Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature. (English) Zbl 1140.53013

Metzger, Jan

J. Differ. Geom. 77, No. 2, 201-236 (2007).

Given a three-dimensional Riemannian manifold \((M, g)\) and an additional symmetric bilinear form \(K\) on \(M\), being sufficiently asymptotically flat, the author constructs a regular foliation of \(M\) by closed convex surfaces \(\Sigma\) with mean curvature \(H\) satisfying the equation \(H + \text{tr}^\Sigma (K) = h\), where \(h\) is a constant and \(\text{tr}^\Sigma (K)\) is the 2-dimensional trace of \(K\). (The precise statement of the result is very technical, hence not provided here.) The author provides interesting interpretation of the result in terms of physics. Among others, \(K\) may be considered as the extrinsic curvature of \(M\) in the surrounding 4-dimensional space-time.

Reviewer: Paweł Walczak (Łódz)

MSC:

53C12

Foliations (differential geometric aspects)

Keywords:

foliation; Riemannian manifold; mean curvature