Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniqueness of asymptotically conical Kähler-Ricci flow

Longteng Chen

TL;DR

This work addresses the uniqueness of the Kahler-Ricci flow starting from a Kahler cone with a smooth canonical model, showing that under a matching conical data, cohomology, Killing-field, and curvature bounds, the forward self-similar flow associated to a given expanding gradient Kahler-Ricci soliton is the unique evolution. The strategy reduces the problem to a complex Monge-Ampère equation by exploiting the soliton symmetry via the Killing field $JX$, and then employs an energy method on the normalized flow to prove a static flow, yielding equality of the two evolutions. The results connect and extend Feldman-Ilmanen-Knopf and Conlon-Deruelle-Sun, providing a partial answer to questions about desingularizations of Kahler cones and the selection principle for AC flows, with implications for uniqueness in non-compact Kahler geometry. The approach combines detailed a priori estimates at spatial infinity, barrier arguments, and a robust interplay between geometric analysis and complex Monge-Ampère techniques, highlighting the role of curvature decay, the obstruction tensor, and soliton potentials in establishing rigidity of AC desingularizations.

Abstract

We study the uniqueness problem for the Kähler-Ricci flow with a conical initial condition. Given a complete gradient expanding Kähler-Ricci soliton on a non compact manifold with quadratic curvature decay, including its derivatives, we establish that any complete solution to the Kahler-Ricci flow emerging from the soliton's tangent cone at infinity--appearing as a Kähler cone--must coincide with the forward self-similar Kähler-Ricci flow associated with the soliton, provided certain conditions hold. Specifically, if its Kähler form remains in the same cohomology class as that of the soliton's self-similar Kähler-Ricci flow, its full Riemann curvature operator is bounded for each fixed positive time, its Ricci curvature is bounded from above by A/t, its scalar curvature is bounded from below by -A/t, and it shares a same Killing vector field with the soliton metric. This paper gives a partial answer to a question in paper of Feldman-Ilmanen-Knopf, and generalizes the earlier work of Conlon-Deruelle and the work of Conlon-Deruelle-Sun.

Uniqueness of asymptotically conical Kähler-Ricci flow

TL;DR

This work addresses the uniqueness of the Kahler-Ricci flow starting from a Kahler cone with a smooth canonical model, showing that under a matching conical data, cohomology, Killing-field, and curvature bounds, the forward self-similar flow associated to a given expanding gradient Kahler-Ricci soliton is the unique evolution. The strategy reduces the problem to a complex Monge-Ampère equation by exploiting the soliton symmetry via the Killing field , and then employs an energy method on the normalized flow to prove a static flow, yielding equality of the two evolutions. The results connect and extend Feldman-Ilmanen-Knopf and Conlon-Deruelle-Sun, providing a partial answer to questions about desingularizations of Kahler cones and the selection principle for AC flows, with implications for uniqueness in non-compact Kahler geometry. The approach combines detailed a priori estimates at spatial infinity, barrier arguments, and a robust interplay between geometric analysis and complex Monge-Ampère techniques, highlighting the role of curvature decay, the obstruction tensor, and soliton potentials in establishing rigidity of AC desingularizations.

Abstract

We study the uniqueness problem for the Kähler-Ricci flow with a conical initial condition. Given a complete gradient expanding Kähler-Ricci soliton on a non compact manifold with quadratic curvature decay, including its derivatives, we establish that any complete solution to the Kahler-Ricci flow emerging from the soliton's tangent cone at infinity--appearing as a Kähler cone--must coincide with the forward self-similar Kähler-Ricci flow associated with the soliton, provided certain conditions hold. Specifically, if its Kähler form remains in the same cohomology class as that of the soliton's self-similar Kähler-Ricci flow, its full Riemann curvature operator is bounded for each fixed positive time, its Ricci curvature is bounded from above by A/t, its scalar curvature is bounded from below by -A/t, and it shares a same Killing vector field with the soliton metric. This paper gives a partial answer to a question in paper of Feldman-Ilmanen-Knopf, and generalizes the earlier work of Conlon-Deruelle and the work of Conlon-Deruelle-Sun.
Paper Structure (19 sections, 30 theorems, 170 equations)

This paper contains 19 sections, 30 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main result
  4. Strategy of proof
  5. Outline of Paper
  6. Properties of expanding gradient Kähler-Ricci soliton
  7. Ricci flow coming out of Riemannian cone
  8. Set up of a complex Monge-Ampère equation
  9. Symmetries of the Kähler potential
  10. Reduction to a complex Monge-Ampère equation
  11. A priori estimates at spatial infinity
  12. Polynomial decay
  13. Obstruction tensor
  14. Hamiltonian potentials of $X$
  15. Maximum principle
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $(C_0,g_0)$ be a Kähler cone of complex dimension $n\ge2$ which admits a smooth canonical model $M$. Let $\pi: M\mapsto C_0$ be a smooth Kähler resolution with exceptional set $E$ such that the canonical line bundle $K_M|_E$ is $\pi-$ample, i.e. $c_1(K_M|_E)>0$. Let $(M,g,X)$ be the unique (up t where $d_{g}(p,\,\cdot)$ denotes the distance to a fixed point $p\in M$ with respect to $g$, with t

Theorems & Definitions (69)

  • Theorem 1.2: Uniqueness theorem
  • Remark 1.3
  • Theorem 2.1: Strong uniqueness for expanders
  • Lemma 2.2: Soliton identities
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Definition 3.1: Ricci flow coming out of cone
  • ...and 59 more