Geometry of SU(3) Manifolds

Abstract

I study differential geometry of 6-manifolds endowed with various $SU(3)$ structures from three perspectives. The first is special Lagrangian geometry; The second is pseudo-Hermitian-Yang-Mills connections or more generally, $\omega$-anti-self dual instantons; The third is pseudo-holomorphic curves.For the first perspective, I am interested in the interplay between $SU(3)$ structures and their special Lagrangian submanifolds. More precisely, I study $SU(3)$-structures which locally support as `nice' special Lagrangian geometry as Calabi-Yau 3-folds do. Roughly speaking, this means that there should be a local special Lagrangian submanifold tangent to any special Lagrangian 3-plane. I call these $SU(3)$-structures {\it admissible}. By employing Cartan-K\"ahler machinery, I show that locally such admissible $SU(3)$-structures are abundant and much more general than local Calabi-Yau structures. However, the moduli space of the compact special Lagrangian submanifolds is not so well-behaved in an admissible $SU(3)$-manifold as in the Calabi-Yau case. For this reason, I narrow attention to {\it nearly Calabi-Yau} manifolds, for which the special Lagrangian moduli space is smooth. I compute the local generality of nearly Calabi-Yau structures and find that they are still much more general than Calabi-Yau structures. I also discuss the relationship between nearly Calabi-Yau and half flat $SU(3)$-structures. To construct complete or compact admissible examples, I study the twistor spaces of Riemannian 4-manifolds. It turns out that twistor spaces over self-dual Einstein 4-manifolds provide admissible and nearly Calabi-Yau manifolds. I also construct some explicit special Lagrangian examples in nearly K\"ahler $\mathbf{CP}^3$ and the twistor space of $H^4$. For the second perspective, we are mainly interested in pseudo-Hermitian-Yang-Mills connections on nearly K\"ahler six manifolds. Pseudo-Hermitian-Yang-Mills connections were introduced by R. Bryant in \cite{BryantAlmCplx} to generalize Hermitian-Yang-Mills concept in K\"ahler geometry to almost complex geometry. If the $SU(3)$ structure is nearly K\"ahler, I show that pseudo-Hermitian-Yang-Mills connections (or, more generally, $\omega$-anti-self-dual instantons) enjoy many nice properties. For example, they satisfies the Yang-Mills equation and thus removable singularity results hold for such connections. Moreover, they are critical points of a Chern-Simons functional. I derive a Weitzenb\"ock formula for the deformation and discuss some of its application. I construct some explicit examples which display interesting singularities. For the third perspective, I study pseudo-holomorphic curves in nearly K\"ahler $\mathbf{CP}^3$. I construct a one-to-one correspondence between {\it null torsion} curves in the nearly K\"ahler $\mathbf{CP}^3$ and contact curves in the K\"ahler $\mathbb{CP}^3$ (considered as a complex contact manifold). From this, I derive a Weierstrass formula for all {\it null torsion} curves by employing a result of R. Bryant in \cite{BryantS^4}. In this way, I classify all pseudo-holomorphic curves of genus~$0$.