Deformations of associative submanifolds with boundary

Details Deformations of associative submanifolds with boundary Deformations of associative submanifolds with boundary

2008-02-09

arxiv.org/abs/0802.1283v4

Adv. Math. 226 (2011), 2351-2370

Mathematics

Differential Geometry

19 pages

Scientific paper

10.1016/j.aim.2010.09.014

Let $M$ be a topological $G_2$-manifold. 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. For a connected boundary $\partial Y$ of genus $g$, the index is given by $\int_{\partial Y}c_1(\nu_X)+1-g$, where $\nu_X$ denotes 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. Further, we exhibit explicit examples of non-trivial index.

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