Deformations of calibrated subbundles of Euclidean spaces via twisting by special sections

Abstract

We extend the "bundle constructions" of calibrated submanifolds, due to Harvey-Lawson in the special Lagrangian case, and to Ionel-Karigiannis-Min-Oo in the cases of exceptional calibrations, by "twisting" the bundles by a special (harmonic, holomorphic, or parallel) section of a complementary bundle. The existence of such deformations shows that the moduli space of calibrated deformations of these "calibrated subbundles" includes deformations which destroy the linear structure of the fibre.

A Classification of Twisted Austere 3-Folds

“…We say that (M, µ) is a twisted-austere pair if L = N * M +µ is a special Lagrangian submanifold inside T * R n with respect to some phase. Following [11], we refer to this as the Borisenko construction.…”

“…It is shown in [11] that L is Lagrangian if and only if ∇µ is a symmetric tensor on M , that is dµ = 0. The conditions under which L is special Lagrangian are more involved.…”

“…(For example, B ij = B(e i , e j ), and the Lagrangian condition is equivalent to B being a symmetric matrix.) Theorem 2.2 (Karigiannis-Leung [11]). Fix a phase angle θ ∈ [0, 2π).…”

“…A generalization of the conormal bundle construction was introduced by Borisenko [1] and later significantly extended by Karigiannis-Leung [11]. The idea is as follows.…”

“…This is the total space of an affine bundle over M whose fibres are affine translates of the conormal spaces, translated by the 1-form µ. In [11] it was proved that N * M + µ is special Lagrangian if and only if the second fundamental form A of M in R n and the 1-form µ satisfy a system of coupled second order fully nonlinear PDE, which we call the twisted austere equations. This result is stated explicitly in Theorem 2.2.…”