On Free Boundary Minimal Planes in the Riemannian Reissner-Nordstrom Manifold

Understanding the geometry of black holes is crucial for physicists. Montezuma's groundbreaking work computed the index of a totally geodesic plane in the three-dimensional Riemannian Schwarzschild manifold. Extending this analysis to the Reissner-Nordström manifold, a broader framework encompassing charged black holes, presents an opportunity to deepen our understanding of the shape of the ambient black hole. In this thesis, we will first review the necessary definitions from the theory of minimal submanifolds and introduce Reissner–Nordström manifold. After that, we will discuss Montezuma's results and attempt to extend some of them to the case of the Reissner-Nordström manifold. Our goal is to estimate the Morse index depending on the charge of the black hole. In particular, we will show that the Morse index of the plane is at least 1.