The Hopf conjecture states that the Euler characteristic of a compact Riemannian $2n$-manifold $\overline{M}$ of negative sectional curvature satisfies $(-1)^{n}\chi(\overline{M})>0[6]$ . Applying the Chern-Gauss-Bonnet theorem gives the conjecture for $n=1,2$ , for spaces of constant curvature, and for spaces of sufficiently pinched curvature [5]. Singer’s idea of instead using the $L^{2}$ index theorem to establish the Hopf conjecture has been successfully carried out for K\ahler manifolds by Gromov [18] (cf. [11]). It is worth noting that the first examples of negatively curved manifolds not admitting metrics of constant negative curvature are rather recent [20], [19]. Singer’s method depends on the vanishing of $L^{2}$ harmonic forms (except in the middle dimension) on the universal cover of a compact negatively curved manifold, as explained in \S 4. This raised the question of such vanishing for arbitrary simply connected negatively curved manifolds. Anderson’s paper [1] shows that such vanishing results are not possible without a pinching condition; however, his examples admit no compact quotient, so Singer’s approach is not ruled out. One of our main results (Corollary 4.4) is that for one-forms vanishing occurs except in the pinching region ruled out by Anderson’s examples. In general, we obtain vanishing results and hence $(-1)^{n}\chi(\overline{M})\geq 0$ (Theorem 4.5) for manifolds of pinched negative curvature, where the pinching constant is more relaxed than in previous work, e.g. [5]. The vanishing theorems depend upon Witten’s deformation $\coprod_{\tau}$ of the Laplacian $\cdot$ on forms on $M[21]$ . In contrast to Witten’s work, in which the Morse inequalities are recovered by letting the deformation parameter $\tau$ go to infinity, the vanishing theorems arise through the study of small deformations. Moreover, instead of deforming the Laplacian by a Morse function as in [21], we use the distance function to a point. The