Longtime existence of Kähler-Ricci flow and holomorphic sectional curvature (Q6159280)

The authors provide an intricate analysis of the Kähler-Ricci flow on noncompact Kähler manifolds. They build upon previous work, focusing on the long-time behavior of the flow, especially concerning its convergence to Kähler-Einstein metrics with negative scalar curvature. Their main contributions, Theorems 1.1 and 1.2, elucidate conditions under which a Kähler manifold, equipped with specific metrics having particular curvature and torsion properties, allows for a long-time solution to the flow. Notably, Theorem 1.1 establishes that under a constraint on the holomorphic sectional curvature \(H_h\) and the torsion \(\hat{T}\), the normalized Kähler-Ricci flow converges to the unique Kähler-Einstein metric. Theorem 1.2 then furnishes conditions on the initial metric, its associated Hermitian structure, and its curvature bounds to ensure the existence of a complete solution to the flow. When contextualized with prior results, such as those from [\textit{D. Wu} and \textit{S.-T. Yau}, J. Am. Math. Soc. 33, No. 1, 103--133 (2020; Zbl 1509.32007)], this paper presents a significant refinement and generalization in understanding the long-time existence of Kähler-Ricci flows on noncompact manifolds.