C ~ 4 Special Lagrangian Submanifolds Constructed

Special Lagrangian submanifolds of complex Euclidean space Cn have been studied widely and deeply over the last few years.These sub-manifolds are volume minimizing,in particular,they are minimal sub-manifolds.Special Lagrangian n-folds in Cn should provide local models for how singularities develop in Special Lagrangian n-folds in Calabi-Yau n-folds.This is the most important motivation for this paper.When ri=2.Special Lagrangian surface of C2 are exactly complex surface with respect to another orthogonal complex structure on R4 = C2. When n=3,D.D.Joyce has constructed and studied deeply a lot of families of Special Lagrangian 3-folds in C3 .The second part of this paper is the preliminary knowledge for the later construction of Special Lagrangian 4-folds in C4 .The mainly part of this paper is the third part.In this part,we construct and study several families of Special Lagrangian 4-folds in C4. using the evolution equation construction method.The construction requires a set of evolution data (P. \).including a 3-submanifold P in /".Then Special Lagrangian 4-folds X in C4 is the subset of C4 swept out by the image of P under a 1-parameter family of linear or affine maps Φt : R4 -?C4.which satisfy a first-order nonlinear o.d.e. in t.The very equation is(-}" = (0().(\)ftl-Qm-I(n)a1...am_lftfflffa-6At last, we show a new method of construction of Special Lagrangian n-folds of Cn using an (n-l)-dimemsional oriented minimal Legendrian submanifold of S'2n~l and certain plane curves [4],making use of this method,we include a new example of Special Lagrangian 4-folds in C4.