Hopf Differential And A Direct Proof Of Non-existence Of λ₂-eigenmap S²�'S²

This paper has two parts,the first part discusses Hopf Conjecture of an immersed surface in R³.It studies the origins,processes and functions of Hopf differential.It gives a proof that Hλ² is analystic by using the moving frame method,where Hλ² is the coefficient of dz² inφ,φis Hopf differential.The second part discussesλ_p-eigenmap between sphere surfaces.It also discusses theλ₂ eigenmap f: S²â†’S².We prove that such eigenmap doesn't exist.Our proof is direct and new.Our method can be generalized to the ease f:S³â†’Sⁿ and S⁴→Sⁿ.