Flat Minimal Tori With Cyclic Harmonic Sequence In CP~n

The flat minimal torus with cyclic harmonic sequence in the complex projective space CPn is studied. It is proved that if the associated harmonic sequence of a minimal immersionψfrom a flat torus T2 into the complex projective space CPn is cyclic, then there is a finite covering p:T2→T2 of the flat torus T2 and a totally isotropic minimal immersionφ:T2→CPn such thatψ(?)p=A(?)φ, where A:CPn→CPn is a linearly holomorphic homeomorphism.The thesis is divided into three sections. In section 1, as an introduction, the historical background of the problem and the principal method used in this thesis are stated, and the main result as well. In section 2, as the preliminaries for further use, we proved for (?)p∈T2, there is local complex coordinate z and a local unitary frame e0,e1,…,en of harmonic bundle sequence L0,L1,…,Ln ofψsuch thatλj are positive real function in equation (2.3) and∑j=0nρj=0 (lemma 2.4). In section 3, the proof of the main theorem(theorem 3.2) is given.