An optimal gap theorem
This paper studies complete non-compact Kähler manifolds with non-negative holomorphic bisectional curvature. The main theorem asserts that such a Kähler manifold \(M\) is flat if the average of scalar curvature \(S\) over the metric ball \(B_{p}(r)\) of radius \(r\) centered at a fixed point \(p\in M\) is a function of \(o(r^{-2})\). This theorem generalizes a earlier result in [the author and \textit{L.-F. Tam}, J. Differ. Geom. 64, No. 3, 457--524 (2003; Zbl 1088.32013)]. The proof involves the Cauchy problem of the Hodge-Laplacian heat equation for closed positive \((1,1)\)-forms. A `positive mass' type corollary is obtained by combining the main result and a monotonicity property, which says that a non-negative non-flat complete Kähler manifolds satisfies \[ \liminf_{r \to \infty} \frac{r^{2}}{\text{Vol}(B_{p}(r))}\int_{B_{p}(r)}Sd\mu >0 \] for any fixed \(p\).
non-negative holomorphic bisectional curvature
scientific article; zbMATH DE number 6084024
scientific article