Deformations and Moduli of Superminimal Surfaces in the Four-Sphere

Abstract

In this article we derive an expression for the second variation of area of a superminimal immersion f:Σ2 → S4 and the corresponding Jacobi operator. We then show the only minimal deformations of the given immersion are superminimal deformations with the implication that the superminimal immersions forms an open component of the moduli of minimal immersions. Also, one may then address questions of nullity and integrability of Jacobi fields with fairly standard methods from algebraic geometry.