[Paper Review] A mean value formula and a Liouville theorem for the complex Monge-Ampère equation

Chao Li, Jiayu Li+1 more|arXiv (Cornell University)|Sep 18, 2017

Geometry and complex manifolds18 references3 citations

TL;DR

This paper establishes a mean value formula for bounded subharmonic Hermitian matrix-valued functions on complete Riemannian manifolds with nonnegative Ricci curvature, proving that their average over large balls converges to a constant Hermitian matrix. As an application, it proves a Liouville-type theorem for the complex Monge-Ampère equation on product manifolds, showing that under certain curvature and volume growth conditions, the Kähler metric must be a sum of a flat metric on $\mathbb{C}^m$ and a Ricci-flat metric on a compact manifold.

ABSTRACT

In this paper, we prove a mean value formula for bounded subharmonic Hermitian matrix valued function on a complete Riemannian manifold with nonnegative Ricci curvature. As its application, we obtain a Liouville type theorem for the complex Monge-Ampère equation on product manifolds.

Motivation & Objective

• To extend the classical mean value property for subharmonic functions to the setting of bounded subharmonic Hermitian matrix-valued functions on complete Riemannian manifolds with nonnegative Ricci curvature.
• To establish a Liouville-type theorem for the complex Monge-Ampère equation on product manifolds $\mathbb{C}^m \times Y$, where $Y$ is a compact Kähler manifold with nonnegative Ricci curvature.
• To generalize previous results on the complex Monge-Ampère equation by removing the need for Calabi $\mathcal{C}^3$ estimates and instead using mean value convergence of matrix-valued functions.
• To show that under volume growth and curvature bounds, the Kähler metric on $\mathbb{C}^m \times Y$ must split as a sum of a flat metric on $\mathbb{C}^m$ and a Ricci-flat metric on $Y$.

Proposed method

• Define subharmonicity for Hermitian matrix-valued functions via the subharmonicity of the quadratic form $\xi A \xi^*$ for all $\xi \in \mathbb{C}^m$.
• Prove a mean value formula: the average of a bounded subharmonic Hermitian matrix function $A$ over large geodesic balls converges to a constant Hermitian matrix $A_0$ as the radius $r \to \infty$, i.e., $\lim_{r\to\infty} \fint_{B_r(p)} A \, dV_g = A_0$.
• Use the convergence of averages and the maximum principle to show that $A \leq A_0$ on $M$.
• Apply the mean value formula to the inverse of the metric components $u^{i\bar{j}}$ of a Kähler metric $\omega$ on $\mathbb{C}^m \times Y$, showing that their average converges to a constant matrix $A$.
• Use volume comparison and metric comparison estimates to relate the $\omega$-ball averages to the $g_0$-ball averages on $\mathbb{C}^m \times Y$, establishing $\fint_{B_r(z_0)\times Y} \det(u^{i\bar{j}}) \, dV_{g_0} = \det A$.
• Derive that $\det(u^{i\bar{j}}) \equiv \det A = c$, and from this, deduce $u^{i\bar{j}} \equiv A$, implying the metric is constant in the $\mathbb{C}^m$ directions, and hence $\nabla_{g_0} g = 0$.

Experimental results

Research questions

• RQ1 Can a mean value formula be established for bounded subharmonic Hermitian matrix-valued functions on complete Riemannian manifolds with nonnegative Ricci curvature?
• RQ2 Does the convergence of averages of such matrix functions imply rigidity in the geometry of the underlying manifold?
• RQ3 Under what conditions does the complex Monge-Ampère equation on $\mathbb{C}^m \times Y$ force the Kähler metric to split into a flat metric on $\mathbb{C}^m$ and a Ricci-flat metric on $Y$?
• RQ4 Can the Liouville theorem for the complex Monge-Ampère equation be proven without relying on Calabi $\mathcal{C}^3$ estimates?
• RQ5 What is the role of the mean value formula in proving rigidity for Kähler metrics on product spaces?

Key findings

• The average of a bounded subharmonic Hermitian matrix-valued function $A$ over large geodesic balls converges to a constant Hermitian matrix $A_0$, i.e., $\lim_{r\to\infty} \fint_{B_r(p)} A \, dV_g = A_0$.
• The matrix $A$ satisfies $A \leq A_0$ on $M$, with equality in the limit of averages.
• For the complex Monge-Ampère equation on $\mathbb{C}^m \times Y$, under the given curvature and volume growth bounds, $\det(u^{i\bar{j}}) \equiv \det A = c$, a positive constant.
• The inverse metric components $u^{i\bar{j}}$ are constant matrices, so $u^{i\bar{j}} \equiv A$.
• The pullback of the Kähler form $\omega$ to each fiber $\{z\} \times Y$ satisfies $i_z^* \omega = \omega_Y$, showing the metric is constant along $\mathbb{C}^m$.
• The metric connection $\nabla_{g_0} g$ vanishes, implying $\nabla_{g_0} g = 0$, so the metric is flat in the $\mathbb{C}^m$ directions and the full metric splits as $\omega = \omega_Y + S^* \omega_{\mathbb{C}^m}$.