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≲ 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 W1-distance between conjugate heat flows to construct a canonical singular-time pseudo-distance. The distances dt (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.
Related
- Gromov-Hausdorff instability of the positive mass theorem for Kähler manifolds
- Finite-Shift Inverse-Concavity and Exterior Noncollapsing for Fully Nonlinear Curvature Flows of Nonconvex Hypersurfaces
- Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class
- Finite time singularities of the Kähler–Ricci flow
- Finite-Time Singularities of the Kähler--Ricci Flow on a ℂPm-Bundle over a Product of Kähler--Einstein Manifolds
BibTeX
@misc{zhang2026finite,
title = {{Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits}},
author = {Lei Zhang and Zhenlei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21824},
eprint = {2610.21824},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21824},
}
RIS
TY - UNPB
TI - Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits
AU - Lei Zhang
AU - Zhenlei Zhang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21824
UR - https://arxiv.org/abs/2610.21824
AB - 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≲ 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.
ER -