An austere submanifold of Euclidean space is one such that each of the quadratic forms in the second fundamental form has its eigenvalues occuring in oppositely signed pairs. In particular, an austere submanifold is minimal, but, except in the case of surfaces, austerity is much more restrictive than minimality. The term austere was coined by Harvey and Lawson in their fundamental paper Calibrated Geometries and characterises those submanifolds whose conormal bundle is special Lagrangian, and hence absolutely minimizing.
The largest known class of examples of austere submanifolds are the complex submanifolds of complex n-space regarded as real submanifolds of Euclidean 2n-space.
In the first part of this paper, I classify the possible second fundamental forms of 3- and 4-dimensional austere submanifolds of Euclidean space and in the remaining parts of the paper, I determine the generality of the 3-dimensional austere submanifolds corresponding to each possible type of second fundamental form.
The classification of the possible austere second fundamental forms in higher dimensions is still unknown and it is also unknown whether or not there exist austere 4-manifolds corresponding to each of the possible algebraic types of austere second fundamental forms found in the first part of the paper. For further progress in the analysis of some examples of austere submanifolds, consult the work of Dajczer and Gromoll, The Weierstrass representation for complete minimal real Kähler submanifolds of codimension two, Inventiones Mathematicae 119 (1995), 235242.