# Stability of the conical K\"ahler-Ricci flows on Fano manifolds

## TL;DR

This paper proves the stability of conical K"ahler-Ricci flows on Fano manifolds near a conical K"ahler-Einstein metric, under a weaker energy condition, and provides new proofs of key existence and openness results.

## Contribution

It establishes stability of conical K"ahler-Ricci flows under a bounded below Log Mabuchi energy, relaxing previous properness assumptions.

## Key findings

- Flow converges to conical K"ahler-Einstein metrics for nearby cone angles.
- Provides parabolic proofs of Donaldson's openness theorem.
- Confirms existence conjecture for positive Ricci curvature cases.

## Abstract

In this paper, we study the stability of the conical K\"ahler-Ricci flows on Fano manifolds. That is, if there exists a conical K\"ahler-Einstein metric with cone angle $2\pi\beta$ along the divisor, then for any $\beta'$ sufficiently close to $\beta$, the corresponding conical K\"ahler-Ricci flow converges to a conical K\"ahler-Einstein metric with cone angle $2\pi\beta'$ along the divisor. Here, we only use the condition that the Log Mabuchi energy is bounded from below. This is a weaker condition than the properness that we have adopted to study the convergence before. As corollaries, we give parabolic proofs of Donaldson's openness theorem and his existence conjecture for the conical K\"ahler-Einstein metrics with positive Ricci curvatures.