Ruled austere submanifolds of dimension four

Abstract We classify 4-dimensional austere submanifolds in Euclidean space ruled by 2-planes. Austere submanifolds in Euclidean space were introduced by Harvey and Lawson in connection with their study of calibrated geometries. The algebraic possibilities for second fundamental forms of austere 4-folds M were classified by Bryant, falling into three types which we label A, B, and C. We show that if M is 2-ruled of Type A, then the ruling map from M into the Grassmannian of 2-planes in R n is holomorphic, and we give a construction for M starting with a holomorphic curve in an appropriate twistor space. If M is 2-ruled of Type B, then M is either a generalized helicoid in R 6 or the product of two classical helicoids in R 3 . If M is 2-ruled of Type C, then M is either one of the above, or a generalized helicoid in R 7 . We also construct examples of 2-ruled austere hypersurfaces in R 5 with degenerate Gauss map.