CSU Physics Colloquium

Solution of the 35-Year-Old Far-From-CMC Existence Problem for the Einstein Constraint Equations

Dr. Mike Holst

University of California, San Diego

Wed,  11/4

**Note the special date and time**

There is currently tremendous interest in geometric partial differential equations (PDE). They play a primary role in general relativity, where the (constrained) Einstein evolution equations describe the propagation of gravitational waves generated by collisions of massive objects such as black holes. The need to validate this model has led to the construction of gravitational wave detectors such as the NSF-funded LIGO project. In this lecture, we consider the non-dynamical subset of the Einstein equations called the Einstein constraints; this coupled nonlinear elliptic system must be solved numerically to produce initial data, and to enforce the constraints during dynamical evolution. The constraints have been studied intensively for half a century; our focus is on a thirty-five-year-old open question involving existence of solutions to the constraint equations on space-like hyper-surfaces with arbitrarily prescribed mean extrinsic curvature. All known existence results have involved assuming either constant (CMC) or nearly-constant (near-CMC) mean extrinsic curvature. After giving a survey of known results through 2007, we outline a new theoretical framework for establishing existence of solutions that is fundamentally free of both CMC and near-CMC conditions, and allows for “weak” background metrics of low differentiability. We then use the framework to give the first known existence results for arbitrarily prescribed mean extrinsic curvature. This is joint work with Gabriel Nagy and Gantumur Tsogtgerel. (Physical Review Letters, Vol. 100 (2008), No. 16, pp. 161101.1, http://arxiv.org/abs/0712.0798; and Comm. Math. Phys., Vol. 288 (June 2009), No. 2, pp. 547-613, http://arXiv:gr-qc/0712.0798)