Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space. (January 2017) - Lin, Hezi
Abstract: Let M n be a complete non-compact submanifold in the hyperbolic space H n + m . We first give an estimate for the bottom of the spectral of the Laplace operator on M n, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L 2 harmonic forms in certain degrees if the total mean curvature of M n is less than an explicit constant and its total curvature is less than a suitable related constant. In addition, we obtain some vanishing results under certain pointwise restrictions on the traceless second fundamental form. Moreover, according to the nonexistence of nontrivial L 2 harmonic 1 -forms, we can further prove some one-end theorems.