The dissertation is done!

Actually I finished it yesterday around 2:30 pm. But I haven't had the chance to blog about it until now. Something about having a late afternoon seminar yesterday, more seminars today, and working on a few new ideas. The topic is "Uniqueness of Kerr-Newman Black Holes." Here's the abstract:

The uniqueness of the Kerr-Newman family of black hole metrics as stationary asymptotically flat solutions to the Einstein equations coupled to a free Maxwell field is a crucial ingredient in the study of final states of the universe in general relativity. If one imposes the additional requirement that the space-time is axial-symmetric, then said uniqueness was shown by the works of B. Carter, D.C. Robinson, G.L. Bunting, and P.O. Mazur during the 1970s and 80s. In the real-analytic category, the condition of axial symmetry can be removed through S. Hawking's Rigidity Theorem. The necessary construction used in Hawking's proof, however, breaks down in the smooth category as it requires solving an ill-posed hyperbolic partial differential equation. The uniqueness problem of Kerr-Newman metrics in the smooth category is considered here following the program initiated by A. Ionescu and S. Klainerman for uniqueness of the Kerr metrics among solutions to the Einstein vacuum equations. In this work, a space-time, tensorial characterization of the Kerr-Newman solutions is obtained, generalizing an earlier work of M. Mars. The characterization tensors are shown to obey hyperbolic partial differential equations. Using the general Carleman inequality of Ionescu and Klainerman, the uniqueness of Kerr-Newman metrics is proven, conditional on a rigidity assumption on the bifurcate event horizon.

The paper will not appear on the internet anytime soon. Definitely not before I get reader's reports from the professors. And perhaps even not before my defense. My officemate, Pin Yu, just told me yesterday of an idea that he had (thinking about which took up much of my day today) that if successful, can slightly reduce the complexity of the proof used in my dissertation. If we manage to work this out, then the published version will probably use this new framework. I'll have to think about whether or not to put my dissertation on arXiv a bit more.

I still have some proof reading to do in regards to the thesis. I'll need to get a final PDF to the printers before next Wednesday so I can have the bound copy in time.