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Bounded and $L^2$ Harmonic Forms on Universal Covers

We relate the positivity of the curvature term in the Weitzenbock formula for the Laplacian on p-forms on a complete manifold to the existence of bounded and $L^2$ harmonic forms. In the case where the manifold is the universal cover of a compact manifold, we obtain topological and geometric information about the compact manifold. For example, we show that a compact manifold cannot admit one metric with pinched negative curvature and another metric with positive Weitzenbock term on two-forms. Many of these results can be thought of as differential form analogues of Myers' theorem. We also give pinching conditions on certain sums of sectional curvatures which imply the positivity of the curvature term, and hence yield vanishing theorems. In particular, we construct a compact manifold with planes of negative sectional curvature at each point and which satisfies the hypothesis of our vanishing theorems.