Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits

Abstract

In this paper, we study heat kernel estimates along the Kähler-Ricci flow with finite time noncollapsing singularities over a compact Kähler manifold. The starting point is the type-II scalar-curvature estimate, which gives the natural parabolic scale $r\lesssim T-t$ near the singular time. By combining the distance distortion and space-time cut-off estimates of Bamler--Zhang with entropy, mean-value inequality, and conjugate heat-kernel estimates, we obtain two-sided Gaussian bounds and gradient bounds that are uniform on this scale. Under the global type-I scalar-curvature estimate, the same estimates hold on every scale. We then use these estimates and the monotonicity of the $W_1$-distance between conjugate heat flows to construct a canonical singular-time pseudo-distance. The distances $d_t$ (induced by $ω(t)$) converge uniformly to this pseudo-distance. Its metric quotient is the unique Gromov--Hausdorff limit. Finally, we apply Bamler's theory and the recent work of Hallgren--Zhang to obtain the following. Every fixed-base-point type-I blow-up admits a subsequential metric-flow limit whose time slices are singular shrinking Kähler--Ricci soliton spaces with normal complex-analytic and log-terminal structure, unique tangent cones, and a polarized Fano fibration. For moving base points, the limiting metric flow, equipped with its limiting conjugate heat flow, is a metric soliton precisely under an entropy-pinching condition.