[Paper Review] Longtime existence of the Kähler-Ricci flow on $\Bbb C ^n$

Albert Chau, Ka-Fai Li+1 more|arXiv (Cornell University)|Sep 5, 2014

Geometry and complex manifolds10 references3 citations

TL;DR

This paper establishes the longtime existence and convergence of the Kähler-Ricci flow on $\mathbb{C}^n$ for complete $U(n)$-invariant Kähler metrics with non-negative holomorphic bisectional curvature, even without bounded curvature assumptions. It proves that such solutions exist for all time, remain uniformly equivalent to the initial metric, and converge to the standard Euclidean metric after rescaling, extending previous results to unbounded curvature settings.

ABSTRACT

We produce longtime solutions to the Kähler-Ricci flow for complete Kähler metrics on $\Bbb C ^n$ without assuming the initial metric has bounded curvature, thus extending results in [3]. We prove the existence of a longtime bounded curvature solution emerging from any complete $U(n)$-invariant Kähler metric with non-negative holomorphic bisectional curvature, and that the solution converges as $t o \infty$ to the standard Euclidean metric after rescaling. We also prove longtime existence results for more general Kähler metrics on $\Bbb C ^n$ which are not necessarily $U(n)$-invariant.

Motivation & Objective

• To establish longtime existence of the Kähler-Ricci flow on $\mathbb{C}^n$ for complete $U(n)$-invariant Kähler metrics with non-negative holomorphic bisectional curvature.
• To prove that the flow solution remains uniformly equivalent to the initial metric and has bounded non-negative bisectional curvature.
• To show that the solution converges to the standard Euclidean metric after rescaling as $t \to \infty$.
• To extend the results beyond $U(n)$-invariant metrics to more general Kähler metrics that are equivalent to $U(n)$-invariant ones with non-negative bisectional curvature and maximum volume growth.
• To resolve the uniqueness and long-term behavior of the flow in the absence of curvature bounds, generalizing prior results by Shi and others.

Proposed method

• The authors use the $U(n)$-invariance of the metric to reduce the Kähler-Ricci flow to an evolution equation for a radial function $\xi(r)$, where $r = |z|^2$, via the metric representation $g_{i\bar{j}} = f_\xi(r)\delta_{ij} + f_\xi'(r)\bar{z}_i z_j$.
• They analyze curvature components using the formulas from Wu-Zheng and Yang, expressing the holomorphic bisectional curvature in terms of $\xi'$, $h_\xi$, and $f_\xi$, and establish conditions for non-negativity.
• The proof relies on the maximum volume growth condition and the fact that complete $U(n)$-invariant metrics with non-negative bisectional curvature satisfy $\int_0^\infty \frac{\sqrt{h_\xi(s)}}{\sqrt{s}} ds = \infty$, ensuring completeness.
• The authors apply a priori estimates and curvature decay control, using results from Shi and Cabezas-Rivas–Wilking to control curvature growth and ensure long-time existence.
• They use a comparison argument between equivalent initial metrics, showing that if $g_0$ and $G_0$ are equivalent and have non-negative bisectional curvature, then their flow solutions remain uniformly equivalent for all time.
• The convergence to the Euclidean metric is established via rescaling: $\frac{1}{|V|_t^2}g(t)$ converges smoothly and uniformly on compact sets to the standard metric as $t \to \infty$.

Experimental results

Research questions

• RQ1 Can the Kähler-Ricci flow be shown to exist for all time on $\mathbb{C}^n$ for complete $U(n)$-invariant Kähler metrics with non-negative holomorphic bisectional curvature, even when the initial metric has unbounded curvature?
• RQ2 Does the flow solution remain uniformly equivalent to the initial metric and preserve non-negative bisectional curvature over time?
• RQ3 Does the flow solution converge to the standard Euclidean metric after appropriate rescaling as $t \to \infty$?
• RQ4 Can the longtime existence and convergence results be extended to non-$U(n)$-invariant Kähler metrics that are equivalent to $U(n)$-invariant ones with non-negative bisectional curvature and maximum volume growth?
• RQ5 Is the solution unique in the class of uniformly equivalent solutions to the initial metric?

Key findings

• The Kähler-Ricci flow admits a smooth longtime $U(n)$-invariant solution on $\mathbb{C}^n$ for any complete $U(n)$-invariant Kähler metric with non-negative holomorphic bisectional curvature, even without bounded curvature.
• The solution remains uniformly equivalent to the initial metric and has bounded non-negative bisectional curvature for all time.
• The solution converges to the standard Euclidean metric on $\mathbb{C}^n$ after rescaling at the origin as $t \to \infty$, with convergence occurring smoothly and uniformly on compact subsets.
• For more general Kähler metrics equivalent to a $U(n)$-invariant metric with non-negative bisectional curvature and maximum volume growth, the flow still admits a unique longtime solution with bounded curvature.
• The solution is unique in the class $\mathcal{S}(g_0)$ of solutions uniformly equivalent to the initial metric $g_0$, under the stated conditions.
• The curvature decay estimate $t \cdot \mathcal{R}(x,t) \leq c_1$ holds uniformly for all $t > 0$, which is key to the comparison argument and long-time control.