On Some Variational Problems in Curves and Surfaces

In this dissertation, we primarily investigate two variational problems involving curves and surfaces. We begin by introducing the variational problems in the context of curves and surfaces, followed by an overview of the spaces in which we work---specifically, spheroidal surfaces and Lorentzian 2-spaces. We then derive the first variation formula for specific Lorentzian spaces, as well as both the first and second variation formulas for surfaces, focusing on the study of special types of surfaces known as minimal surfaces within a spheroid. The first problem concerns the study of $p$-elastic curves in two space forms. Specifically, we explore the existence and conditions for closed $p$-elastic curves in the hyperbolic plane and de Sitter 2-space. Additionally, we investigate connections to closed $p$-elastic curves in the 2-sphere, as well as in hyperbolic and pseudo-hyperbolic planes.The second problem focuses on free boundary minimal surfaces and their Morse index in spheroidal domains. The computation of the Morse index for free boundary minimal surfaces in convex domains remains an open problem. Here, we consider a special case as a step toward generalizing results to convex domains. We first introduce free boundary minimal surfaces in a spheroid and compute the Morse index for specific examples. To address this problem, we separate it into two simpler subproblems: the fixed boundary problem and the Dirichlet-to-Neumann map. By leveraging these, we seek an approximation of the Morse index for a spheroid with radii $1$ and $a$, where $a$ is any positive real number. In particular, we prove the existence of free boundary minimal surfaces and compute the Morse index of equatorial disks. Furthermore, we provide numerical evidence toward the Morse index of the critical catenoid.