The First Variation Of The M?bius Scalar Curvature Functional Of Submanifolds In S~n

Let x:M→Sn be a m-dimensional submanifold in Sn without umbilical points, the Mobius metric g, the Mobius second fundamental form B, the Mobius form Φ and the Blaschke tensor A are the four fundamental Mobius invariants of x. The normalized scalar curvature with respect to Mobius metric g is called the Mobius normalized scalar curvature R.In this paper, we define R(M) as the functional of Mobius normalized scalar curvature R, which is called Mobius scalar curvature functional. Using the relations between Mobius invariants and the divergence theorem, we can calculate the first variation of the Mobius scalar curvature functional and gain the corresponding Euler-Lagrange equation. At last, we discuss the critical submanifolds to our studied functional by calculating some classical examples.