Sobre princípios do máximo relacionados ao crescimento de volume e suas aplicações

Abstract

This work is divided into three parts. In the first part, the objective is to generalize the maximum principle for smooth functions f in complete and noncompact Riemannian manifolds M with polynomial or exponential Volume growth, for which there is a vector field X whose norm is estimated by the distance function to the power of k, for k ∈ [0, 1], such that ⟨∇f, X⟩ ≥ 0 in M and div X ≥ af outside a closed subset of M, for some positive function a ∈ C1(M) such that ⟨∇a, X⟩ ≥ 0. With this result at hand, we will prove Bernstein-type rigidity theorems for oriented hypersurfaces immersed in a Riemannian manifold with a closed conformal field. For the second part, we will extend this maximum principle to weighted Riemannian manifolds σ, in addition to proving Bernstein-type rigidity theorems for oriented hypersurfaces embedded in a weighted Riemannian manifold endowed with a closed conformal field. Finally, in the last part, we will study applications of these maximum principles in Lorentzian manifolds, starting with spacelike hypersurfaces of conformally stationary spacetimes, passing through spacelike hypersurfaces in generalized Robertson-Walker spacetimes and ending with spacelike hypersurfaces in ppwave spacetimes.