On Yau Rigidity Theorem for Submanifolds in Pinched Manifolds

In this paper, we investigate Yau's rigidity problem for compact submanifolds with parallel mean curvature in pinched Riemannian manifolds. Firstly, we prove that if M-n is an oriented closed minimal submanifold in an (n + p)-dimensional complete simply connected Riemannian manifold Nn(+p), then there exists a constant delta(0)(n, p) is an element of (0, 1) such that if the sectional curvature of N satisfies (K) over bar (N) is an element of[delta(0)(n, p), 1], and if M has a lower bound for the sectional curvature and an upper bound for the normalized scalar curvature, then N is isometric to Sn+p. Moreover, M is either a totally geodesic sphere, one of the Clifford minimal hypersurfaces S-k(root k/n) x S-n (k)(root n-k/n) in Sn+1 for k = 1,..., n-1, or the Veronese submanifold in Sn+d, where d = 1/2n(n + 1) - 1. We then generalize the above theorem to the case where M ia a compact submanifold with parallel mean curvature in a pinched Riemannian manifold.