Generalization of Liouville's theorem in higher dimensions

Abstract

The generalisation of Liouville’s theorem in higher dimensions expands the classical principle that bounded entire functions must be constant, extending its influence to complex variables, harmonic functions, nonlinear elliptic equations, and nonlocal operators. This broader framework demonstrates how structural conditions such as boundedness, growth restrictions, integrability, and geometric constraints enforce rigidity and nonexistence of nontrivial solutions in multidimensional settings. Modern Liouville-type results play a crucial role in classifying entire solutions, preventing blow-up phenomena, and understanding asymptotic behaviour in partial differential equations. They also provide essential tools in geometric analysis, especially on manifolds where curvature and topology influence solution properties. This study synthesises key developments and highlights the analytical mechanisms that underpin the persistence of Liouville-type rigidity across diverse mathematical contexts.