On a twisted conical K\"ahler-Ricci flow

TL;DR

This paper studies the behavior of a twisted conical K"ahler-Ricci flow, establishing diameter bounds, convergence properties, and scalar curvature bounds, with implications for understanding singularities and flow dynamics on complex manifolds.

Contribution

It provides new results on diameter bounds, Gromov-Hausdorff convergence, and scalar curvature bounds for the twisted conical K"ahler-Ricci flow without smooth approximation.

Findings

Diameter bounds for the flow.

Gromov-Hausdorff convergence results.

Instant bounded scalar curvature for $t>0$.

Abstract

In this paper, we discuss diameter bound and Gromov-Hausdorff convergence of a twisted conical K\"ahler-Ricci flow on the total spaces of some holomorphic submersions. We also observe that, starting from a model conical K\"ahler metric with possibly unbounded scalar curvature, the conical K\"ahler-Ricci flow will instantly have bounded scalar curvature for $t>0$, and the bound is of the form $\frac{C}{t}$. Several key results will be obtained by direct arguments on the conical equation without passing to a smooth approximation. In the last section, we present several remarks on a twisted K\"ahler-Ricci flow and its convergence.