Twisted-Austere Submanifolds in Euclidean Space

Abstract

A twisted-austere 𝑘-fold (𝑀, 𝜇) in ℝⁿ consists of a 𝑘-dimensional submanifold 𝑀 of ℝⁿ 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+𝜇 is a special Lagrangian submanifold of ℂⁿ. 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 ℝ⁵ discovered by Bryant, there are no other possibilities for the base 𝑀. This gives a complete classification of twisted-austere 3-folds in Rⁿ.