A Note on Existence and Non-existence of Horizons in Some Asymptotically Flat $3$-manifolds

We consider asymptotically flat manifolds of the form $(S^3 \setminus \{ P \}, G^4 g)$, where $G$ is the Green’s function of the conformal Laplacian of $(S^3, g)$ at a point $P$. We show if $Ric(g) \geq 2 g$ 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 $(S^3 \setminus \{ P \}, G^4 g)$. We also give an example of $(S^3, g)$ where $Ric(g) > 0$ but $(S^3 \setminus \{ P \}, G^4 g)$ does have closed minimal surfaces.

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Published 1 January 2007