Proving local volume noncollapsing for the Kähler-Ricci flow, indicating broader implications for Kähler manifolds.
We prove a local volume noncollapsing property for the long-time Kähler-Ricci flow on a minimal Kähler manifold, improving previous results of Guo-Phong-Song-Sturm, Guedj-Tô and Vu. Similar result is also proved for a twisted Kähler-Ricci flow on non-minimal Kähler manifolds. In the course of proof, we develop new Green's function estimates for a general Kähler family under the Lp>1 and L¹(log L)q>n volume density conditions.
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Zhang et al. (2025) studied this question.
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