On the Lifts of Minimal Lagrangian Submanifolds

Bryant and Salamon constructed metrics with holonomy G2 and Spin(7) on spin bundles of 3-dimensional space forms, and spin bundles and bundles of anti-self-dual 2-forms on self-dual Einstein 4-manifolds [BrS]. Since, apart from holonomy, the construction of integrable G2(respectively Spin(7)) structures amounts to finding differential 3(4)forms of generic type on 7(8) manifolds satisfying appropriate differential equations, the tautological and canonical forms on these bundles serve as the basis of their construction. In this paper, we show the total space of the canonical line bundle L of a KahlerEinstein manifold X supports integrable SU(n+1) structures, or Calabi-Yau structures. The parallel holomorphic volume form in each case is the exterior derivative of the canonical (n, 0) form on L. Since each of integrable SU(n+1), G2, and Spin(7) structures gives rise to a calibration on the underlying manifold, a question arises as to what the calibrated submanifolds are with regard to the above constructions. We show in SU(n + 1) case, the canonical real line bundle L ⊂ L over a minimal Lagrangian submanifold M ⊂ X is calibrated and hence can be considered as the special Lagrangian lift of M . As a corollary, we show the area of compact connected minimal Lagrangian submanifolds in a complex projective space with the standard Kahler structure admits a uniform lower bound. In G2, and Spin(7) cases, minimal surfaces in self-dual Einstein 4-manifolds with