Calabi flow on polarized Kähler surfaces
Abstract
We study global smooth Calabi flows on polarized Kähler surfaces (M,L), assuming uniformly bounded Sobolev constants and the algebraic conditions χ(𝒪M)=1 and c1(M)· c1(L)>0. Using the orbifold compactness theorem, we prove a uniform Bergman estimate, construct a normal test configuration for each geometric limit, and establish uniqueness of the extremal orbifold limit up to polarized holomorphic isometry. K-polystability then identifies the limit with the original surface and gives a constant scalar curvature Kähler metric. We also prove finite-group equivariant versions of these results. Independently of the flow, we obtain a uniform Bergman lower bound for polarized cscK surfaces with bounded volume and Sobolev constants and |S|≤1. We illustrate the hypotheses with nonstationary flows near a product cscK metric on ℙ1×ℙ1 and near Calabi's non-cscK extremal metric on the first Hirzebruch surface. Their controlled-cone energy inequalities follow from elementary calculations and continuity.
Related
- Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants
- Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants
- Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I
- Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces
- Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I
BibTeX
@misc{li2026calabi,
title = {{Calabi flow on polarized Kähler surfaces}},
author = {Haozhao Li and Bing Wang and Kai Zheng},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01968},
eprint = {2611.01968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01968},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Calabi flow on polarized Kähler surfaces
AU - Haozhao Li
AU - Bing Wang
AU - Kai Zheng
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01968
UR - https://arxiv.org/abs/2611.01968
AB - We study global smooth Calabi flows on polarized Kähler surfaces (M,L), assuming uniformly bounded Sobolev constants and the algebraic conditions χ(𝒪_M)=1 and c_1(M)· c_1(L)>0. Using the orbifold compactness theorem, we prove a uniform Bergman estimate, construct a normal test configuration for each geometric limit, and establish uniqueness of the extremal orbifold limit up to polarized holomorphic isometry. K-polystability then identifies the limit with the original surface and gives a constant scalar curvature Kähler metric. We also prove finite-group equivariant versions of these results. Independently of the flow, we obtain a uniform Bergman lower bound for polarized cscK surfaces with bounded volume and Sobolev constants and |S|≤1. We illustrate the hypotheses with nonstationary flows near a product cscK metric on ℙ^1×ℙ^1 and near Calabi's non-cscK extremal metric on the first Hirzebruch surface. Their controlled-cone energy inequalities follow from elementary calculations and continuity.
ER -