$L^1$-Integrability of $L^2$-Harmonic Forms and the Hopf Conjecture

Abstract

In this note, we study $L^2$-harmonic forms on complete simply-connected Riemannian manifolds with non-positive sectional curvature. We first establish an a priori $L^{\infty }$-estimate for such forms via Moser iteration, under the curvature bounds $-K\le {\sec }_g\le 0$. We then prove that any $L^2$-harmonic form which is also $L^1$-integrable must vanish identically. Consequently, on the universal cover of a closed non-positively curved manifold, the $k$-th $L^2$-Betti number vanishes if and only if every $L^2$-harmonic $k$-form is $L^1$-integrable. This criterion reformulates a topological vanishing statement as an analytic integrability condition.

16 pages. All comments are welcome!