Rigidity of Willmore submanifolds and extremal submanifolds in the unit sphere

Let M be an n-dimensional (n = 4) compact Willmore (or extremal) submanifold in the unit sphere Sn+p. Denote by Ric and H the Ricci curvature and the mean curvature of M, respectively. It is proved that if (?(M) (Ric(?)-) (n)/(2)) (2)/(n) < A(n, ?, H, ?) (or B(n, ?, H, ?)), then M is a totally umbilical sphere, where A(n, ?, H, ?) and B(n, ?, H, ?) are two explicit positive constants depending on n, ?, H, and ?. This extends parts of the results from a pointwise Ricci curvature lower bound to an integral Ricci curvature lower bound.