Geometrical Properties of Solutions to −Δg u=f(u)

The study of level sets of solutions to partial differential equations, particularly for the semilinear elliptic problem (Formula presented.) on a Riemannian manifold (Formula presented.), has been a fundamental area in mathematical analysis. This type of equation appears in various physical models, including reaction-diffusion processes and phase transition phenomena. In this survey, we provide a historical overview and discussion of the main properties and results related to the level sets of solutions under zero Dirichlet boundary conditions. Additionally, we address the related problem of uniqueness of the critical point of the solution (Formula presented.).