A Liouville-type theorem for the p-Laplacian on complete non-compact Riemannian manifolds
Matheus Nunes Soares, Fábio Reis dos Santos
TL;DR
This work proves a Liouville-type theorem for the $p$-Laplacian on complete non-compact Riemannian manifolds: if the first eigenvalue $λ_{1,p}(Σ)$ vanishes, every bounded strongly $p$-subharmonic function must be unbounded, extending known results to $p\ge2$. The authors develop a linearization $P_u$ of the $p$-Laplacian, decompose it into a linear part $L_u$ and a nonlinear remainder, and establish a key eigenvalue lower bound (Lemma 2.1) for functions with $Δ_p f\ge1$. They then apply this Liouville-type result to geometric settings of hypersurfaces in warped product manifolds, deriving expressions for the height and warped-height functions and proving nonexistence or rigidity results for complete non-compact hypersurfaces contained in slabs under various curvature and volume-growth conditions, including pseudo-hyperbolic and helix-type cases. The results provide a bridge between spectral properties of the $p$-Laplacian and geometric constraints on isometric immersions in warped products, with potential implications for rigidity phenomena in geometric analysis.
Abstract
A Liouville-type result for the p-Laplacian on complete Riemannian manifolds is proved. As an application are present some results concerning complete non-compact hypersurfaces immersed in a suitable warped product manifold.