The Stability Of Willmore Submanifold

Let x0:M→n is a conformal submanifold in the sphere space Sn.We give the relationship between conformal invariants and isometric ones for submanifold in the sphere space Sn and calculate again the first and second variation formulae of Willmore functional of x0.We also calculate the Willmore stability of the Clifford torus and the Veronese surface in Sn.This text is organized as follows.In Section 1,we introduce the problem of back-ground and research content.In Section 2,we recall the submanifold theory in the sphere space Sn and give the relationship between conformal invariants and isometric ones in Lorentz space forms.In Section 3 and 4,we present the first and second varia-tion formulae in the sphere space Sn.In Section 5,we give the proof of the stability of the Veronese surface.In the end of this text,we present related problems and prospects.