[Paper Review] The Calabi-Yau equation, symplectic forms and almost complex structures

Valentino Tosatti, Ben Weinkove|arXiv (Cornell University)|Jan 12, 2009

Geometry and complex manifolds31 references16 citations

TL;DR

This paper investigates a conjecture by Donaldson on the existence of solutions to the Calabi-Yau equation for symplectic forms compatible with non-integrable almost complex structures in four dimensions. Using a monotonicity formula for harmonic maps, the authors establish a new local $L^1$-bound on the trace of the metric tensor, which implies a key $C^0$ estimate and supports the conjectured $C^∞$ regularity of solutions.

ABSTRACT

We discuss a conjecture of Donaldson on a version of Yau's Theorem for symplectic forms with compatible almost complex structures and survey some recent progress on this problem. We also speculate on some future possible directions, and use a monotonicity formula for harmonic maps to obtain a new local estimate in the setting of Donaldson's conjecture.

Motivation & Objective

• To investigate Donaldson's conjecture on the existence of solutions to the Calabi-Yau equation for symplectic forms compatible with non-integrable almost complex structures in dimension four.
• To extend Yau's Theorem—originally valid in Kähler geometry—to the setting of symplectic forms and almost complex structures.
• To establish a new local $C^0$ estimate for the almost-Kähler potential, which is crucial for proving $C^∞$ bounds.
• To explore the possibility that the blow-up set of solutions has real codimension at least two, possibly corresponding to a $J$-holomorphic curve.

Proposed method

• Adapts the continuity method used in Yau’s original proof of the Calabi-Yau theorem to the non-Kähler, almost complex setting.
• Uses a monotonicity formula for harmonic maps to derive a new $L^1$-bound on the trace of the metric tensor $\mathrm{tr}_g \tilde{g}$.
• Applies the $L^1$ trace bound in conjunction with the monotonicity inequality (5.6) to control the growth of the trace over geodesic balls.
• Relies on the fact that $\tilde{\omega}^2 = \sigma$ implies uniform control between $\mathrm{tr}_g \tilde{g}$ and $\mathrm{tr}_{\tilde{g}} g$, enabling $L^1$ bounds via Stokes’ Theorem.
• Derives a new differential inequality involving the $\tilde{g}$-Laplacian of $\mathrm{tr}_g \tilde{g}$, with constants depending only on fixed geometric data.
• Proposes a conjectural $\varepsilon$-regularity result analogous to that in harmonic map theory, though the standard proof fails due to uncontrolled Sobolev constants on $\tilde{g}$.

Experimental results

Research questions

• RQ1 Can the Calabi-Yau equation $\tilde{\omega}^2 = \sigma$ be solved for symplectic forms $\tilde{\omega}$ cohomologous to a given symplectic form $\Omega$ on a compact 4-manifold with a tame almost complex structure $J$?
• RQ2 Does the blow-up set of solutions to the Calabi-Yau equation have real codimension at least two, and could it be represented by a $J$-holomorphic curve?
• RQ3 Can a new $C^0$ estimate for the almost-Kähler potential be derived using geometric monotonicity formulas?
• RQ4 Is there a viable $\varepsilon$-regularity result for the trace $\mathrm{tr}_g \tilde{g}$ in the non-Kähler setting, despite the failure of standard harmonic map techniques?
• RQ5 To what extent can the $C^\infty$ estimates from Yau’s Theorem be extended to the almost complex, symplectic setting?

Key findings

• A new monotonicity formula (5.6) is established for the $L^1$-norm of $\mathrm{tr}_g \tilde{g}$ over geodesic balls, depending only on the background metric $g$.
• Corollary 5.1 proves that $\int_{B_g(p,r)} \mathrm{tr}_g \tilde{g} \, dV_g \leq C r^2$ for all $p \in M$ and small $r > 0$, under the Calabi-Yau equation $\tilde{\omega}^2 = \sigma$.
• The $L^1$ trace bound (5.9) is derived using the monotonicity formula and the uniform $L^1$ bound on $\mathrm{tr}_{\tilde{g}} g$ via Stokes’ Theorem.
• The trace $\mathrm{tr}_g \tilde{g}$ is uniformly controlled in $L^1$-norm on small balls, suggesting that the blow-up set of solutions has codimension at least two.
• A conjectural $\varepsilon$-regularity estimate is proposed, which would imply local $C^0$ bounds on $\mathrm{tr}_g \tilde{g}$, but the standard harmonic map strategy fails due to uncontrolled Sobolev constants on $\tilde{g}$.
• The differential inequality $\tilde{\Delta} \mathrm{tr}_g \tilde{g} \geq -C_2 - C_3 (\mathrm{tr}_g \tilde{g})^2$ is used, with constants depending only on fixed data, indicating a potential path to higher regularity.