|DATE||November 16 (Mon), 2015|
|TITLE||[Differential and Complex Geometry Seminar] Non-associative algebras of minimal cubic cones I|
Quadratic minimal cones played the crucial role in both the solution of the famous Bernstein problem and later on in the construction of the higher dimensional counterexamples by Bombieri, de Giorgi and Giusti. Despite the fact that the minimal cones of degree 3 and higher have been the subject of considerable recent interest, there is still very little known about their structure and their classification is a long-standing open problem. In my talk, I discuss a recent progress in classification of the so-called Hsiang cubic minimal cones by using a non-associative algebra approach. The method is outlined in [1,2] and goes back to the Freudenthal-Springer construction of exceptional simple algebras. We associate to any homogeneous cubic polynomial solution of the minimal surface equation a commutative nonassociative algebra, called Hsiang algebra. The key result of our classification is the existence of a formally real rank three Jordan algebra J inside any Hsiang algebra V. We show that the structural properties of V and the geometry of the corresponding minimal cone are completely determined by J. With this Jordan algebra structure in hand, we are able to show that any Hisang algebra belongs to either of two distinguished classes: a) an infinite and well understood family admitting a natural Z^2-grading compatible with a certain Clifford algebra structure, or b) a finite family of the so-called exceptional Hsiang algebras. The classification of the latter class is much more delicate task and require the study of a finer, the so called tetrad structure of a Hsiang algebra. In , a partial solution of another problem posed in 1980 by John Lewis was obtained by using similar methods.