Deformations of associative submanifolds with boundary

Damien Gayet (1) , Frederik Witt (2)

1 ICJ - Institut Camille Jordan
2 FB Mathematik und Informatik

Let M be a compact smooth manifold of holonomy G_2. We prove that the space of infinitesimal associative deformations of a compact associative submanifold Y with boundary in a coassociative submanifold X is the solution space of an elliptic problem. Further, we compute its virtual dimension. For \partial Y connected it is given by \int_{\partial Y}c_1(\nu_X)+1-g, where g denotes the genus of \partial Y, \nu_X the orthogonal complement of T\partial Y in TX_{|\partial Y} and c_1(\nu_X) the first Chern class of \nu_X with respect to its natural complex structure.

Mots clés: G2 holonomy; calibrated manifolds; elliptic boundary problems on manifolds

Domaines: Géométrie différentielle [math.DG]

Damien Gayet, Frederik Witt. Deformations of associative submanifolds with boundary. Advances in Mathematics, 2011, 226 (3), pp.2351-2370. ⟨hal-00258055v2⟩