mirage

# New Families of Embedded Triply Periodic Minimal Surfaces of Genus Three in Euclidean Space

## DSpace/Manakin Repository

 dc.contributor.advisor Weber, Matthias en_US dc.contributor.author Weyhaupt, Adam Gilbert en_US dc.date.accessioned 2010-06-01T22:03:51Z dc.date.available 2011-04-16T17:06:07Z dc.date.issued 2010-06-01T22:03:51Z dc.date.submitted 2006 en_US dc.identifier.uri http://hdl.handle.net/2022/7765 dc.description Thesis (PhD) - Indiana University, Mathematics, 2006 en_US dc.description.abstract Until 1970, all known examples of embedded triply periodic minimal surfaces (ETPMS) contained either straight lines or curves of planar symmetry. In 1970, Alan Schoen discovered the gyroid, an ETPMS that contains neither straight lines nor planar symmetry curves. Meeks discovered in 1975 a 5-parameter family of genus 3 ETPMS that contained all known examples of genus 3 ETPMS except the gyroid. A second example lying outside the Meeks family was proposed by Lidin in 1990. Grosse-Brauckmann and Wohlgemuth showed in 1996 the existence and embeddedness of the gyroid and Lidinoid''. In a series of investigations the scientists, Lidin, et. al., numerically indicate the existence of two 1-parameter families of ETPMS that contain the gyroid and one family that contains the Lidinoid. In this thesis, we prove the existence of these families. To prove the existence of these families, we describe the Riemann surface structure using branched covers of non-rectangular tori. The holomorphic 1-forms Gdh, 1/G dh, and dh each place a cone metric on the torus; we develop the torus with this metric into the plane and describe the periods in terms of these flat structures. Using this description of the periods, we define moduli spaces for the horizontal and vertical period problems so that Weierstrass data (X, G, dh) solves the period problem if the flat structures of X induced by these 1-forms are in the moduli spaces. To show that there is a curve of suitable data, we use an intermediate value type argument. en_US dc.language.iso EN en_US dc.publisher [Bloomington, Ind.] : Indiana University en_US dc.subject triply periodic minimal surfaces en_US dc.subject gyroid en_US dc.subject Lidinoid en_US dc.subject minimal surfaces en_US dc.subject deformations of minimal surfaces en_US dc.subject.classification Mathematics (0405) en_US dc.title New Families of Embedded Triply Periodic Minimal Surfaces of Genus Three in Euclidean Space en_US dc.type Doctoral Dissertation en_US
﻿