## Geometry Seminar

**Date:** February 17, 2020

**Time:** 3:00PM - 4:00PM

**Location:** BLOC 628

**Speaker:** Da Rong Cheng, University of Chicago

**Title:** *Bubble tree convergence of cross product preserving maps*

**Abstract:** We study a class of weakly conformal 3-harmonic maps, called Smith maps, which parametrize associative 3-folds in 7-manifolds equipped with G2-structures. These maps satisfy a first-order system of PDEs generalizing the Cauchy-Riemann equation for J-holomorphic curves, and we are interested in their bubbling phenomena. Specifically, we first prove an epsilon-regularity theorem for Smith maps in W^{1, 3}, and then explain how that combines with conformal invariance to yield bubble trees of Smith maps from sequences of such maps with uniformly bounded 3-energy. When the G2-structure is closed, we show that both 3-energy and homotopy are preserved in the bubble tree limit. The result can be viewed as an associative analogue of the bubble tree convergence theorem for J-holomorphic curves. This is joint work with Spiro Karigiannis and Jesse Madnick.