3
$\begingroup$

I am trying to read the paper by Schoen and Yau, Proof of the Positive Mass Theorem II. The notation is very different from what I am familiar with (basically Robert Wald's book on general relativity). In particular, they use Cartan's moving frame method and their structure equation is of different sign from what I have seen. Can anyone point me to a differential geometry reference that is of same notation as Schoen&Yau paper?

Any references on PDE or related toward understanding Schoen-Yau paper is also greatly appreciated!

Just in case it would be helpful to anyone else. For Schoen and Yau 1979 proof, Fernado Marques did a very helpful lecture covering it (recording available at https://video-archive.fields.utoronto.ca/list/event/1367) and also his lecture note (https://web.math.princeton.edu/~rcabral/pdfs/minimalsurfaces.pdf).

Thanks!

Improve this question
$\endgroup$
3
  • 5
    $\begingroup$ I think Dan Lee's Geometric Relativity book (AMS GTM201) has an entire chapter (chapter 8) devoted to the space-time PMT. I don't know how closely it follows Schoen and Yau's original proof, as Dan has also worked on a more modern version recently so that may have influenced his writing. (Certainly for the Riemannian PMT Dan "cheats" by using Lohkamp's reduction so did not fully follow Schoen and Yau I.) $\endgroup$
    – Willie Wong
    Commented Dec 6, 2023 at 4:50
  • $\begingroup$ Thank you! I will check it out. $\endgroup$
    – Bowen Zhao
    Commented Dec 6, 2023 at 8:10
  • $\begingroup$ It's exactly what I wanted. Thank you @WillieWong $\endgroup$
    – Bowen Zhao
    Commented Dec 7, 2023 at 4:14

1 Answer 1

Reset to default
3
$\begingroup$

Already found one that might be helpful so I share it here, notes for a short course by Schoen at Tsinghua unversity, https://web.math.ucsb.edu/~zhou/Math_GR_2012.pdf

Also, Yau's own explanation in a lecture at Harvard https://www.youtube.com/watch?v=qCIpxab8WDc&list=PL5ahlY3eFvHYrtDc_8Y-iO2uOQjz8sslC&index=3

Thanks to Willie Wong for pointing out Dan Lee's Geometric Relativity book, which gives the most up-to-date and complete summary on PMT proof. (Still Schoen's own lecture note is very helpful for building intuition about Jang's equation.)

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .