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

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.

### MSC:

 53C12 Foliations (differential geometric aspects)

### Keywords:

foliation; Riemannian manifold; mean curvature
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