Publications [#243400] of Robert Bryant

Papers Published

1. Bryant, RL, Some remarks on the geometry of austere manifolds, Boletim da Sociedade Brasileira de Matemática, vol. 21 no. 2 (September, 1991), pp. 133-157, Springer Nature
   (last updated on 2024/04/18)

   Author's Comments:
   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), 235–242.

   Abstract:
   We prove several structure theorems about the special class of minimal submanifolds which Harvey and Lawson have called "austere" and which arose in connection with their foundational work on calibrations. The condition of austerity is a pontwise condition on the second fundamental form and essentially requires that the non-zero eigenvalues of the second fundamental form in any normal direction at any point occur in oppositely signed pairs. We solve the pointwise problem of describing the set of austere second fundamental forms in dimension at most four and the local problem of describing the austere three-folds in Euclidean space in all dimensions. © 1991 Sociedade Brasileira de Matemática.