An inequality between Willmore functional and Weyl functional for submanifolds in space forms

Let φ : M → Rn+p(c) be an n-dimensional submanifold in an (n + p)dimensional space form Rn+p(c) with the induced metric g. Willmore functional of φ is W (φ) = ∫M (S − nH2)n/2dv, where S = ∑ α,i, j (h α i j ) 2 is the square of the length of the second fundamental form, H is the mean curvature of M . The Weyl functional of (M, g) is ν(g) = ∫M |Wg|n/2dv, where |Wg|2 = ∑ i, j,k,l W 2 i jkl and Wi jkl are the components of the Weyl curvature tensor Wg of (M, g). In this paper, we discover an inequality relation between Willmore functional W (φ) and Weyl funtional ν(g).