[Paper Review] Complex geodesics and complex Monge-Ampère equations with boundary singularity

Jimmy Xiangji Huang, Xieping Wang|arXiv (Cornell University)|Feb 2, 2020

Geometry and complex manifolds37 references4 citations

TL;DR

This paper establishes the uniqueness of complex geodesics in bounded strongly linearly convex domains in $\mathbb{C}^n$ with $C^3$-smooth boundary, even under minimal regularity assumptions. Building on this uniqueness and Lempert's existence results, the authors construct a pluricomplex Poisson kernel that solves a homogeneous complex Monge-Ampère equation with prescribed boundary singularity, generalizing the classical Poisson kernel to higher dimensions.

ABSTRACT

We study complex geodesics and complex Monge-Ampère equations on bounded strongly linearly convex domains in $\mathbb C^n$. More specifically, we prove the uniqueness of complex geodesics with prescribed boundary value and direction in such a domain, when its boundary is of minimal regularity. The existence of such complex geodesics was proved by the first author in the early 1990s, but the uniqueness was left open. Based on the existence and the uniqueness proved here, as well as other previously obtained results, we solve a homogeneous complex Monge-Ampère equation with prescribed boundary singularity, which was first considered by Bracci et al. on smoothly bounded strongly convex domains in $\mathbb C^n$.

Motivation & Objective

• To resolve the long-standing open problem of uniqueness for complex geodesics in bounded strongly linearly convex domains with minimal boundary regularity ($C^3$).
• To extend the theory of complex geodesics beyond $C^{m,\alpha}$ smoothness to domains with only $C^3$ boundary, ensuring robustness of the construction.
• To apply the uniqueness result to solve a homogeneous complex Monge-Ampère equation with prescribed boundary singularity, generalizing the classical Poisson kernel.
• To establish a connection between the pluricomplex Poisson kernel and the pluricomplex Green function via the Busemann function and boundary derivatives.
• To demonstrate that the constructed solution coincides with known solutions under stronger smoothness assumptions, validating consistency with prior work.

Proposed method

• Prove a boundary uniqueness result for complex geodesics in $C^3$ strongly linearly convex domains using deformation techniques and the Kobayashi-Royden metric.
• Construct a foliation of the domain by complex geodesics emanating from a fixed boundary point, leveraging the uniqueness result.
• Define the pluricomplex Poisson kernel $P_{\Omega,p}$ as the solution to the homogeneous complex Monge-Ampère equation $\mathrm{MA}(u) = 0$ with a prescribed singularity at $p \in \partial\Omega$.
• Use the dual mapping $\varphi^*$ of a complex geodesic $\varphi$ to derive the boundary behavior of the solution via the inner product $\langle \varphi', \overline{\varphi^*} \rangle = 1$.
• Establish the identity $P_{\Omega,p} \circ \varphi_v = -P / \langle v, \nu_p \rangle^2$ for all $v \in L_p$, linking the solution to geodesic dynamics.
• Relate the solution to the pluricomplex Green function via the Busemann function and the derivative $\partial g_\Omega / \partial \nu_p$, proving $P_{\Omega,p}(z,p) = -\partial g_\Omega / \partial \nu_p(z,p)$.

Experimental results

Research questions

• RQ1 Is the complex geodesic with prescribed boundary value and direction unique in a bounded strongly linearly convex domain with $C^3$ boundary?
• RQ2 Can the uniqueness of complex geodesics be established under minimal regularity assumptions, beyond $C^{m,\alpha}$ smoothness?
• RQ3 Does the existence and uniqueness of complex geodesics allow for the construction of a solution to a homogeneous complex Monge-Ampère equation with prescribed boundary singularity?
• RQ4 How does the constructed pluricomplex Poisson kernel relate to the classical Poisson kernel in the unit disc?
• RQ5 Is the solution $P_{\Omega,p}$ to the Monge-Ampère equation consistent with known solutions in the literature, such as those by Bracci-Patrizio or BPT?

Key findings

• The complex geodesic with a given boundary point and direction is uniquely determined in a bounded strongly linearly convex domain with $C^3$ boundary.
• The solution $P_{\Omega,p}$ to the homogeneous complex Monge-Ampère equation with a prescribed singularity at $p \in \partial\Omega$ is constructed via a foliation of complex geodesics.
• The constructed solution $P_{\Omega,p}$ satisfies $P_{\Omega,p} \circ \varphi_v = -P / \langle v, \nu_p \rangle^2$ for all $v \in L_p$, where $\varphi_v$ is the preferred complex geodesic.
• The solution $P_{\Omega,p}$ coincides with the one constructed by Bracci-Patrizio when the domain is $C^\infty$-smooth and strongly convex.
• The solution satisfies $P_{\Omega}(z,p) = -\partial g_\Omega / \partial \nu_p(z,p)$, linking it to the pluricomplex Green function and the Busemann function.
• The boundary of the pluripotential compactification of a strongly linearly convex domain with $C^3$ boundary is homeomorphic to the Euclidean boundary, extending a result of Poletsky to less regular domains.