Outermost apparent horizons diffeomorphic to unit normal bundles (Q2194548)

In this paper, the authors study asymptotically Euclidean metrics, which are Riemannian manifolds which have an end where the metric converges to that of Euclidean flat space. In particular, they study the \emph{outermost apparent horizon}, which is defined to be a bounding minimal hypersurface which encloses all other bounding minimal hypersurfaces. The main motivation of this paper is to study the following question: Is the existence of a positive scalar curvature metric sufficient for a compact, bounding manifold to be the outermost apparent horizon in an asymptotically Euclidean manifold of nonnegative scalar curvature? This question is related to the large-scale geometry of spacetime in general relativity, so has applications in mathematical physics. The authors are not able to answer this question, but are able to construct many new examples of outermost apparent horizons in a systematic way. The main result of this paper is to show that given a smooth submanifold \(S \subset \mathbb{R}^n\) of codimension at least three, it is possible to construct a Riemannian metric of \(\mathbb{R}^n \backslash S\) which is asymptotically Euclidean and for which the outermost apparent horizon is diffeomorphic to the unit normal bundle of \(S\). The authors do this using an explicit metric which is defined in terms of the Green's function of the Euclidean Laplacian. As a result of this theorem, they show that horizons can be diffeomorphic to products of spheres, have many components, or even have arbitrary fundamental group. Going further, they find that given compact a embedded smooth submanifold of codimension at least 3, then there is an asymptotically Euclidean metric on \(\mathbb{R}^n\) with non-negative scalar curvature for which the outermost apparent horizon is diffeomorphic to the unit normal bundle. Furthermore, the horizon is the graph of a smooth function on the unit normal bundle in terms of normal coordinates for \(S\).