Asymptotically Conical Associative 3-folds

TL;DR

This paper studies the deformation space of asymptotically conical associative 3-folds in R^7, revealing conditions under which they are isolated or have a positive-dimensional moduli space based on their decay rates.

Contribution

It provides a local description of the moduli space of such 3-folds and computes its virtual dimension, depending on the decay rate, advancing understanding of their deformation theory.

Findings

Moduli space is locally homeomorphic to the kernel of a smooth map.

Virtual dimension is non-negative for rates > -1.

Associative 3-folds are expected to be isolated for rates ≤ -1.

Abstract

Given an associative 3-fold in R^7 which is asymptotically conical with generic rate less than 1, we show that its moduli space of deformations is locally homeomorphic to the kernel of a smooth map between smooth manifolds. Moreover, the virtual dimension of the moduli space is computed and shown to be non-negative for rates greater than -1, whereas the associative 3-fold is expected to be isolated for rates less than or equal to -1.