What We Know and Don’t Know about the Space of Solutions of the Einstein Constraint Equations

Wednesday, November 9, 2016 -
3:00pm to 4:00pm
Ten years ago, Robert Bartnick and I wrote a review article summarizing what was known at the time about the Einstein constraint equations and their solutions. In that article, we noted that while much  was known about solutions of the constraints which have constant mean curvature (CMC) or are nearly CMC, very little was known about solutions which are far from CMC. The hope at the time was  that the effectiveness of the conformal method for constructing CMC and near-CMC solutions would (perhaps after much work) extend to far-from-CMC solutions.   In the years since that article appeared, new results have slowly been obtained. While some have been consistent with this optimistic view, many others have shown that the picture for far-from CMC solutions is likely to be much more complicated. After a brief survey of what the conformal method is and what it has shown us for CMC and for near-CMC solutions, we survey a variety of new results which show how complicated things can become for far-from-CMC solutions, and for solutions which include a cosmological constant (of the DeSitter type). We also note some very recent results which fill in some holes in our understanding of CMC and near-CMC solutions. 
Jim Isenberg
University of Oregon
Event Location: 
Fine Hall 314