Minimal surfaces and Heegaard splittings of the three-torus
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.
scientific article; zbMATH DE number 3975884