[Paper Review] On the lower bound of energy functional E_1 (I)-- a stability theorem on the Kaehler Ricci flow

Xiuxiong Chen|ArXiv.org|Feb 9, 2005

Geometry and complex manifolds8 references3 citations

TL;DR

This paper establishes a stability theorem for the Kähler Ricci flow on compact Kähler manifolds with positive first Chern class, proving that if the initial metric is sufficiently close to a Kähler-Einstein metric in terms of energy functional $E_1$, Ricci curvature bounded below, and Riemann curvature bounded in $L^∞$, then the flow converges exponentially fast to a Kähler-Einstein metric. The key contribution is a novel iteration scheme that recovers curvature pinching after initial loss, enabling uniform curvature bounds and convergence without requiring the pinching condition to be preserved throughout.

ABSTRACT

In the present paper, we prove a stability theorem for the Kaehler Ricci flow near the infimum of the functional E_1 under the assumption that the initial metric has Ricci > -1 and |Riem| bounded. At present stage, our main theorem still need a topological assumption (1.2) which we hope to be removed in subsequent papers. The underlying moral is: if a Kaehler metric is sufficiently closed to a Kaehler Einstein metric, then the Kaehler Ricci flow converges to it. The present work should be viewed as a first step in a more ambitious program of deriving the existence of Kaehler Einstein metrics with an arbitrary energy level, provided that this energy functional has a uniform lower bound in this Kaehler class.

Motivation & Objective

• To establish a stability theorem for the Kähler Ricci flow near the infimum of the energy functional $E_1$ in Kähler classes with positive first Chern class.
• To prove that metrics sufficiently close to a Kähler-Einstein metric in $E_1$-energy and curvature norms will evolve under the Kähler Ricci flow to a Kähler-Einstein metric.
• To provide conditions under which the $L^∞$ norm of curvature remains uniformly bounded, enabling convergence without requiring initial pointwise curvature pinching.
• To generalize classical curvature pinching theorems in Ricci flow to the Kähler setting, where pinching may be lost initially but recovered periodically.

Proposed method

• Introduce a subspace $\mathcal{A}(\delta,\Lambda,\epsilon)$ of Kähler metrics satisfying $Ric > -1 + \delta$, $|Riem| < \Lambda$, and $E_1$ close to its infimum.
• Use the decreasing property of $E_1$ along the Kähler Ricci flow to control the $L^2$ norm of the traceless Ricci tensor.
• Apply Sobolev and Poincaré inequalities to estimate the $L^2$ norm of the traceless Ricci tensor and derive pointwise bounds.
• Implement a Moser iteration scheme in a 'move' time interval to recover pointwise Ricci pinching after initial loss.
• Establish a compactness lemma to control the $L^p$ norm of the Riemann curvature tensor over time.
• Use iterative application of curvature control to show that the Ricci curvature converges uniformly to the Kähler form, implying exponential convergence of the flow.

Experimental results

Research questions

• RQ1 Under what geometric conditions does the Kähler Ricci flow converge to a Kähler-Einstein metric when the initial metric is near the $E_1$-infimum?
• RQ2 Can curvature pinching be recovered dynamically in the Kähler Ricci flow even if it is lost immediately after initiation?
• RQ3 Is the $E_1$ functional bounded from below in Kähler classes without Kähler-Einstein metrics, and what geometric conditions imply such a lower bound?
• RQ4 How does the stability of the complex structure affect the convergence of the Kähler Ricci flow in the absence of a Kähler-Einstein metric?
• RQ5 Can the $L^p$ norm of curvature for $p > n+1$ replace the $L^\infty$ bound on curvature to ensure long-time existence and convergence?

Key findings

• For any $\delta, \Lambda > 0$, there exists $\epsilon(\delta, \Lambda) > 0$ such that if $E_1(g) \leq \inf E_1 + \epsilon$ and $Ric(g) > -1 + \delta$, $|Riem(g)| < \Lambda$, then the Kähler Ricci flow converges exponentially fast to a Kähler-Einstein metric.
• The estimate $\epsilon(\delta, \Lambda) \leq \left(\frac{1}{\Lambda}\right)^{2n} \delta \cdot \epsilon_0(n)^2$ is provided, though it is not optimal.
• The $L^p$ norm of the Riemann curvature tensor is uniformly bounded for $p > n+1$, ensuring non-singular flow evolution.
• A Moser iteration scheme recovers pointwise Ricci pinching after an initial time interval, enabling control of the $L^\infty$ norm of curvature.
• The Ricci curvature converges uniformly to the Kähler form: $\max_{x \in M} |Ric(g(t)) - \omega_{g(t)}| \to 0$ as $t \to \infty$, with exponential decay.
• In complex surfaces, the stability of the complex structure under diffeomorphism ensures convergence without additional assumptions, completing the proof of Theorem 6.