A note on \(L^ 2\) harmonic forms on a complete manifold
Motivated by a result of \textit{K. D. Elworthy} and \textit{St. Rosenberg} [Tokyo J. Math. 16, No. 2, 513-524 (1993; Zbl 0799.53046)], the author derives a nonexistence result for harmonic forms with values in a Riemann-connected vector bundle \(E\) over a complete Riemannian manifold \(M\) that possesses a pole. One main result states that there are no \(L^2\)-harmonic \(E\)-valued \(p\)-forms on \(M\) for \(p < \dim M/2\) provided that the radial curvature \(K_r\) satisfies \[ - \left( {m - p - 1\over p} \right)^2 \leq K_r \leq -1. \]
scientific article; zbMATH DE number 737444