A pinching problem on submanifolds with parallel mean curvature vector field in a sphere

Abstract

Let M be a closed oriented submanifold with nonzero parallel mean curvature vector field immersed into a unit sphere S. Denote by S the square of the length of the second fundamental form. We consider a pinching problem on S. We give a pinching constant C on S which depends only on n and p. It is better than one given by Xu [12]. When p = 1,2 or n > 8, we show that it is the best possible among this kind of pinching constants. We also characterize those M with S = C.