On Milnor's criterion for deciding whether a surface is hyperbolic or parabolic for biharmonic functions

TL;DR

This work extends Milnor's criterion from harmonic to biharmonic functions on rotationally symmetric surfaces by analyzing $M=\mathbb{R}^2$ with $g=dr^2+\phi(r)^2 d\theta^2$. It uses a Fourier/separation-of-variables approach to represent biharmonic functions via radial factors $\varphi_m$ and $\psi_m$, and leverages Plancherel theory and curvature-driven growth estimates to constrain the coefficients. The authors prove a Milnor-type theorem: bounded biharmonic functions are constant under $K\ge -\dfrac{1}{r^2\log r}$ with $\phi(r)\to\infty$, and they show a complementary regime where curvature decay $K\le -r^{2+\epsilon}$ admits bounded nonharmonic biharmonics, while an intermediate regime enforces harmonicity. These results illuminate how curvature decay governs polyharmonic behavior on rotationally symmetric surfaces and extend Liouville-type principles to biharmonic equations.

Abstract

In this paper we generalise a celebrated result of Milnor that characterises whether a rotationally symmetric surface is parabolic or hyperbolic to the case of biharmonic functions.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1: Comparison Lemma
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 4