchi-Hanson gravitational instanton and, to the best of our knowledge, have not been previously considered in the literature. *