Cohn-Vossen inequality on certain noncompact Kähler manifolds (Q2172502)

Let \((M^n,\omega)\) be an \(n\)-dimensional complete noncompact Kähler manifold with nonnegative Ricci curvature and Euclidean volume growth and \(B(r)\) be a geodesic ball of radius \(r\). Furthermore, \(M^n\) has either a nonnegative bisectional curvature or the asymptotic volume ratio is a small number relying on the dimension. Then the author gives a positive answer on the following question raised by \textit{B. Yang} [Math. Ann. 355, No. 2, 765--781 (2013; Zbl 1264.53043)]: for all \(k\), is it true that \(r^{-2n+2k}\int_{B(r)}\mathrm{Ric}^k\wedge\omega^{n-k}\) is bounded as \(r\) goes to infinity?