Second-Order Families of Minimal Lagrangians in CP3

Abstract

In this thesis we analyze families of minimal Lagrangian submanifolds of complex projective 3-space CP3 whose fundamental cubic forms satisfy geometrically natural conditions at every point, namely that their fundamental cubic form be preserved by a proper subgroup of SO(3). There is a classification of SO(3)-stabilizer types of such fundamental cubics, which shows there are precisely five families of such cubic forms: Those with stabilizers contained in SO(2), A4, S3, Z2, and Z3. We use the method of moving frames, along with exterior differential systems techniques to prove existence of minimal Lagrangian submanifolds with each stabilizer type. We also attempt to integrate the resulting structure equations to give explicit examples of each.