Publication Date:
2014-11-20
Description:
Let M 7 be a smooth manifold equipped with a G 2 -structure , and Y 3 be a closed compact -associative submanifold. McLean [Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705–747] proved that the moduli space M Y , of the -associative deformations of Y has vanishing virtual dimension. In this paper, we perturb into a G 2 -structure in order to ensure the smoothness of M Y , near Y . If Y is allowed to have a boundary moving in a fixed coassociative submanifold X , it was proved in Gayet and Witt [Deformations of associative submanifolds with boundary, Adv. Math. 226 (2011), 2351–2370] that the moduli space M Y , X of the associative deformations of Y with boundary in X has finite virtual dimension. We show here that a generic perturbation of the boundary condition X into X ' gives the smoothness of M Y , X ' . In another direction, we use Bochner's technique to prove a vanishing theorem that forces M Y or M Y , X to be smooth near Y . For every case, some explicit families of examples will be given.
Print ISSN:
0033-5606
Electronic ISSN:
1464-3847
Topics:
Mathematics
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