On the geometry of surfaces in manifolds with density

Abstract

This thesis explores the geometry of surfaces in pseudo-Riemmennian manifolds with density, an important extension of classical geometric analysis that provides a natural framework for studying variational problems, curvature, and minimal surfaces under weighted volume measures. A manifold with density is a Riemannian or pseudo-Riemannian manifold (M, g) endowed with a smooth weight function ρ = eφ, which modifies the standard geometric structure and leads to generalized notions of curvature, mean curvature, and divergence. Such weighted structures arise naturally in probability theory, diffusion processes, and the study of geometric flows. In this work, we investigate the geometric properties of surfaces in Euclidean space R3 and in Minkowski space R31 , both endowed with a density function ρ = eφ. After reviewing the necessary preliminaries on differential geometry and pseudo-Riemannian manifolds, we develop and analyze weighted curvature formulas, including the generalized Gauss curvature and mean curvature. We also examine Plateau-type problems and study the behavior of the weighted divergence operator in manifolds with density, supported by explicit examples. The main contributions of this thesis focus on the classification and geometric analysis of weighted flat translation surfaces in Minkowski 3-space R31 and weighted flat radial surfaces in both Euclidean R3 and Minkowski R31 spaces. By deriving the corresponding flatness conditions and curvature relations, we extend previous results on linear densities and establish new characterizations of surfaces governed by weighted geometric structures. These results provide deeper insight into minimal and flat surfaces in weighted settings and offer a foundation for future research in differential geometry with density.