Weakly Lagrangian Minimal S~4 With Constant Curvature In CP~n

In the present paper the weakly Lagrangian minimal immersionφ:S4→CPn from the Euclidean sphere S4 with constant curvature into the complex projective space CPn is studied. It is proved that if the induced metric has constant sectional curvature c, than there is an integer s≥1 such that c=4/[s(s+3)].The immersionφis uniquely determined by two homogeneous polynomials fs and fs-1 of four variables. When s=1, that is, c=1 or s=2, that is, c=2/5 the immersionφis absolute real.The thesis is divided into four 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. Firstly, in section 2 the spectrum 0=λ0<λ1<…<λs<…of the LaplacianΔon the unit sphere Sn is given. And the dimension of the eigenspace Vλs corresponding to eigenvalueλs is computed. In section 3 weakly Lagrangian minimal immersions from S4 with constant curvature into CPn are studied, and the proof of the main theorem is given in this part. In section 4 some examples of weakly Lagrangian minimal S4 with constant curvature in the complex projective space CPn are provided.