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

• Autores: Pengzi Miao
• Localización: Mathematical research letters, ISSN 1073-2780, Vol. 14, Nº 3, 2007, págs. 395-402
• Idioma: inglés
• DOI: 10.4310/mrl.2007.v14.n3.a4
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• Resumen

  • 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.