Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities

Longteng Chen, Max Hallgren, Lucas Lavoyer

Abstract

Let $(Y,g_0)$ be a compact Kähler space with a finite number of singular points, where the metric at each singular point is modelled on an admissible Kähler cone. We show that the Kähler-Ricci flow with such initial data satisfies a $C/t$ curvature bound, and that the flow near each singular point is modelled on the unique Kähler-Ricci expander asymptotic to the corresponding cone. Our motivation is to give a geometric description of the Kähler--Ricci flow emerging from singularities arising in the analytic minimal model program.

Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities

Abstract

Let be a compact Kähler space with a finite number of singular points, where the metric at each singular point is modelled on an admissible Kähler cone. We show that the Kähler-Ricci flow with such initial data satisfies a curvature bound, and that the flow near each singular point is modelled on the unique Kähler-Ricci expander asymptotic to the corresponding cone. Our motivation is to give a geometric description of the Kähler--Ricci flow emerging from singularities arising in the analytic minimal model program.

Paper Structure

This paper contains 19 sections, 62 theorems, 387 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Admissible Kähler cones
  4. Asymptotically conical gradient Kähler--Ricci expanders
  5. The approximating solution
  6. Localisation of the problem and the modified solutions to CMA equations
  7. Rough estimates on the normalised conical region
  8. Uniform estimates on the expanding region
  9. $C^2$-estimates on the expanding region
  10. Higher order and improved estimates
  11. Curvature control and local stability
  12. Flowing metrics with conical singularities
  13. Taking the limit
  14. Convergence to the initial data
  15. Tangent flow at singular points
  16. ...and 4 more sections

Key Result

Theorem A

Let $(Y,g_0)$ be a compact analytic space with isolated conical singularities at $\{y_i\}_{i=1}^Q\subset Y$, each modelled on an admissible Kähler cone $(\mathcal{C}(S_i),g_{\mathcal{C}_i})$ in the sense of Definition def of good kahler cone. Then there exists a smooth Kähler manifold $M$, a smooth $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: unnormalised and normalised space-time
  • Figure 2: modified space-time

Theorems & Definitions (135)

  • Definition 1.1
  • Theorem A
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1: Ishii
  • Theorem 2.2: ConlonDeruelleSun
  • Remark 2.3
  • Remark 2.4
  • ...and 125 more