Embedding Misner and Brill–Lindquist Initial Data for Black-Hole Collisions

In this article we consider the isometrical immersions into Euclidean three-space of two-dimensional slices of the Misner and the Brill–Lindquist initial data for black-hole collisions. We show negativity of curvature and deduce other geometric properties of the slices. Under the assumption that ends behave strongly like paraboloid of revolution, we prove that Misner and the Brill–Lindquist slices cannot be isometrically immersed into R³. This condition on an end is natural in general relativity because it holds for each end of a slice of the Schwartzschild metric where it is embedded as a paraboloid of revolution.