On the existence of minimal Heegaard surfaces
| Autor: | Ketover, Daniel, Liokumovich, Yevgeny, Song, Antoine |
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| Rok vydání: | 2019 |
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| Druh dokumentu: | Working Paper |
| Popis: | Let $H$ be a strongly irreducible Heegaard surface in a closed oriented Riemannian $3$-manifold. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a non-orientable minimal surface with a vertical handle attached. This confirms a long-standing conjecture of J. Pitts and J.H. Rubinstein. In the case of positive scalar curvature, we show for spherical space forms not diffeomorphic to $\mathbb{S}^3$ or $\mathbb{RP}^3$ that any strongly irreducible Heegaard splitting is isotopic to a minimal surface, and that there is a minimal Heegaard splitting of area less than $4\pi$ if $R\geq 6$. Comment: This preprint supersedes both arXiv:1706.01037 [math.DG] and arXiv:1709.09744 [math.DG] |
| Databáze: | arXiv |
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