Gap theorems for submanifolds in [formula omitted

Abstract

In this paper, we obtain some gap theorems for complete immersed submanifolds in Hn×R. For this purpose, we first prove the lower bound estimate of the first eigenvalue of submanifolds in a product space satisfying some curvature conditions. Based on this estimate, we get some Bernstein type theorems for submanifolds in Hn(−1)×R under integral curvature pinching conditions.