Two new embedded triply periodic minimal surfaces of genus 4
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Daniel Freese · Matthias Weber
· A. Thomas Yerger · Ramazan Yol
Two new embedded triply periodic minimal surfaces of genus 4 Received: 29 January 2020 / Accepted: 8 September 2020 Abstract. We add two new 1-parameter families to the short list of known embedded triply periodic minimal surfaces of genus 4 in R3 . Both surfaces can be tiled by minimal pentagons with two straight segments and three planar symmetry curves as boundary. In one case (which has the appearance of the CLP surface of Schwarz with an added handle) the two straight segments are parallel, while they are orthogonal in the second case. The second family has as one limit the Costa surface, showing that this limit can occur for triply periodic minimal surfaces. For the existence proof we solve the 1-dimensional period problem through a combination of an asymptotic analysis of the period integrals and geometric methods.
1. Introduction We construct two new, closely related 1-parameter families of embedded triply periodic minimal surface of genus 4 in Euclidean space. These surfaces are interesting for several reasons: First, by a result of Meeks [5], a triply periodic minimal surface of genus 4 cannot be hyperelliptic, limiting the known construction methods for these surfaces. In fact, the available list of examples is rather small: They consist of Alan Schoen’s H –T, I-WP, and S −S surfaces [4,6], as well as several numerically constructed examples that to the authors’ knowledge have never been described in detail. One particularly effective construction method that is still available is due to Traizet [8,9]: He is able to construct triply periodic minimal surfaces of any genus g > 2 that resemble horizontal planes joined by catenoidal necks. Our surfaces, however, have more complicated limits. Indeed, one of the families limits on one side in the Costa surface so that one could call it a triply periodic Costa surface. There exist other examples (of higher genus) with the appearance of a triply periodic Costa surface (see Batista’s surface [1] and Alan Schoen’s I6 surface, called Figure 8 annulus in [4]), but these examples do not truly limit in the Costa surface but rather in the singly periodic Callahan–Hoffman–Meeks surface ([2]). This is significant if one wants to extend Traizet’s regeneration construction to employ more general necks than the catenoidal ones: Our example M. Weber (B): Department of Mathematics, Indiana University, Bloomington, IN 47405, USA. e-mail: [email protected] Mathematics Subject Classification: 49Q05
https://doi.org/10.1007/s00229-020-01244-9
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Fig. 1. Translational fundamental pieces
suggests it should be possible to use Costa necks joining three consecutive planes. A Callahan–Hoffman–Meeks limit would require an entirely different gluing procedure, involving cutting off a Callahan–Hoffman–Meeks surface by a cylinder, glued to the complement of a solid vertical cylinder in a family of horizontal planes at finite distance fr
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