Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 5

Conformal fields and the structure of the space of solutions of the Einstein constraint equations

Pages: 1157 – 1201



Michael Holst (Department of Mathematics, University of California at San Diego, La Jolla, Calif., U.S.A.)

David Maxwell (Department of Mathematics and Statistics, University of Alaska, Fairbanks, Ak., U.S.A.)

Rafe Mazzeo (Department of Mathematics, Stanford University, Stanford, Calif., U.S.A.)


The drift method, introduced in [22], provides a new formulation of the Einstein constraint equations, either in vacuum or with matter fields. The natural of the geometry underlying this method compensates for its slightly greater analytic complexity over, say, the conformal or conformal thin sandwich methods. We review this theory here and apply it to the study of solutions of the constraint equations with non-constant mean curvature. We show that this method reproduces previously known existence results obtained by other methods, and does better in one important regard. Namely, it can be applied even when the underlying metric admits conformal Killing (but not true Killing) vector fields. We also prove that the absence of true Killing fields holds generically.

This work was supported by NSF grant 1263544. The first author was also supported by NSF grants DMS-1262982 and DMS-1620366. The third author was also supported by NSF grant DMS-1608223.

Published 30 March 2023