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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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