A new deformation family of Schwarz’ D surface

  title={A new deformation family of Schwarz’ D surface},
  author={Hao Chen and Matthias J. Weber},
  journal={arXiv: Differential Geometry},
We prove the existence of a new 2-parameter family o$\Delta$ of embedded triply periodic minimal surfaces of genus 3. The new surfaces share many properties with classical orthorhombic deformations of Schwarz' D surface, but also exotic in many ways. In particular, they do not belong to Meeks' five-dimensional family. Nevertheless, o$\Delta$ meets classical deformations in a 1-parameter family on its boundary. 

Figures from this paper

An orthorhombic deformation family of Schwarz’ H surfaces
The classical H surfaces of H. A. Schwarz form a 1-parameter family of triply periodic minimal surfaces (TPMS) that are usually described as close relatives to his more famous P surface. However, a
Stacking Disorder in Periodic Minimal Surfaces
The construction of 1-parameter families of non-periodic embedded minimal surfaces of infinite genus in T denotes a flat 2-tori and can be interpreted as disordered stacking of layers of periodically arranged catenoid necks.
Existence of the tetragonal and rhombohedral deformation families of the gyroid
We provide an existence proof for two 1-parameter families of embedded triply periodic minimal surfaces of genus three, namely the tG family with tetragonal symmetry that contains the gyroid, and the


Genera of minimal balance surfaces
The genus of a three-periodic intersection-free surface in R3 refers to a primitive unit cell of its symmetry group. Two procedures for the calculation of the genus are described: (1) by means of
On the genus of triply periodic minimal surfaces
we prove the existence of embedded minimal surfaces of arbitrary genus g 3 in any at 3-torus. In fact we construct a sequence of such surfaces converging to a planar foliation of the 3-torus. In
On bifurcation and local rigidity of triply periodic minimal surfaces in $\mathbb R^3$
We use bifurcation theory to determine the existence of infinitely many new examples of triply periodic minimal surfaces in $\mathbb R^3$. These new examples form branches issuing from the H-family,
Uniqueness of the Riemann minimal examples
Abstract. We prove that a properly embedded minimal surface in R 3 of genus zero with infinite symmetry group is a plane, a catenoid, a helicoid or a Riemann minimal example. We introduce the
Parametrization of triply periodic minimal surfaces. II. Regular class solutions
A derivation is given of the set of triply periodic minimal surfaces of monoclinic symmetry and higher that fall within the regular class (including those containing self-intersections). The Gauss
Deformations of the gyroid and lidinoid minimal surfaces
The gyroid and Lidinoid are triply periodic minimal surfaces of genus 3 embedded in R that contain no straight lines or planar symmetry curves. They are the unique embedded members of the associate
New Families of Embedded Triply Periodic Minimal Surfaces of Genus Three in Euclidean Space
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
Generalizations of the gyroid surface
The deformation of Schoen's gyroid — one of the three examples of triply-periodic minimal surfaces possessing cubic symmetry and genus 3 — is discussed. Lower-symmetry variants (similarly of genus 3)
The classification of doubly periodic minimal tori with parallel ends
Let K be the space of properly embedded minimal tori in quotients of R 3 by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove
Classification of doubly-periodic minimal surfaces of genus zero
Abstract.We prove that if the quotient surface of a properly embedded doubly–periodic minimal surface in ℝ3 has genus zero, then the surface is one of the classical Scherk examples.