$C^{k}$-regular extremal black holes in maximally-symmetric spacetime and the third law of black hole thermodynamics

In this work we extend the proof of Ryan Unger and Christoph Kehle's work, "Gravitational collapse to extremal black holes and the third law of black hole thermodynamics", to construct examples of black hole formation from regular, one-ended asymptotically flat Cauchy data for the Einstein-Maxwell charged scalar field system in maximally-symmetric 3+1 dimensional spacetime which are exactly isometric to $dS_{4}/AdS_{4}$ Reissner N\"ordstrom black holes after a finite advanced time along the event horizon. Furthermore, the apparent horizon coincides with that of a vacuum at finite advanced time. This paper exists as an extension of the aforementioned work done by Unger and Kehle which disproves the "third law of black hole thermodynamics" originally posed by Hawking and Bardeen's "The Four Laws of Black Hole Mechanics". We begin with a brief introduction to the history of black hole thermodynamics, tracing the the lineage of the third law of black hole thermodynamics up to Kehle & Unger's 2022 work; the basepoint for our extension to dS/AdS spacetime. We adapt the machinery from Kehle and Unger to Schwarzschild and Reissner-N\"ordstrom solutions in $dS_{4}$ and $AdS_{4}$ spacetime. Then, we study the relevant manifold gluing theory and reprove necessary gluing theorems in $dS_{4}$ and $AdS_{4}$. Finally, we provide analysis of the third law of black hole thermodynamics in these maximally symmetric spacetimes.