Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics

Abstract

Abstract. We prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the Regge-Teitelboim condition near infinity. It is known that the (Hamiltonian) center of mass is well-defined for manifolds satisfying this condition. We also show that the foliation is asymptotically concentric, and its geometric center is the center of mass. The construction of the foliation generalizes the results of Huisken-Yau, Ye, and Metzger, where strongly asymptotically flat manifolds and their small perturbations were studied.

On the center of mass in general relativity

“…In [12], we generalized the earlier results of the constant mean curvature foliation to asymptotically flat manifolds with the RT condition, which is a natural condition to impose when center of mass is discussed. Furthermore, the foliation that we constructed is unique under some conditions.…”

“…One may consider to choose a better asymptotically flat coordinate chart, but it seems hard to handle and simplify all the summation terms in (3.1). Instead, we explicitly construct a family of approximate spheres S(p, R) which reflect the asymptotics better than the coordinate spheres for each p and large R. Moreover, from our construction, the approximate spheres also adapt better the RT condition [12].…”

Equality in the Spacetime Positive Mass Theorem

“…It is well-known that we can express E as a flux integral involving the Ricci curvature (see, for example, [14,18]) and thus S∞ −aR ij x i ν j dH n−1 = (n − 1)(n − 2)ω n−1 aE.…”

Foliations of asymptotically flat manifolds by surfaces of Willmore type

“…Then using the Gauss equation (4), the Codazzi equation (6) and equation (3), the Simons identity [22] becomes…”