Cayley submanifolds of Calabi-Yau 4-folds

Our main results are: (1) The complex a Lagrangian points of a non-complex Lagrangian $2n$-dimensional submanifold $F:M\ra N$, immersed with parallel mean curvature and with equal Kaehler angles into a Kaehler-Einstein manifold $(N,J,g)$ of complex dimension $2n$, are zeros of finite order of $\sin^2\theta$ and $\cos^2\theta$ respectively, where $\theta$ is the common $J$-Kaelher angle. (2) If $M$ is a Cayley submanifold of a Calabi-Yau (CY) manifold $N$ of complex dimension 4, then $\bigwedge^2_+NM$ is naturally isomorphic to $\bigwedge^2_+TM$. (3) If $N$ is Ricci-flat (not necessarily CY) and $M$ is a Cayley submanifold, then $ p_1(\bigwedge^2_+NM)= p_1(\bigwedge^2_+TM)$ still holds, but $p_1(\bigwedge^2_-NM)- p_1(\bigwedge^2_-TM)$ may describe a residue on the $J$-complex points, in the sense of Harvey and Lawson. We describe this residue by a PDE on a natural morphism $\Phi:TM \to NM$, $\Phi(X)=(JX)^{\bot}$, with singularities at the complex points. We give an explicit formula of this residue in a particular case. When $(N,I,J,K,g)$ is an hyper-Kaehler manifold and $M$ is an $I$-complex closed 4-submanifold, the first Weyl curvature invariant of $M$ may be described as a residue on the $J$-Kaehler angle at the $J$_Lagrangian points by a Lelong-Poincar\'{e} type formula. We study the almost complex structure $\Jw$ on $M$ induced by $F$.