A Classification of Twisted Austere 3-Folds

Abstract

A twisted-austere k-fold (M, µ) in R n consists of a k-dimensional submanifold M of R n together with a closed 1-form µ on M , such that the second fundamental form A of M and the 1-form µ satisfy a particular system of coupled nonlinear second order PDE. Given such an object, the "twisted conormal bundle" N * M + dµ is a special Lagrangian submanifold of C n. We review the twisted-austere condition and give an explicit example. Then we focus on twisted-austere 3-folds. We give a geometric description of all solutions when the "base" M is a cylinder, and when M is austere. Finally, we prove that, other than the case of a generalized helicoid in R 5 discovered by Bryant, there are no other possibilities for the base M. This gives a complete classification of twisted-austere 3-folds in R n .