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arXiv · 0802.1283

Deformations of associative submanifolds with boundary

Damien Gayet·Frederik Witt
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Abstract

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 $\int_{\partial Y}c_1(ν_X)+1-g$, where $ν_X$ denotes the orthogonal complement of $T\partial Y$ in $TX_{|\partial Y}$ and $c_1(ν_X)$ the first Chern class of $ν_X$ with respect to its natural complex structure. Further, we exhibit explicit examples of non-trivial index.

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Damien Gayet, Frederik Witt. 2010-11-02. Deformations of associative submanifolds with boundary. https://doi.org/10.1016/j.aim.2010.09.014

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