Rigidity results on Liouville equation

TL;DR

This work provides a complete rigidity classification for bounded-above solutions to the Liouville equation $- abla^2 u = e^{2u}$ on $\mathbb{R}^2$ by linking $u$ to developing maps $f$ and employing Nevanlinna theory to analyze the associated linear ODEs. The authors establish a discrete growth spectrum for $k(u)=\limsup_{|z|\to\infty} u(z)/\log|z|$, show that radiality at infinity, axis-symmetric monotonicity, concavity, and metric diameter constraints impose strong one- or two-dimensional structures, and extend a concavity rigidity to higher dimensions via a constant rank framework. They classify all bounded-above two-dimensional solutions (up to conformal transformations) and describe the asymptotic and geometric behavior of finite-growth solutions through the Schwarzian, ODEs, and asymptotic analysis. In higher dimensions, they prove that concave solutions with an attaining maximum are necessarily one-dimensional, illustrating a powerful dimension-reduction principle for Liouville-type equations. These results illuminate connections between conformal geometry, meromorphic function theory, and rigidity phenomena in nonlinear elliptic PDEs with constant curvature metrics.

Abstract

We give a complete classification of solutions bounded from above of the Liouville equation $$-Δu=e^{2u}\quad\mbox{in}\quad {\mathbf{R}}^2.$$ More generally, solutions in the class $$N:=\{ u:\limsup_{z\to\infty} u(z)/\log|z|:=k(u)<\infty\}$$ are described. As a consequence, we obtain five rigidity results. First, $k(u)$ can take only a discrete set of values: either $k=-2$, or $2k$ is a non-negative integer. Second, $u\to-\infty$ as $z\to\infty$, if and only if $u$ is radial about some point. Third, if $u$ is symmetric with respect to $x$ and $y$ axes and $u_x<0,\; u_y<0$ in the first quadrant then $u$ is radially symmetric. Fourth, if $u$ is concave and bounded from above, then $u$ is one-dimensional. Fifth, if $u$ is bounded from above, and the diameter of ${\mathbf{R}}^2$ with the metric $e^{2u}δ$ is $π$, where $δ$ is the Euclidean metric, then $u$ is either radial about a point or one-dimensional. In addition, we extend the concavity rigidity result on Liouville equation in higher dimensions.

Theorems & Definitions (36)

• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Theorem 1.10
• Remark 2.1
• Remark 2.2
• Proposition 2.3
• Remark 2.4