On Milnor’s Criterion for Deciding Whether a Surface is Hyperbolic or Parabolic for Biharmonic Functions

In 1977, John Milnor established a celebrated and nearly sharp criterion describing how curvature controls the existence of nonconstant bounded harmonic functions on R2 equipped with a rotationally symmetric metric. Biharmonic functions, natural higher-order analogues of harmonic functions and fundamental in elasticity theory and slow fluid flow, also exhibit a very rich behavior, yet no counterpart to Milnor’s theorem has been known. In this paper, we extend Milnor’s analysis to biharmonic functions on rotationally symmetric surfaces. We show that curvature bounds alone determine a complete trichotomy: whether bounded nonconstant biharmonic functions exist, whether every bounded biharmonic function must in fact be harmonic, or whether nontrivial bounded biharmonic functions arise that are not harmonic. As in Milnor’s original theorem, unexpected phenomena emerge, revealing a new and intriguing intermediate curvature regime in which biharmonic functions behave in a qualitatively different way.