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Kähler-Ricci flows coming out of metric spaces

Alix Deruelle, Vincent Guedj, Henri Guenancia, Ahmed Zeriahi

Abstract

Given a compact Kähler manifold $X$ and a closed, positive $(1,1)$-current $T$ on $X$, we find sufficient conditions for $T$ to induce a metric structure $(X,d_T)$ which is the Gromov-Hausdorff limit of compact Kähler manifolds either in a "static" way or at time zero of smooth Kähler-Ricci flows. In dimension $1$ we extend works of T. Richard and M. Simon, showing that any oriented compact Alexandrov surface with bounded integral curvature and without cusp is the initial datum of a Kähler-Ricci flow.

Kähler-Ricci flows coming out of metric spaces

Abstract

Given a compact Kähler manifold and a closed, positive -current on , we find sufficient conditions for to induce a metric structure which is the Gromov-Hausdorff limit of compact Kähler manifolds either in a "static" way or at time zero of smooth Kähler-Ricci flows. In dimension we extend works of T. Richard and M. Simon, showing that any oriented compact Alexandrov surface with bounded integral curvature and without cusp is the initial datum of a Kähler-Ricci flow.

Paper Structure

This paper contains 36 sections, 36 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Outline of the paper
  5. Preliminaries
  6. Quasi-plurisubharmonic functions
  7. Regularization
  8. Monge-Ampère measures
  9. Diameter of Kähler metrics with Hölder potentials
  10. Complex Monge-Ampère flows
  11. A priori estimates along the flow
  12. ${\mathcal{C}}^0$-bounds
  13. Bounds on $\dot{\varphi}_t$
  14. Laplacian bounds
  15. Parabolic Chern-Lu inequality
  16. ...and 21 more sections

Key Result

Theorem A

Let $(X,\omega_X)$ be a compact Kähler manifold and let $T$ be a current with geometric singularities. Then the following holds:

Theorems & Definitions (67)

  • Definition 1.1
  • Theorem A
  • Corollary B
  • Theorem C
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 57 more