Bubble Tree Convergence of Conformally Cross Product Preserving Maps

Abstract

We study a class of weakly conformal $3$-harmonic maps, called associative Smith maps, from $3$-manifolds into $7$-manifolds that parametrize associative $3$-folds in Riemannian $7$-manifolds equipped with $\mathrm{G}_2$-structures. Associative Smith maps are solutions of a conformally invariant nonlinear first order PDE system, called the Smith equation, that may be viewed as a $\mathrm{G}_2$-analogue of the Cauchy-Riemann system for $J$-holomorphic curves. In this paper, we show that associative Smith maps enjoy many of the same analytic properties as $J$-holomorphic curves in symplectic geometry. In particular, we prove: (i) an interior regularity theorem, (ii) a removable singularity result, (iii) an energy gap result, and (iv) a mean-value inequality. While our approach is informed by the holomorphic curve case, a number of nontrivial extensions are involved, primarily due to the degeneracy of the Smith equation. At the heart of above results is an $\varepsilon$-regularity theorem that gives quantitative $C^{1,\beta}$-regularity of $W^{1,3}$ associative Smith maps under a smallness assumption on the $3$-energy. The proof combines previous work on weakly $3$-harmonic maps and the observation that the associative Smith equation demonstrates a certain "compensation phenomenon" that shows up in many other geometric PDEs. Combining these analytical properties and the conformal invariance of the Smith equation, we explain how sequences of associative Smith maps with bounded $3$-energy may be conformally rescaled to yield bubble trees of such maps. When the $\mathrm{G}_2$-structure is closed, we prove that both the $3$-energy and the homotopy are preserved in the bubble tree limit. This result may be regarded as an associative analogue of Gromov's Compactness Theorem in symplectic geometry.