The maximally symmetric surfaces in the 3-torus

• Published in: Geometriae dedicata Vol. 189; no. 1; pp. 79 - 95
• Main Authors: Bai, Sheng, Robins, Vanessa, Wang, Chao, Wang, Shicheng
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-08-2017; Springer Nature B.V
• Subjects: Algebraic Geometry; Convex and Discrete Geometry; Differential Geometry; Hyperbolic Geometry; Mathematics; Mathematics and Statistics; Minimal surfaces; Original Paper; Projective Geometry; Symmetry; Topology; Toruses; Triply periodic minimal surfaces; 57M60; 20F65; 05C10; 57S25; 53A10; Knotted surfaces in the 3-torus; Maximal surface symmetry; 57N10
• ISSN: 0046-5755; 1572-9168
• DOI: 10.1007/s10711-017-0218-0

Suppose an orientation-preserving action of a finite group
G
on the closed surface
Σ
g
of genus
g
>
1
extends over the 3-torus
T
3
for some embedding
Σ
g
⊂
T
3
. Then
|
G
|
≤
12
(
g
-
1
)
, and this upper bound
12
(
g
-
1
)
can be achieved for
g
=
n
2
+
1
,
3
n
2
+
1
,
2
n
3
+
1
,
4
n
3
+
1
,
8
n
3
+
1
,
n
∈
Z
+
. The surfaces in
T
3
realizing a maximal symmetry can be either unknotted or knotted. Similar problems in the non-orientable category are also discussed. The connection with minimal surfaces in
T
3
is addressed and the situation when the maximally symmetric surfaces above can be realized by minimal surfaces is identified.