Abstract
In this paper, we first shows a lower bound estimate for the first eigenvalue of the p-Laplacian on a complete submanifold in a simply connected manifold of negative sectional curvature, in terms of Ln-norm of the mean curvature. In the second part, we prove some vanishing theorems of Lqp-harmonic ℓ-forms on complete noncompact submanifolds in Hadamard manifolds, by assuming the index of certain Schrödinger operators are zero, or by assuming integral or pointwise pinching conditions on the mean curvature and the traceless second fundamental form.
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