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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 4-dimensional austere submanifolds were classified by Bryant, into three types which we label A, B, and C. In this paper, we show that type A submanifolds correspond exactly to real Kahler submanifolds, we construct new examples of such submanifolds in R^6 and R^10, and we obtain classification results on submanifolds of types B and C with maximal second fundamental forms.

Austere submanifolds of dimension four: Examples and maximal types