The author first classifies the defining functions of 3-dimensional real analytic submanifolds in , up to third order terms. Excluding the trivial totally real case, there are 13 types of quadratic terms. The author then finds normal forms under a formal change of coordinates for two types of quadratic terms.
More precisely, if is a graph over the -subspace of the form , and , then the normal form of under formal holomorphic transformation is defined by . If is defined by and , then the normal form of under a formal transformation is defined by , or by with .
The author conjectures that the normal form of can be achieved by a convergent transformation.