Topology and
geometry
A well-known conjecture of Hopf asserts that the Euler number of a compact manifold of non-positive curvature is either zero or its sign is $(-1)^n$. If the sectional curvature is strictly negative then one expects the Euler number to be non-zero and have sign (-1)^n$. Gromov had proved this second conjecture when the manifold in question is homeomorphic to a Kahler manifold of negative curvature. In [19] we extended his result to cover manifolds of non-positive curvature. As in Gromov's proof, we also establish a vanishing theorem for square integrable harmonic forms on the universal cover. The result then follows from the Atiyah index theorem for covers. While Gromov isolates the right concept of Kahler-hyperbolicity, our work singles out the appropriate notion of Kahler-parabolicity (similar results were also obtained by Jost-Zuo). The paper [5] also addresses the question of vanishing of L^2 harmonic forms.
Related Publications
[5] F. Xavier (with H. Donnelly), On the differential form spectrum of negatively curved manifolds, American Journal of Mathematics, 106, 169-187, (1984).
[19] F. Xavier (with J.Cao), Kahler parabolicity and the Euler number of compact manifolds of non-positive curvature, Math. Ann., 319, 483-491 (2001).
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