In this article, we define a new class of middle dimensional submanifolds of a Hyperkahler manifold which contains the class of complex Lagrangian submanifolds, and show that this larger class is invariant under the mean curvature flow. Along the flow, the complex phase map satisfies the generalized harmonic map heat equation, tt is also related to the mean curvature vector via a first order differential equation. Moreover, we proved a result on nonexistence of Type I singularity.

In this paper we study the geometry of manifolds with vector cross products and its complexification. First we develop the theory of instantons and branes and study their deformations. For example they are (i) holomorphic curves and Lagrangian submanifolds in symplectic manifolds and (ii) associative submanifolds and coassociative submanifolds in G2-manifolds. Second we classify K¨ahler manifolds with the complex analog of the vector cross product, namely they are Calabi-Yau manifolds and hyperk¨ahler manifolds. Furthermore we study instantons, Neumann branes and Dirichlet branes on these manifolds. For example they are special Lagrangian submanifolds with phase angle zero, complex hypersurfaces and special Lagrangian submanifolds with phase angle pi/2 in Calabi-Yau manifolds.

It is known that the hard Lefschetz action, together with K¨ahler identities for K¨ahler (resp. hyperk¨ahler) manifolds, determines a su(1, 1)_sup (resp. sp(1, 1)_sup) Lie superalgebra action on differential forms. In this paper, we explain the geometric origin of this action, and we also generalize it to manifolds with other holonomy groups. For semi-flat Calabi-Yau (resp. hyperk¨ahler) manifolds, these symmetries can be enlarged to a so(2, 2)_sup (resp. su(2, 2)_sup) action.

We study the moduli space M of torsion-free G2-structures on a fixed compact manifold M7, and define its associated universal intermediate Jacobian J . We define the Yukawa coupling and relate it to a natural pseudo-K¨ahler structure on J . We consider natural Chern–Simons-type functionals, whose critical points give associative and coassociative cycles (calibrated submanifolds coupled with Yang–Mills connections), and also deformed Donaldson–Thomas connections. We show that the moduli spaces of these structures can be isotropically immersed in J by means of G2-analogues of Abel–Jacobi maps.

This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson–Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We expect that the Yukawa coupling characterizes (co-)associative fibrations on these manifolds. We discuss the Fourier transformation along such fibrations and the analog of the Strominger–Yau–Zaslow mirror conjecture for G2-manifolds. We also discuss similar structures and transformations for Spin(7)- manifolds.