A note on existence and non-existence of horizons in some asymptotically flat 3-manifolds

摘要

We consider asymptotically flat manifolds of the form (S-3 backslash{P}, G(4)g), where G is the Green's function of the conformal Laplacian of (S-3, g) at a point P. We show if Ric(g) >= 2g and the volume of (S-3, g) is no less than one half of the volume of the standard unit sphere, then there are no closed minimal surfaces in (S-3 backslash{P}, G(4)g). We also give an example of (S-3, g) where Ric(g) > 0 but (S-3 backslash{P}, G(4)g) does have closed minimal surfaces.