A Note on Existence and Non-existence of Minimal Surfaces in Some Asymptotically Flat 3-manifolds

Motivated by problems on apparent horizons in general relativity, we prove the following theorem on minimal surfaces: Let g be a metric on the three-sphere S satisfying Ric(g) ≥ 2g. If the volume of (S, g) is no less than one half of the volume of the standard unit sphere, then there are no closed minimal surfaces in the asymptotically flat manifold (S \ {P}, Gg). Here G is the Green’s function of the conformal Laplacian of (S, g) at an arbitrary point P . We also give an example of (S, g) with Ric(g) > 0 where (S \ {P}, Gg) does have closed minimal surfaces.