An analysis of the height and distance functions through new types of p-parabolic operators

Abstract

The objective of this thesis is to study the rigidity of submanifolds of codimension greater than or equal to 1 using two of the most intuitive functions: the height function and the distance function. In the first part of this thesis, we study the p-parabolicity of submanifolds immersed in the Riemannian warped product Mn+k ×ρ R. As the main result, we establish a rigidity result that verifies the existence or absence of submanifolds within slices of the warped

product through the height function. The results were obtained by generalizing the drift- Laplacian, here referred to as the (p, ψ)-Laplacian. With this generalization, we present

new conditions for obtaining rigidity results for p-parabolic submanifolds.

We dedicate the second part of this thesis to studying the implications of the infinity- Laplacian operator in the context of rigidity theorems. Initially, we analyze the application

of the operator alongside the height function within the Riemannian framework of the two main cases of warped products: Mn+k ×ρ R and I ×f Mn+k

. Thus, we introduce new characterizations for obtaining rigidity results into the theory of complete submanifolds. In the last part of this thesis, we study the behavior of the distance function through the infinity-Laplacian operator in two settings: the Riemannian and the Lorentzian, focusing on spacelike submanifolds. Here, we impose certain conditions on the submanifolds and use one of the most classical tools of geometric analysis, the Hessian comparison theorem. The rigidity results obtained here involve the determination of submanifolds within level curves.