Power Convexity of Solutions to the Complex Monge-Ampère Equation $\det(u_{i\overline{j}})=1$ in Complex Dimension Two

Technical Summary: Power Convexity of Solutions to the Complex Monge-Ampère Equation in Complex Dimension Two

Problem Statement
This paper investigates the convexity properties of solutions to the Dirichlet problem for the complex Monge-Ampère equation:
$$\det(u_{i\bar{j}}) = 1 \quad \text{in } \Omega, \quad u = 0 \quad \text{on } \partial\Omega,$$
where $\Omega$ is a strictly convex bounded domain in $\mathbb{C}^n$. It is well-established that the solution $u$ itself is generally not convex. The central question addressed is whether a suitable power transformation of the solution, specifically $g(u) = -(-u)^\alpha$ for some $0 < \alpha < 1$, yields a convex function. The authors focus on the specific case of complex dimension $n=2$ and the exponent $\alpha = 1/2$, aiming to prove that $v := -\sqrt{-u}$ is strictly convex in $\Omega$.

Methodology
The authors employ a method developed in prior works ([H25], [H24B], [H24A], [HS26]) which utilizes complex derivatives to study the real convexity (positivity of the real Hessian) of solutions. The core of the approach involves the following steps:

1. Auxiliary Function Construction: The authors introduce a real-valued auxiliary function $\sigma$ that quantifies the convexity of $v = -\sqrt{-u}$. Specifically, they define matrices $A = (u_{i\bar{j}} + \frac{u_i u_{\bar{j}}}{-2u})$ and $B = (u_{ij} + \frac{u_i u_j}{-2u})$. The convexity of $v$ is equivalent to the positive definiteness of $A$ and the matrix $M = A - BA^{-1}B$. The function $\sigma$ is defined as the minimum eigenvalue of the matrix $K = MA^{-1}$.

2. Differential Equations for Second Derivatives: The authors derive detailed differential equations for the second derivatives of $u$ and $v$. By applying complex differential operators ($\partial_i, \partial_{\bar{j}}$) to the Monge-Ampère equation and the definitions of $A$ and $B$, they establish relations governing the evolution of $M$ and $K$. A key result is the derivation of an inequality for the Laplacian of the minimum eigenvalue $\sigma$:
   $$u_{\bar{q}p} \partial_{p\bar{q}} \sigma \leq C(\sigma + |\nabla \sigma|),$$
   under the condition that the minimum eigenvalue of $K$ is strictly smaller than the second minimal eigenvalue.

3. Algebraic Reduction and Dimensional Analysis: The proof relies heavily on algebraic manipulations of the quadratic forms arising from the second derivatives. The authors reduce the problem to analyzing a specific quadratic form involving a skew-Hermitian matrix $W$ and the matrix $K$.

   • Crucial Dimensional Constraint: The paper demonstrates that the non-positivity of a specific quadratic form (related to $|tr(W)|^2 - \sum |W_{p\bar{s}}|^2$) holds if and only if the complex dimension is 2. In dimensions $n > 2$, this form is not necessarily non-positive, which explains why the result is currently restricted to $n=2$.

4. Approximation and Perturbation: Since the minimum eigenvalue $\sigma$ is generally only $C^0$ and the strict inequality between the first and second eigenvalues may not hold everywhere, the authors use a perturbation method. They perturb the solution $u$ by pluriharmonic quadratic polynomials $q$ to create a family of functions $u+q$. By analyzing the lower envelope of the minimum eigenvalues of the perturbed systems, they extend the result from the generic case (distinct eigenvalues) to the general case.

5. Continuity Argument: Finally, a continuity argument (deformation method) connects the strictly convex domain $\Omega$ to a ball, where the convexity of $-\sqrt{-u}$ is known, to establish the result for the general domain.

Key Contributions and Results

• Main Theorem: The paper proves that for a strictly convex bounded domain $\Omega \subset \mathbb{C}^2$ with smooth boundary, the classical solution $u \in C^4(\Omega) \cap C^2(\bar{\Omega})$ to the complex Monge-Ampère equation satisfies the property that $-\sqrt{-u}$ is strictly convex in $\Omega$.
• Methodological Advancement: The work extends the technique of using complex derivatives to analyze real convexity to the complex Monge-Ampère equation. It provides a systematic way to handle the non-convexity of the solution $u$ by transforming it into a convex function via a power law.
• Dimensional Limitation: The authors explicitly identify that the proof relies on a specific algebraic inequality that is valid only in complex dimension 2. They provide a counter-example construction showing that the key quadratic form fails to be non-positive in higher dimensions, clarifying the current boundary of this specific approach.

Significance and Claims
The paper claims to establish the power convexity of solutions to the complex Monge-Ampère equation in complex dimension 2. The authors note that this result was recently proved by Zhang and Zhou ([ZZ26]) using a different method, but their approach offers a distinct perspective by leveraging complex derivatives to study real Hessian properties.

The authors state that while the convexity result is currently limited to dimension 2, the underlying approach (using complex derivatives to study real convexity) is applicable to all dimensions. The paper serves as a foundational step in understanding the convexity properties of complex Monge-Ampère solutions, bridging the gap between real and complex elliptic PDE theory. The work builds upon a lineage of research into power convexity, citing foundational works by Caffarelli and Friedman ([CF85]) and subsequent generalizations by various authors, positioning this result as a natural extension of those techniques to the complex setting.