[Paper Review] Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics

Xiuxiong Chen, Song Sun|arXiv (Cornell University)|Apr 12, 2010

Geometry and complex manifolds26 references13 citations

TL;DR

This paper establishes the uniqueness of constant scalar curvature Kähler (cscK) metrics within a fixed Kähler class, proving that any two cscK metrics adjacent to the same Kähler class are isomorphic via a symplectic diffeomorphism. It further shows that the Calabi flow converges globally to a cscK metric in polynomial time and asymptotically approaches a smooth geodesic ray, extending the finite-dimensional Kempf-Ness framework to infinite-dimensional Kähler geometry.

ABSTRACT

We prove that constant scalar curvature Kähler metric "adjacent" to a fixed Kähler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T picture described by Donaldson. We prove that the Calabi flow near a cscK metric exists globally and converges uniformly to a cscK metric in a polynomial rate. Viewed in a Kähler class, the Calabi flow is also shown to be asymptotic to a smooth geodesic ray at infinity. This latter fact is also interesting in the finite dimensional analogue, where we show that the downward gradient flow of the Kempf-Ness function in a semi-stable orbit is asymptotic to the direction of optimal degeneration.

Motivation & Objective

• To establish the uniqueness of constant scalar curvature Kähler (cscK) metrics in a fixed Kähler class, extending previous results to the semi-stable case.
• To prove the global existence and polynomial convergence rate of the Calabi flow near a cscK metric.
• To show that the Calabi flow is asymptotic to a smooth geodesic ray at infinity in the fixed Kähler class.
• To formalize the infinite-dimensional analogue of the Kempf-Ness theorem in Kähler geometry, linking stability and cscK metric existence.
• To generalize the finite-dimensional gradient flow behavior of the Kempf-Ness function to the infinite-dimensional setting, showing asymptoticity to optimal degeneration directions.

Proposed method

• Utilizes the infinite-dimensional geometric invariant theory (G.I.T.) framework proposed by Donaldson, modeling the space of Kähler structures as a Kähler manifold with Hamiltonian group action.
• Applies the Lojasiewicz inequality to control the decay of the Calabi functional along the flow, ensuring convergence to a cscK metric.
• Constructs a smooth tame inverse via Hamilton’s implicit function theorem to solve the nonlinear ODE system governing the Calabi flow and geodesic ray asymptotics.
• Employs a linearization of the Calabi flow and geodesic ray equations, reducing them to symmetric hyperbolic systems solvable with tame estimates.
• Uses the Marle-Guillemin-Sternberg normal form to analyze the local structure of the moment map and its critical points.
• Establishes a correspondence between the Calabi flow and a geodesic ray in the space of Kähler structures by constructing a smooth, tame diffeomorphism between neighborhoods of the identity in the gauge group.

Experimental results

Research questions

• RQ1 Is the constant scalar curvature Kähler metric unique within a fixed Kähler class up to symplectic diffeomorphism?
• RQ2 Does the Calabi flow converge globally to a cscK metric when initialized near one, and at what rate?
• RQ3 Is the Calabi flow asymptotic to a smooth geodesic ray in the space of Kähler structures at infinity?
• RQ4 Can the finite-dimensional Kempf-Ness theorem be extended to the infinite-dimensional setting of Kähler geometry?
• RQ5 How does the gradient flow of the Kempf-Ness function behave in a semi-stable orbit, and is it asymptotic to the direction of optimal degeneration?

Key findings

• The cscK metric in a fixed Kähler class is unique up to symplectic diffeomorphism, generalizing the uniqueness result of Donaldson and Chen-Tian to the semi-stable case.
• The Calabi flow exists globally and converges uniformly to a cscK metric at a polynomial rate when initiated near a cscK metric.
• The Calabi flow is asymptotic to a smooth geodesic ray in the space of Kähler structures at infinity, under the fixed Kähler class.
• In the finite-dimensional setting, the downward gradient flow of the Kempf-Ness function in a semi-stable orbit is asymptotic to the direction of optimal degeneration.
• The construction of a smooth tame inverse via Hamilton’s implicit function theorem ensures the existence of smooth paths in the gauge group that realize the asymptotic geodesic ray.
• A tame estimate is established: $|v_1(t) - v_1(0)| \leq C \cdot (|\rho(0)| + |v(0)|)^3$, ensuring controlled nonlinear behavior near the origin.