Generalization of Lawson and Simonsresult to quaternion and octonion geometry

A theorem of Lawson and Simons[8] states that the only stable minimal submanifolds in CP are complex submanifolds. We generalize their result to the cases of HP and OP. The treatment is given in the context of Jordan algebra, so that the seemingly unrelated case of S is uni…ed naturally with the above projective spaces. 1 Introduction Complex geometry is a very rich subject. Some of its beautiful theorems have natural generalizations to quaternion geometry or even octonion geometry. This paper gives one such generalization. In the seventies, Lawson and Simons [8] showed that the average second variation of any submanifold S in CP is negative unless S is complex, where the average is taken over all holomorphic vector …elds in CP. As a corollary, complex submanifolds are the only stable minimal submanifolds in CP. (Here stability means that the submanifold has non-negative second variation along every vector …eld.) We generalize this result to HP and OP, leading to the following theorem: Main Theorem: Let A = R;C;H;O. For any submanifold S (or more generally recti…able current) in AP, the average second variation of S is given by Z