The topological uniqueness of complete one-ended minimal surfaces and Heegaard surfaces in \( \mathbb{R}^ 3\). (English) Zbl 0886.57015
The authors prove two fundamental theorems on the topological uniqueness of certain surfaces in \(\mathbb{R}^3\). The first of these theorems, which will depend on the second theorem, shows that a properly embedded minimal surface in \(\mathbb{R}^3\) with one end is unknotted. More precisely,
Theorem 1.1. Two properly embedded one-ended minimal surfaces in \(\mathbb{R}^3\) of the same genus are ambiently isotopic.
Theorem 1.2. Heegaard surfaces of the same genus in \(\mathbb{R}^3\) are ambiently isotopic. Equivalently, given two diffeomorphic Heegaard surfaces in \(\mathbb{R}^3\), there exists a diffeomorphism of \(\mathbb{R}^3\) that takes one surface to the other surface.
Theorem 1.1. Two properly embedded one-ended minimal surfaces in \(\mathbb{R}^3\) of the same genus are ambiently isotopic.
Theorem 1.2. Heegaard surfaces of the same genus in \(\mathbb{R}^3\) are ambiently isotopic. Equivalently, given two diffeomorphic Heegaard surfaces in \(\mathbb{R}^3\), there exists a diffeomorphism of \(\mathbb{R}^3\) that takes one surface to the other surface.
Reviewer: He Baihe (Changchun)
MSC:
| 57N12 | Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010) |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
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