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Minimal surfaces and Heegaard splittings of the three-torus. (English) Zbl 0604.57006
The author proves two theorems on minimal surfaces and Heegaard splittings of the three-torus T. Theorem 2.1. A one-sided Heegaard splitting of T of the (minimal possible) genus 4 is unique up to homeomorphism. Theorem 3.1. An orientable closed genus 3 surface in T that can be a minimal surface in a flat metric on T is unique up to homeomorphism. (Note that such a surface is automatically a two-sided Heegaard surface of minimal genus.)
The proof of both theorems is based on the following unknotting criterion. Let F be a closed surface of positive genus, and let K be a proper arc in \(F\times I\) connecting \(F\times 0\) with \(F\times 1\). The arc K is isotopic to *\(\times I\) if and only if the exterior of K in \(F\times I\) is a handlebody.
Reviewer: V.Turaev

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
57R40 Embeddings in differential topology
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