Minimal surfaces and Heegaard splittings of the three-torus
type: Item
instance of: scholarly article
title: Minimal surfaces and Heegaard splittings of the three-torus
zbMATH Open document ID: 0604.57006
published in: Pacific Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 57R40
zbMATH DE Number: 3975884
zbMATH Keywords: three-torus
zbMATH Keywords: one-sided Heegaard splitting
zbMATH Keywords: minimal surface in a flat metric
zbMATH Keywords: minimal genus
author: Charles Frohman
MaRDI profile type: Publication
full work available at URL: pjm.1986.124.119
OpenAlex ID: W2064635200
DOI: 10.2140/PJM.1986.124.119
description: scientific article; zbMATH DE number 3975884