All Theses, Dissertations, and Capstone Projects
Year of Award
Doctor of Philosophy (PhD)
College of Arts & Sciences
Math and Computer Science
cone, structure, points, quotient, symmetry, geometry, gyroid, corresponding
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 Gdh, 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.
Open Access Dissertation
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.
Weyhaupt, Adam G., "New Families of Embedded Triply Periodic Minimal Surfaces of Genus Three in Euclidean Space" (2006). All Theses, Dissertations, and Capstone Projects. 160.