[Paper Review] Deforming symplectomorphisms of complex projective spaces by the mean curvature flow

Ivana Medoš, Mu‐Tao Wang|arXiv (Cornell University)|Jul 15, 2009

Geometric Analysis and Curvature Flows6 references3 citations

TL;DR

This paper applies the mean curvature flow to deform symplectomorphisms of complex projective space $\mathbb{CP}^n$ with a pinching condition on their pullback metrics. It proves that any $\Lambda$-pinched symplectomorphism is symplectically isotopic to a biholomorphic isometry via the flow, establishing a smooth deformation retract of the symplectomorphism group onto its isometry subgroup for $n \geq 1$. The result generalizes earlier work on Riemann surfaces and provides a new geometric approach to symplectic topology.

ABSTRACT

We apply the mean curvature flow to deform symplectomorphisms of $\mathbb{CP}^n$. In particular, we prove that, for each dimension n, there exists a constant $Λ$, explicitly computable, such that any $Λ$-pinched symplectomorphism of $\mathbb{CP}^n$ is symplectically isotopic to a biholomorphic isometry.

Motivation & Objective

• To study the topology of the symplectomorphism group of $\mathbb{CP}^n$ using geometric flows.
• To extend previous results on Riemann surfaces to higher-dimensional complex projective spaces.
• To establish that $\Lambda$-pinched symplectomorphisms are symplectically isotopic to biholomorphic isometries.
• To prove long-time existence and smooth convergence of the mean curvature flow on the graph of a symplectomorphism.
• To provide a geometric deformation retract of the symplectomorphism group onto its isometry subgroup for $\mathbb{CP}^n$.

Proposed method

• The graph of a symplectomorphism $f: \mathbb{CP}^n \to \mathbb{CP}^n$ is embedded in $\mathbb{CP}^n \times \mathbb{CP}^n$ as a Lagrangian submanifold with respect to the symplectic form $\pi_1^*\omega - \pi_2^*\omega$.
• The mean curvature flow is applied to this graph, evolving it as a Lagrangian submanifold under the Fubini-Study metric.
• The flow is shown to preserve the graphical nature of the submanifold via the maximum principle applied to the Jacobian of the projection $\pi_1$.
• Blow-up analysis is used to rule out finite-time singularities by bounding the second fundamental form uniformly.
• A comparison principle is applied to show that the pinching condition improves over time, driving $f_t^*g$ toward the Fubini-Study metric $g$.
• Uniform bounds on the second fundamental form are established as $t \to \infty$, ensuring smooth convergence to a minimal Lagrangian submanifold.

Experimental results

Research questions

• RQ1 Can the mean curvature flow deform any $\Lambda$-pinched symplectomorphism of $\mathbb{CP}^n$ to a biholomorphic isometry?
• RQ2 Does the mean curvature flow preserve the graphical property of the symplectomorphism graph in $\mathbb{CP}^n \times \mathbb{CP}^n$?
• RQ3 Is there a uniform $\Lambda(n) > 1$ such that all $\Lambda$-pinched symplectomorphisms are symplectically isotopic to isometries?
• RQ4 Does the flow converge smoothly to a minimal Lagrangian submanifold as $t \to \infty$?
• RQ5 Can the symplectomorphism group of $\mathbb{CP}^n$ for $n > 2$ be deformation retracted onto its isometry group via geometric flow?

Key findings

• For each $n$, there exists a computable $\Lambda(n) > 1$ such that any $\Lambda$-pinched symplectomorphism with $1 < \Lambda < \Lambda(n)$ is symplectically isotopic to a biholomorphic isometry.
• The mean curvature flow of the graph of $f$ exists smoothly for all time $t \geq 0$ under the $\Lambda$-pinching condition.
• The flow remains graphical throughout, so $\Sigma_t$ is the graph of a symplectomorphism $f_t$ for all $t \geq 0$.
• As $t \to \infty$, $f_t$ converges smoothly to a biholomorphic isometry of $\mathbb{CP}^n$.
• The second fundamental form of $\Sigma_t$ is uniformly bounded in $t$, ensuring smooth convergence.
• The pinching condition improves over time due to curvature properties of $\mathbb{CP}^n$, driving $f_t^*g$ toward $g$.