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

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

2006-01-19

arxiv.org/abs/math/0601480v1

Math. Res. Lett. 14, no. 3, 395-402 (2007)

Mathematics

Differential Geometry

10 pages

Scientific paper

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 $S^3$ satisfying $Ric(g) \geq 2 g$. If 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 the asymptotically flat manifold $(S^3 \setminus \{P \}, G^4 g)$. Here $G$ is the Green's function of the conformal Laplacian of $(S^3, g)$ at an arbitrary point $P$. We also give an example of $(S^3, g)$ with $Ric(g) > 0$ where $(S^3 \setminus \{P \}, G^4 g)$ does have closed minimal surfaces.

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