An orthorhombic deformation family of Schwarz' H surfaces
Hao Chen Matthias Weber



The  family]An orthorhombic deformation family
 of Schwarz' H surfaces


[Chen]Georg-August-Universität Göttingen, Institut für Numerische und Angewandte Mathematik
h.chen@math.uni-goettingen.de
H. Chen is supported by Individual Research Grant from Deutsche Forschungsgemeinschaft within the project “Defects in Triply Periodic Minimal Surfaces”, Projektnummer 398759432.


[Weber]Indiana University, Department of Mathematics
matweber@indiana.edu


[2010]Primary 53A10





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 crucial distinction between these
surfaces is that the P surface belongs to a 5-dimensional smooth family of
embedded TPMS of genus three discovered by W. Meeks, while the H surfaces are
among the few known examples outside this family.  We construct a 2-parameter
family of embedded TPMS of genus three that contains the H family and meets the
Meeks family.  In particular, we prove that H surfaces can be deformed
continuously within the space of TPMS of genus three into Meeks surfaces.


[
    Matthias Weber
    December 30, 2023
=====================





§ INTRODUCTION


This is the second of two papers dealing with new 2-dimensional families of
embedded triply periodic minimal surfaces (TPMS) of genus three whose
1-dimensional “intersections” with the well-known Meeks family exhibit
singularities in the moduli space of TPMS.

Among the many TPMS discovered by H. A. Schwarz <cit.> is a
1-parameter family $̋ that can be constructed by extending the Plateau
solution for the boundaries of the two triangular faces of a triangular prism.
Such a Plateau solution does not exist for all heights of the prism.  For small
heights, there are two distinct solutions. One of them limits in the most
symmetric singly periodic Scherk surfaces with 6 annular ends. The other
degenerate to a foliation of^3by horizontal parallel planes that are
joined by catenoidal necks, placed in a hexagonal lattice.

This family is remarkable because it does not belong to the 5-dimensional Meeks
familyof TPMS of genus 3 <cit.>.  Members of that family have
the eight branched values of the Gauss map forming four antipodal pairs, while
for an$̋ surface, they are located at the north and south poles of the
2-sphere and the six vertices of a triangular prism.  The only other known TPMS
of genus 3 outside  are the Gyroid-Lidinoid family <cit.>, and the
recently discovered  family <cit.>.

We will exhibit a 2-parameter family  of embedded TPMS of genus 3 that
can be understood as an orthorhombic deformation family of Schwarz' $̋
surfaces.  The closure of this family has a 1-dimensional intersection with thefamily, a classical 2-parameter orthorhombic deformation family of
Schwarzsurface.  However, surfaces inare not in.  This is,
after thefamily in <cit.>, another 2-parameter non-Meeks
family of TPMS of genus 3.



Consider an embedded minimal surfacesSinside an axes parallel box[-A,A]×[-B,B]×[-1,1]such that

  * S satisfies free boundary condition on the vertical planes x=± A
		and y=± B, and fixed (Plateau) boundary condition on the horizontal
		segments {(x,0,± 1) | -A≤ x ≤ A}.

	
  * S intersects the edges of the box in eight vertices, but disjoint
		from the vertical lines with (x,y)=±(+A,+B).  Hence, apart from the
		four ends of the fixed boundaries, S intersects the vertical lines
		(x,y)=±(+A,-B) in two vertices each.

	
  * S is symmetric under the inversion in the origin.

ThereforeSis a right-angled minimal octagon, with its inversion center at
the origin.  See Figure <ref> (left) for an example.  If the
vertices are labeled as in this figure, then the fixed boundaries are the
segmentsV_8V_1andV_4V_5.

Because the two horizontal segments are in the middle of the top and bottom
faces of the box, rotations about them and reflections in the lateral faces of
the box extendSto a TPMSΣ̃.  More specifically,Σ̃is invariant under the latticeΛspanned by(2A,0,2),(-2A,0,2)and(0,4B,0).  In the 3-torus^3 / Λ,Σ= Σ̃/ Λis a compact surface of genus3.  In
Figure <ref> (right) we show part ofΣ̃consisting
of eight copies ofS.






	For crystallographers, the orthorhombic lattice spanned by (4A,0,0),
	(0,4B,0) and (0,0,4) is probably more convenient.  This is responsible
	for the letter “o” in our naming.  The part shown in
	Figure <ref> (right) is actually a translational fundamental
	domain of this orthorhombic lattice.  The quotient of Σ̃ by this
	lattice is a double cover of Σ, hence of genus 5.