Finite-dimensionality conjecture for bounded biharmonic functions on manifolds with nonnegative Ricci curvature

Let MM be an open manifold with nonnegative Ricci curvature, and let uu be a bounded biharmonic function on MM, meaning that abla4u=0 abla^4 u=0 in the source's terminology via the biharmonic equation Delta2u=0Delta^2u=0. Bounded biharmonic-function conjecture. Every bounded biharmonic function on MM is constant. The source presents this as a natural conjecture because bounded biharmonic functions can be nonconstant on some complete noncompact manifolds, while the resolution status is not specified.

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Primary source

Lin Wang and Miaomiao Zhu, “The qualitative behavior for biharmonic functions on open manifolds”, arXiv:2511.09393 (2025).

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