A gap theorem for complete submanifolds with parallel mean curvature in the hyperbolic space

Abstract

Let M be an n(≥7)-dimensional complete submanifold with parallel mean curvature in the hyperbolic space Hn+p, whose mean curvature satisfies H2−1≤0. Denote by A˚ and BR(q) the trace free second fundamental form of M and the geodesic ball of radius R centered at q∈M, respectively. We prove that if lim supR→∞∫BR(q)|A˚|2dMR2=0, and if (∫M|A˚|ndM)2/n+2n(n−2)3n(n−1)H(∫M|A˚|n/2dM)2/n≤C(n), then M is congruent to an n-dimensional hyperbolic space or the Euclidean space Rn. Here C(n) is an explicit positive constant depending only on n. We also obtain a similar gap theorem in the case where n=5,6.