Comptes Rendus Mathématique

Herold Dehling

Davide Giraudo

Dalibor Volný

Some remarks on the ergodic theorem for $U$-statistics

Baptiste Chantraine

Noémie Legout

Comptes Rendus
Mathématique
Baptiste Chantraine and Noémie Legout
Doubly slice knots and obstruction to Lagrangian concordance
Volume 361 (2023), p. 1605-1609
https://doi.org/10.5802/crmath.478
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 1605-1609
https://doi.org/10.5802/crmath.478
Geometry and Topology / Géométrie et Topologie
Doubly slice knots and obstruction to
Lagrangian concordance
Noeuds doublement bordant et obstruction aux
concordances lagrangiennes
Baptiste Chantraine a and Noémie Legout ∗,b
a Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, UMR
6629, F-44000 Nantes, France
b Uppsala University, Department of Mathematics, Box 480, 751 06 Uppsala, Sweden
E-mails: baptiste.chantraine@univ-nantes.fr (B. Chantraine),
noemie.legout@math.uu.se (N. Legout)
Abstract. In this short note we observe that a result of Eliashberg and Polterovitch allows to use the doubly
slice genus as an obstruction for a Legendrian knot to be a slice of a Lagrangian concordance from the trivial
Legendrian knot with maximal Thurston–Bennequin invariant to itself. This allows to obstruct concordances
from the Pretzel knot P (3,−3,−m) when m ≥ 4 to the unknot. Those examples are of interest because the
Legendrian contact homology algebra cannot be used to obstruct such a concordance.
Résumé. Dans cette note, nous remarquons qu’un résultat d’Eliashberg et Polterovitch permet d’utiliser
la notion de nœuds doublement bordant afin d’obstruer la possibilité pour un noeud legendrien d’appa-
raitre comme une tranche dans une concordance lagrangienne du noeud legendrien trivial d’invariant de
Thurston–Bennequin maximal vers lui-même. Cela permet d’obstruer l’existence pour m ≥ 4 de concor-
dances du noeud pretzel P (3,−3,−m) vers le noeud trivial. Ces exemples s’avèrent particulièrement inté-
ressants car l’algèbre d’homologie de contact legendrienne ne permet pas d’obstruer une telle concordance.
2020 Mathematics Subject Classification. 57K33, 57K10.
Funding. The first author is partially supported by the ANR projects ENUMGEOM (ANR-18-CE40-0009) COSY
(ANR-21-CE40-0002) and COSYDY (ANR-CE40-0014).
The second author is supported by the grant KAW 2016.0198 from the Knut and Alice Wallenberg Foundation
and the grant 2016-03338 from the Swedish Research Council.
Manuscript received 15 July 2022, revised 10 January 2023, accepted 16 February 2023.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1606 Baptiste Chantraine and Noémie Legout
1. Introduction and results
Definition 1. LetΛ−,Λ+ be two Legendrian knots in S3. A Lagrangian concordance fromΛ− toΛ+
is a Lagrangian submanifold Σ⊂R×S3 diffeomorphic to a cylinder and such that for some T > 0,
• ((−∞,−T )×S3)∩Σ= (−∞,−T )×Λ− and
• ((T,+∞)×S3)∩Σ= (T,+∞)×Λ+
We denoteΛ− ≺Λ+ if there is a concordance fromΛ− toΛ+.
Up to now it is not known if this relation induces a partial order on the set of Legendrian
isotopy classes of Legendrian knots. It is of course well defined, transitive and reflexive but it
is not known if Λ− ≺ Λ+ and Λ+ ≺ Λ− implies that Λ− and Λ+ are Legendrian isotopic. There is
a related notion of decomposable Lagrangian concordance (denoted ≺dec) that are concordances
built from elementary combinatorial moves. The projection to R of such a concordance only
has critical points of index 0 and 1 hence are ribbon. It follows from recent work of Agol [1]
that if Λ− ≺dec Λ+ and Λ+ ≺dec Λ− then Λ− and Λ+ are smoothly isotopic. Since Lagrangian
concordances preserve both the Thurston–Bennequin number (TB) and the rotation number it
implies that the relation ≺dec is a partial order on the class of Legendrian knots whose isotopy
classes are simple.
In this note we are concerned with Legendrian knots Λ such that Λ0 ≺ Λ ≺ Λ0 where Λ0 is
the maximal TB Legendrian unknot. The Legendrian knot Λ0 is fillable by an exact Lagrangian
disc. Thus it follows from the fact that a Lagrangian disc can be perturbed to a symplectic disc
(because it is an open manifold) and a combination of results of Rudolph [12] and Boileau–
Orevkov [2] that the concordance realisingΛ0 ≺Λ is ribbon. On the other hand in [4, Theorem 3.2]
it is shown that the concordance realising Λ ≺ Λ0 cannot be decomposable, actually following
the proof one observes that they prove that this concordance cannot be ribbon (which is also
implied by a recent result of Zemke [15] or the above mentioned result from [1]). In [4] they
give a list of Lagrangianly slice Legendrian knots for which the existence of a concordance to
Λ0 cannot be obstructed using Legendrian contact homology. The reason is that their Chekanov-
Eliashberg algebras are stable tame isomorphic to the one of Λ0 (see [6, Appendix A] for explicit
computation). The family in question consists of some T B =−1 Legendrian realisations of pretzel
knots P (3,−3,−m), for m ≥ 4. Denote Λm such a Legendrian knot, see Figure 1 for its front
projection. As observed in [4], there is a Lagrangian concordance Λ0 ≺ Λm for all m ≥ 4. Such
a concordance can be built starting with a pinch move along the red Reeb chord depicted in
Figure 1. After some Reidemeister moves, one gets a cobordism from two unlinked copies of Λ0
toΛm . Filling one of theseΛ0 by a Lagrangian disc leads to the desired concordance.
In this note we make the following observation.
Theorem 2. There is no concordanceΛm ≺Λ0.
Theorem 2 follows from a result of Eliashberg-Polterovitch [5] that we recall here (rephrased
in the language that fits our purpose):
Theorem 3. Let D be a filling ofΛ0, then D is Hamiltonian isotopic to the standard filling ofΛ0.
Indeed this result leads to obstructions for Legendrian knots to exist as middle slices of a
concordance from Λ0 to Λ0. The symplectic flavour of the result leads to obstructions from
Legendrian contact homology and variations of it, this has been used in [3, 11] and [14] for
instance. But the unknottedness of the disc in [5] has topological implications that to our
knowledge have not been used in order to study Lagrangian cobordisms. Recall that a smooth
knot K is doubly slice if there is an unknotted 2-sphere S in R4 such that S ∩ S3 = K . As an
immediate corollary of Theorem 3 we obtain:
Baptiste Chantraine and Noémie Legout 1607
Theorem 4. If there are concordancesΛ0 ≺Λ andΛ≺Λ0, thenΛ is doubly slice.
Now the main computation of this note shows that:
Theorem 5. The pretzel knots P (3,−3,−m) for m ≥ 4 are not doubly slice.
This implies Theorem 2.
Remark 6. For the case m = 3, the knot Λ3 has topological type m(946) and it has been shown
previously by the first author in [3] that there is no concordance from Λ3 to Λ0. In this case
computations using Legendrian contact homology or other modern tools for the case are still
necessary because it turns out that the knot m(946) is doubly slice. The fact that the only one
to be doubly slice in this family happens to be the only one to have a rich Legendrian contact
homology algebra is both fortunate and puzzling.
m
Figure 1. Front projection of the Legendrian representative Λm of the pretzel knot
P (3,−3,−m). The red line indicates a Reeb chord that can be pinched in order to produce a
concordanceΛ0 ≺Λm .
Remark 7. In [9] the authors give an (almost complete) classification of doubly slice knots up to
12 crossings. From their work, the pretzel knots P (3,−3,−4) = m(10140), P (3,−3,−5) = 11n139 and
P (3,−3,−6) = 12n582 are not doubly slice.
2. Proof of Theorem 5
2.1. Case P (3,−3,−2k −1), k ≥ 2
In this case the pretzel knots are odd because they have an odd number of parameters which are
all odd. McDonald in [10], using work of Issa and McCoy in [7], proves the following result:
Theorem 8 ([10, Theorem 3]). For K an odd pretzel knot, the following are equivalent:
• Σ2(S3,K ) embeds in S4,
• K is a doubly slice pretzel knot,
• K is a mutant of P (a,−a, a,−a, . . . , a) for some odd a.
It is noted in [10] that it follows from the proof of [7, Theorem 1.11] that an odd pretzel knot
which is a mutant of P (a,−a, a,−a, . . . , a) for some odd a has for parameters a permutation of
(a,−a, a,−a, . . . , a). One concludes that the pretzel knots P (3,−3,−2k−1) for k ≥ 2 are not doubly
slice.
1608 Baptiste Chantraine and Noémie Legout
2.2. Case P (3,−3,−2k), k ≥ 2
For these knots we obstruct doubly sliceness using the signature function as in [9] for P (3,−3,−6).
Given a knot K and a Seifert matrix A for K , the Levine-Tristram signature of K is a function
σK : S1 →Zwhose value at ω ∈ S1 is given by the signature of the matrix
(1−ω)A+ (1− ω̄)AT
where the signature is the number of positive eigenvalues minus the number of negative eigen-
values. If K is slice, then σK (ω) = 0 away of the roots of the Alexander polynomial of K . Moreover,
if K is doubly slice then σK (ω) vanishes for all ω ∈ S1. This is discussed in [9, Proposition 2.2] and
follows from the fact that when K is doubly slice then there exists a Seifert matrix for K that is
hyperbolic (see [13]). Seifert matrices for the knots P (3,−3,−2k) have been computed in [8, Sec-
tion 3]. We denote Ak a Seifert matrix for P (3,−3,−2k). We have:
Ak =

−1 −1 0 0 −1 0
0 −1 0 0 −1 0
0 0 1 1 1 0
0 0 0 1 1 0
0 0 0 0 0 0
0 0 0 0 1 −k
 .
It turns out that the Alexander polynomial of these knots is the same, given by (t 2−t+1)2 and has
for roots the sixth root of unity and its conjugate. One can then compute that σK (ω) ̸= 0 for K =
P (3,−3,−2k) andω= e i π3 . Indeed a direct computation shows that the matrix (1−ω)Ak+(1−ω̄)ATk
has rank 5 and hence cannot have even signature (one can compute that its signature is actually
−1). Thus K is not doubly slice which concludes both the proofs of Theorem 5 and Theorem 2.
Acknowledgement
This work was completed while the authors were at the Institute Mittag–Leffler for the program
“Frontiers of Quantitative Symplectic and Contact Geometry”, they want to thank the organisers
and the staff from the institute for the stimulating environment they provided.
References
[1] I. Agol, “Ribbon concordance of knots is a partial ordering”, Comm. Amer. Math. Soc. 2, p. 374-379.
[2] M. Boileau, S. Orevkov, “Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe”, C. R. Math.
Acad. Sci. Paris 332 (2001), no. 9, p. 825-830.
[3] B. Chantraine, “Lagrangian concordance is not a symmetric relation”, Quantum Topol. 6 (2015), no. 3, p. 451-474.
[4] C. Cornwell, L. Ng, S. Sivek, “Obstructions to Lagrangian concordance”, Algebr. Geom. Topol. 16 (2016), no. 2, p. 797-
824.
[5] Y. M. Eliashberg, L. V. Polterovich, “Local Lagrangian 2-knots are trivial”, Ann. Math. 144 (1996), no. 1, p. 61-76.
[6] J. B. Etnyre, L. Ng, “Legendrian contact homology in R3”, in Surveys in differential geometry 2020. Surveys in 3-
manifold topology and geometry, Surveys in Differential Geometry, vol. 25, International Press, Boston, MA, 2022,
p. 103-161.
[7] A. Issa, D. McCoy, “Smoothly embedding Seifert fibered spaces in S4”, Trans. Am. Math. Soc. 373 (2020), no. 7, p. 4933-
4974.
[8] S. Jabuka, “Rational Witt classes of pretzel knots”, Osaka J. Math. 47 (2010), no. 4, p. 977-1027.
[9] C. Livingston, J. Meier, “Doubly slice knots with low crossing number”, New York J. Math. 21 (2015), p. 1007-1026.
[10] C. McDonald, “Doubly slice odd pretzel knots”, Proc. Am. Math. Soc. 148 (2020), no. 12, p. 5413-5420.
[11] L. Ng, D. Rutherford, V. Shende, S. Sivek, “The cardinality of the augmentation category of a Legendrian link”, Math.
Res. Lett. 24 (2017), no. 6, p. 1845-1874.
[12] L. Rudolph, “Algebraic functions and closed braids”, Topology 22 (1983), p. 191-202.
[13] D. W. L. Sumners, “Invertible knot cobordisms”, Comment. Math. Helv. 46 (1971), p. 240-256.
Baptiste Chantraine and Noémie Legout 1609
[14] A. Wu, “Obstructing Lagrangian concordance for closures of 3-braids”, 2022, https://arxiv.org/abs/2201.08466.
[15] I. Zemke, “Knot Floer homology obstructs ribbon concordance”, Ann. Math. 190 (2019), no. 3, p. 931-947.


Doubly slice knots and obstruction to Lagrangian concordance

Ryo Takahashi

Remarks on complexities and entropies for singularity categories

Mohsen Zivari-Rezapour

Yichen Liu

Behrouz Emamizadeh

A formula for the sum of $n$ weak$^\star $ closed sets in $L^\infty $

Bertrand Deroin

Carlos Matheus

Non-linear bi-algebraic curves and surfaces in moduli spaces of Abelian differentials

Martijn Caspers

Comptes Rendus
Mathématique
Martijn Caspers
On the isomorphism class of q-Gaussian W∗-algebras for infinite variables
Volume 361 (2023), p. 1711-1716
Published online: 21 December 2023
https://doi.org/10.5802/crmath.489
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 1711-1716
https://doi.org/10.5802/crmath.489
Functional analysis / Analyse fonctionnelle
On the isomorphism class of q-Gaussian
W∗-algebras for infinite variables
Martijn Caspers a
a TU Delft, EWI/DIAM, P.O.Box 5031, 2600 GA Delft, The Netherlands
E-mail: M.P.T.Caspers@tudelft.nl
Abstract. Let Mq (HR) be the q-Gaussian von Neumann algebra associated with a separable infinite dimen-
sional real Hilbert space HR where −1 < q < 1. We show that Mq (HR) ̸≃ M0(HR) for −1 < q ̸= 0 < 1. The
C∗-algebraic counterpart of this result was obtained recently in [1]. Using ideas of Ozawa we show that this
non-isomorphism result also holds on the level of von Neumann algebras.
Keywords. q-Gaussian von Neumann algebras, Akemann–Ostrand property.
2020 Mathematics Subject Classification. 46L35, 46L06.
Funding. Supported by the NWO Vidi grant VI.Vidi.192.018 “Non-commutative harmonic analysis and rigid-
ity of operator algebras”.
Manuscript received 23 October 2022, revised 3 March 2023, accepted 6 March 2023.
1. Introduction
Von Neumann algebras of q-Gaussian variables originate from the work of Bożejko and Spe-
icher [3] (see also [2]). To a real Hilbert space HR and a parameter −1 < q < 1 it associates a
von Neumann algebra Mq (HR). At parameter q = 0 this assignment HR 7→ Mq (HR) is known as
Voiculescu’s free Gaussian functor. The dependence of q of these von Neumann algebras has
been an intriguing and very difficult problem. A breakthrough result in this direction was ob-
tained by Guionnet–Shlyakhtenko [8] who showed that for finite dimensional HR for a range of
q close to 0 all von Neumann algebras Mq (HR) are isomorphic. The range for which isomor-
phism is known decreases as the dimension HR becomes larger. The Guionnet–Shlyakhtenko ap-
proach is based on free analogues of (optimal) transport techniques. Their result also relies on
existence and power series estimates of conjugate variables obtained by Dabrowski [5]. In fact
the free transport techniques provide even an isomorphism result of underlying q-Gaussian C∗-
algebras.
In case HR is infinite dimensional the isomorphism question of q-Gaussian algebras was
addressed by Nelson and Zeng [10]. They showed that for mixed q-Gaussians for which the array
(qi j )i j of commutation coefficients decays fast enough to 0 one obtains isomorphism of mixed
q-Gaussian C∗- and von Neumann algebras. However, the isomorphism question for the original
(non-mixed) q-Gaussians remained open, see Questions 1.1 and 1.2 of [10]. In [1] we showed
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1712 Martijn Caspers
that on the level of C∗-algebras there exists a non-isomorphsim result. In the current note we
improve on this result: we show that for an infinite dimensional separable real Hilbert space
HR and −1 < q < 1, q ̸= 0 we have Mq (HR) ̸≃ M0(HR). This then fully answers Questions 1.1
and 1.2 of [10] and provides a stark contrast to the results of Guionnet–Shlyakhtenko for finite
dimensional HR.
The distinguishing property of Mq (HR) and M0(HR) is a variation of the Akemann–Ostrand
property that was suggested in a note by Ozawa [12] (see also [6]) and which we shall call W∗AO.
We formally define it in Definition 1. The most important novelty is that we quotient B(L2(M))
by the C∗-algebra KM which is much larger than the ideal of compact operators on L2(M).
This larger quotient turns out to provide von Neumann algebraic descriptions of the Akemann–
Ostrand property [12]. We use this to distinguish Mq (HR) and M0(HR).
Acknowledgements
The author thanks Mateusz Wasilewski, Changying Ding and Cyril Houdayer for comments on an
earlier draft of this paper.
2. Preliminaries
B(X ,Y ) denotes the bounded operators between Banach spaces X → Y . K (X ,Y ) denotes the
compact operators, meaning that they map the unit ball to a relatively compact set. We set
B(X ) :=B(X , X ) and K (X ) :=K (X , X ).
The algebraic tensor product (vector space tensor product) is denoted by ⊗alg and ⊗min is the
minimal tensor product of C∗-algebras. ⊗ is used for tensor products of elements.
We refer to [13] as a standard reference on von Neumann algebras. For a von Neumann algebra
M we denote by (M ,L2(M), J ,L2(M)+) its standard form. For x ∈ M we write xop := J x∗ J which is
the right multiplication with x on the standard space. For a finite von Neumann algebra M with
trace τwe have M ⊆ L2(M) where L2(M) is the completion of M with respect to the inner product
〈x, y〉 = τ(y∗x). Therefore every T ∈ B(L2(M)) determines a map Q0(T ) ∈ B(M ,L2(M)) given by
x 7→ T (x). Set
Q1 : B(L
2(M)) →B(M ,L2(M))/K (M ,L2(M)) : T 7→Q0(T )+K (M ,L2(M)).
Q1 is clearly continuous and we define the closed left-ideal K LM = ker(Q1) and the hereditary
C∗-subalgebra KM = (K LM )∗ ∩K LM of B(L2(M)) (see also [12]). We let M (KM ) ⊆ B(L2(M)) be
the multiplier algebra of KM ; indeed this multiplier algebra is faithfully represented on L2(M)
by [9, Proposition 2.1]. Then KM is an ideal in the C∗-algebra M (KM ). We have M ⊆ M (KM )
and M op ⊆M (KM ).
2.1. A von Neumann version of the Akemann–Ostrand property
Definition 1. Let M be a finite von Neumann algebra. We say that M has W∗AO if the map
θ : M ⊗alg M op →M (KM )/KM : a ⊗bop 7→ abop +KM . (1)
is continuous with respect to the minimal tensor norm and thus extends to a ∗-homomorphism
M ⊗min M op →M (KM )/KM .
We recall the following from [12, Section 4]. Let Γ be a discrete group and let L (Γ) and
R(Γ) be the left and right group von Neumann algebra respectively acting on ℓ2(Γ). In this case
L2(L (Γ)) ≃ ℓ2(Γ) as bimodules with the natural left and right actions of L (Γ) and R(Γ) on ℓ2(Γ).
We have Jδs = δs−1 which extends to an antilinear isometry on ℓ2(Γ). Then R(Γ) = JL (Γ)J .
Martijn Caspers 1713
Assume Γ is icc so that L (Γ) and R(Γ) are factors, i.e. L (Γ)∩R(Γ) =C1. The map
π : C∗(L (Γ),R(Γ)) →B(ℓ2(Γ)⊗ℓ2(Γ)) : abop 7→ a ⊗bop, a ∈L (Γ),bop ∈R(Γ).
is a well-defined ∗-homomorphism by Takesaki’s theorem on minimality of the spatial tensor
product. In [12, Section 4, Theorem] Ozawa showed the following theorem.
Theorem 2. Let Γ be an exact icc group such that the ∗-homomorphism
C∗r (Γ)⊗alg C∗r (Γ)op →B(ℓ2(Γ))/K (ℓ2(Γ)) : a ⊗bop 7→ abop +K (ℓ2(Γ)),
is continuous with respect to ⊗min. Then L (Γ) has W∗AO.
Proof. By [12, Section 4, Theorem] we have
ker(π) =KL (Γ) ∩C∗(L (Γ),R(Γ)).
Therefore,
L (Γ)⊗min R(Γ) →≃π
−1
C∗(L (Γ),R(Γ))/(KL (Γ) ∩C∗(L (Γ),R(Γ))),
which concludes the theorem. □
Remark 3. It follows that if Γ is an icc group that is bi-exact (or said to be in class S , see [4,
Section 15]) then L (Γ) has W∗AO.
2.2. q-Gaussians
Let −1 < q < 1 and let HR be a real Hilbert space with complexification H := HR⊕ i HR. Set the
symmetrization operator P kq on H
⊗k ,
P kq (ξ1 ⊗ . . .⊗ξn) =
∑
σ∈Sk
q i (σ)ξσ(1) ⊗ . . .⊗ξσ(n), (2)
where Sk is the symmetric group of permutations of k elements and i (σ) := #{(a,b) | a < b,
σ(b) < σ(a)} the number of inversions. The operator P kq is positive and invertible [3]. Define a
new inner product on H⊗k by
〈ξ,η〉q := 〈P kqξ,η〉,
and call the new Hilbert space H⊗kq . Set the Hilbert space direct sum Fq (H) := CΩ⊕ (
⊕∞
k=1 H
⊗k
q )
whereΩ is a unit vector called the vacuum vector. For ξ ∈ H let
lq (ξ)(η1 ⊗ . . .⊗ηk ) := ξ⊗η1 ⊗ . . .⊗ηk , lq (ξ)Ω= ξ,
and then l∗q (ξ) = lq (ξ)∗. These ‘creation’ and ‘annihilation’ operators are bounded and extend to
Fq (H). We define a von Neumann algebra by the double commutant
Mq (HR) := {lq (ξ)+ l∗q (ξ) | ξ ∈ HR}′′.
Then τΩ(x) := 〈xΩ,Ω〉 is a faithful tracial state on Mq (HR) which is moreover normal. Now Fq (H)
is the standard form Hilbert space of Mq (HR) and J xΩ = x∗Ω. For vectors ξ1, . . . ,ξk ∈ H there
exists a unique operator Wq (ξ1 ⊗ . . .⊗ξk ) ∈ Mq (HR) such that
Wq (ξ1 ⊗ . . .⊗ξk )Ω= ξ1 ⊗ . . .⊗ξk .
These operators are called Wick operators. It follows that Wq (ξ)opΩ= ξ.
Remark 4. Let F∞ be the free group with countably infinitely many generators. F∞ is icc and ex-
act [4] and hence Theorem 2 applies. We conclude that L (F∞) has W∗AO. We have that L (F∞) ≃
Γ0(HR) with HR a separable infinite dimensional real Hilbert space (see [7, Theorem 2.6.2]) and
so Γ0(HR) has the W∗AO.
1714 Martijn Caspers
3. Non-isomorphism of q-Gaussian von Neumann algebras
The following theorem provides a necessary condition for W∗AO.
Theorem 5. Let M be a finite von Neumann algebra with finite normal faithful tracial state τ.
Suppose there exists a unital von Neumann subalgebra B ⊆ M and infinitely many subspaces
Mi ⊆ M , i ∈ N that are left and right B-invariant and mutually τ-orthogonal in the sense that
τ(y∗x) = 0 for x ∈ Mi , y ∈ M j , i ̸= j . Suppose moreover that there exists δ > 0 and finitely many
operators b j ,c j ∈ B, with ∑ j b j ⊗ copj non-zero, such that for every i ∈N we have∥∥∥∥∥Q0
(∑
j
b j c
op
j
)∥∥∥∥∥
B(Mi ,L2(Mi ))
≥ (1+δ)
∥∥∥∥∥∑
j
b j ⊗ copj
∥∥∥∥∥
B⊗minB op
. (3)
Then M does not have W∗AO.
Proof. Let X be the set of finite rank operators x ∈ B(L2(M)) such that there exists Ix ⊆ I finite
with ker(x)⊥ ⊆⊕i∈Ix L2(Mi ). Take x ∈ X and choose k ∈ I \Ix . Then,∥∥∥∥∥Q0
(∑
j
b j c
op
j +x
)∥∥∥∥∥
B(M ,L2(M))
≥
∥∥∥∥∥Q0
(∑
j
b j c
op
j +x
)∥∥∥∥∥
B(Mk ,L2(Mk ))
=
∥∥∥∥∥Q0
(∑
j
b j c
op
j
)∥∥∥∥∥
B(Mk ,L2(Mk ))
≥ (1+δ)
∥∥∥∥∥∑
j
b j ⊗ copj
∥∥∥∥∥
B⊗minB op
.
The operators in X are norm dense in K (L2(M)) and by [12, Section 2, Proposition] we have that
Q0(K (L2(M))) is dense in Q0(K LM ) in the norm of B(M ,L
2(M)). As Q0 is contractive Q0(X ) is
dense in Q0(K LM ). It therefore follows that for any x ∈K LM we have∥∥∥∥∥Q0
(∑
j
b j c
op
j +x
)∥∥∥∥∥
B(M ,L2(M))
≥ (1+δ)
∥∥∥∥∥∑
j
b j ⊗ copj
∥∥∥∥∥
B⊗minB op
.
Since Q0 is contractive for every x ∈K LM we have,∥∥∥∥∥∑
j
b j c
op
j +x
∥∥∥∥∥
B(L2(M))
≥ (1+δ)
∥∥∥∥∥∑
j
b j ⊗ copj
∥∥∥∥∥
B⊗minB op
.
Hence, certainly for the Banach space quotient norm we have∥∥∥∥∥∑
j
b j c
op
j +KM
∥∥∥∥∥
B(L2(M))/KM
≥ (1+δ)
∥∥∥∥∥∑
j
b j ⊗ copj
∥∥∥∥∥
B⊗minB op
.
As the left hand side norm is the norm of the C∗-quotient M (KM )/KM this concludes the proof
(see [9, Proposition 2.1] as in the preliminaries). □
The proof of the following theorem essentially repeats its C∗-algebraic counterpart from [1,
Theorem 3.3].
Theorem 6. Assume dim(HR) =∞ and −1 < q < 1, q ̸= 0. Then the von Neumann algebra Mq (HR)
does not have W∗AO.
Proof. Let d ≥ 2 be such that q2d > 1. Let
M := Mq (Rd ⊕HR), B := Mq (Rd ⊕0).
Martijn Caspers 1715
Let { fi }i be an infinite set of orthogonal vectors in 0⊕ HR such that ∥Wq ( fi )∥ = 1. Let Mq,i :=
BWq ( fi )B which is a B-B invariant subset of M . Then Mq,i and Mq, j are τΩ-orthogonal if i ̸= j .
For k ∈N let
B(k) = {Wq (ξ) | ξ ∈ (Rd ⊕0)⊗k }.
It is proved in [1, Equation (3.2)] that for b,c ∈B(k) we have
〈bWq ( fi )cΩ, fi 〉q = 〈bcop fi , fi 〉q = qk〈bcopΩ,Ω〉q .
Then for finitely many b j ,c j ∈B(k) we have∥∥∥∥∥Q0
(∑
j
b j c
op
j
)∥∥∥∥∥
B(Mq,i ,L2(Mq,i ))
≥
∥∥∥∥∥∑
j
b j Wq ( fi )c j
∥∥∥∥∥
L2(Mq,i )
≥
∣∣∣∣∣∣
〈∑
j
b j Wq ( fi )c jΩ, fi
〉
q
∣∣∣∣∣∣=
∣∣∣∣∣∑
j
qk
〈
b jΩc j ,Ω
〉
q
∣∣∣∣∣ . (4)
Now let {e1, . . . ,ed } be an orthonormal basis of R
d ⊕ 0 and for j = ( j1, . . . , jk ) ∈ {1, . . . ,d}k let
e j = e j1 ⊗ . . . ⊗ e jk . Let Jk be the set of all such multi-indices of length k. So #Jk = d k . Set
ξ j = (P kq )−
1
2 e j so that 〈ξ j ,ξ j 〉q = 〈P kqξ j ,ξ j 〉 = 1.
Now (4) yields that for all k ≥ 1 and all i ,∥∥∥∥∥Q0
( ∑
j∈Jk
Wq (ξ j )
∗Wq (ξ j )op
)∥∥∥∥∥
B(Mq,i ,L2(Mq,i ))
≥ ∑
j∈Jk
qk〈Wq (ξ j )∗ΩWq (ξ j ),Ω〉q
= ∑
j∈Jk
qk〈ΩWq (ξ j ),Wq (ξ j )Ω〉q
= ∑
j∈Jk
qk〈ξ j ,ξ j 〉q = qk d k .
From [11, Proof of Theorem 2] (or see [1, Proof of Theorem 3.3]) we find,∥∥∥∥∥ ∑
j∈Jk
Wq (ξ j )
∗⊗Wq (ξ j )op
∥∥∥∥∥
B⊗minB op
≤
( ∞∏
i=1
(1−q i )−1
)3
(k +1)2d k/2.
Therefore, as q2d > 1 there exists δ> 0 such that for k large enough we have for every i ,∥∥∥∥∥Q0
( ∑
j∈Jk
Wq (ξ j )
∗Wq (ξ j )op
)∥∥∥∥∥
B(Mq,i ,L2(Mq,i ))
≥ (1+δ)
∥∥∥∥∥ ∑
j∈Jk
Wq (ξ j )
∗⊗Wq (ξ j )op
∥∥∥∥∥
B⊗minB op
.
Hence the assumptions of Theorem 5 are witnessed which shows that W∗AO does not hold. □
Corollary 7. Let HR be an infinite dimensional real separable Hilbert space. The von Neumann
algebras Γ0(HR) and Γq (HR) with −1 < q < 1, q ̸= 0 are non-isomorphic.
Proof. This is a consequence of Theorem 6 and Remark 4 as W∗AO is preserved under isomor-
phism. □
References
[1] M. Borst, M. Caspers, M. Klisse, M. Wasilewski, “On the isomorphism class of q-Gaussian C∗-algebras for infinite
variables”, to appear in Proc. Am. Math. Soc., 2022, https://arxiv.org/abs/2202.13640.
[2] M. Bożejko, B. Kümmerer, R. Speicher, “q-Gaussian processes: non-commutative and classical aspects”, Commun.
Math. Phys. 185 (1997), no. 1, p. 129-154.
[3] M. Bożejko, R. Speicher, “An example of a generalized Brownian motion”, Commun. Math. Phys. 137 (1991), no. 3,
p. 519-531.
1716 Martijn Caspers
[4] N. P. Brown, N. Ozawa, C∗-algebras and finite-dimensional approximations, Graduate Studies in Mathematics,
vol. 88, American Mathematical Society, 2008, xvi+509 pages.
[5] Y. Dabrowski, “A free stochastic partial differential equation”, Ann. Inst. Henri Poincaré 50 (2014), no. 4, p. 1404-1455.
[6] C. Ding, S. Kunnawalkam Elayavalli, J. Peterson, “Properly Proximal von Neumann Algebras”, 2022, https://arxiv.
org/abs/2204.00517.
[7] K. J. Dykema, A. Nica, D. V. Voiculescu, Free Random Variables, CRM Monograph Series, vol. 1, American Mathemat-
ical Society, 1992, v+70 pages.
[8] A. Guionnet, D. Shlyakhtenko, “Free monotone transport”, Invent. Math. 197 (2014), no. 3, p. 613-661.
[9] E. C. Lance, Hilbert C∗-modules, A toolkit for operator algebraists, London Mathematical Society Lecture Note Series,
vol. 20, London Mathematical Society, 1995, x+130 pages.
[10] B. Nelson, Q. Zeng, “Free monotone transport for infinite variables”, Int. Math. Res. Not. 2018 (2018), no. 17, p. 5486-
5535.
[11] A. Nou, “Non injectivity of the q-deformed von Neumann algebra”, Math. Ann. 330 (2004), no. 1, p. 17-38.
[12] N. Ozawa, “A comment on free group factors”, in Noncommutative harmonic analysis with applications to probabil-
ity. II, Banach Center Publications, vol. 89, Polish Academy of Sciences, 2010, p. 241-245.
[13] M. Takesaki, Theory of operator algebras. I, Springer, 1979, vii+415 pages.


On the isomorphism class of $q$-Gaussian W$^\ast $-algebras for infinite variables

Kwangwoo Lee

Salem numbers of automorphisms of K3 surfaces with Picard number $4$

Pratyoosh Kumar

Sumit Kumar Rano

Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees

Maxim Gurevich

A triangular system for local character expansions of Iwahori-spherical representations of general linear groups

Heayong Shin

Young Wook Kim

Sung-Eun Koh

Hyung Yong Lee

Seong-Deog Yang

Comptes Rendus
Mathématique
Heayong Shin, Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee
and Seong-Deog Yang
Addendum to the paper: Compact embedded minimal surfaces in the
Berger sphere
Volume 361 (2023), p. 257-264
https://doi.org/10.5802/crmath.403
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 257-264
https://doi.org/10.5802/crmath.403
Geometry and Topology / Géométrie et Topologie
Addendum to the paper: Compact embedded
minimal surfaces in the Berger sphere
Addendum à l’article: Surfaces minimales compactes
intégrées dans la sphère Berger
Heayong Shina, YoungWook Kimb, Sung-Eun Koh∗, c, Hyung Yong Leeb
and Seong-Deog Yangb
a Department of Mathematics, Chung-Ang University, Seoul 06974, Korea
b Department of Mathematics, Korea University, Seoul 02841, Korea
c Department of Mathematics, Konkuk University, Seoul 05029, Korea
E-mails: hsin@cau.ac.kr, ywkim@korea.ac.kr, sekoh@konkuk.ac.kr,
distgeo@korea.ac.kr, sdyang@korea.ac.kr
Abstract. We construct a two discrete parameter family of compact minimal surfaces embedded in the Berger
sphere which may be considered as the analogue of the helicoidal Karcher-Scherk surfaces.
Résumé. Nous construisons une famille à deux paramètres discrets de surfaces minimales compactes plon-
gées dans la sphère de Berger qui peut être considérée comme l’analogue de l’hélicoïde de Karcher-Scherk.
Funding. Heayong Shin was supported by NRF 2019R1F1A1046327, Sung-Eun Koh by NRF S202003S00080,
the Konkuk University’s research support program for its faculty on sabbatical leave in 2019 and Seong-Deog
Yang by NRF2017R1E1A1A03070929 and NRF 2020R1F1A1A01074585. The last author would like to thank the
Department of Mathematics at Brown University for its hospitality during his sabatical year.
Manuscript received 9 April 2021, revised 16 February 2022, accepted 29 June 2022.
1. Introduction
Not too many examples of embedded minimal surfaces are known in the Berger sphere, which is
a very interesting homogeneous three manifold with an isometry group of dimension 4. In this
article, we apply again the method of constructing compact or complete embedded minimal sur-
faces by repeatedly reflecting the Plateau solution bounded by an appropriate geodesic polygon.
The surfaces we construct are analogues of the helicoidal-Karcher-Scherk surface [1], which may
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
258 Heayong Shin, Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee and Seong-Deog Yang
be obtained by applying a screw motion, at least in the boundary level, to the compact embedded
minimal surfaces the authors constructed previously in [4]. See Figure 1.
The construction in this article is basically the same as in [4]. A noticeable difference is that we
use geodesic polygons consisting of both horizontal and vertical geodesics, while in [4] we used
only horizontal geodesics. Note that in the standard three sphere S3, the vertical geodesics and
the horizontal geodesics are congruent each other, while they are not in the Berger sphere.
Figure 1. On the left is a half of the minimal surface we construct in this article and on the
right is half of that in [3] with k = l = 2. Each solid cylinder represents a half of the Berger
sphere.
Note that a Karcher-Scherk tower in E3 admits a continuous deformation, but the surface in
the Berger sphere we construct in this article does not. It is an isolated one because the values
of lengths and angles of the polygonal boundary can lie in a discrete set. This discrepancy is
caused by the geometry of the Berger sphere. At any tangent plane of the Clifford torus which
is represented as the cylinder in this article, there are only one horizontal direction and also only
one vertical direction up to sign. See Figure 2. It is said in [1] “it is still an open question whether
these screw motion invariant Karcher-Scherk surfaces together with the helicoid are the only
screw motion invariant embedded minimal surfaces with genus 0 in the quotient.” We wonder
if the embedded minimal surfaces in [3] and in this article may have similar characterizations
but we have no clue on this at this moment.
The Figure 1 would look different if we have used the parametrization of S3 used in [4]. In
particular, the surfaces in [4] in the coordinates of [4] would look like the one on the left of
Figure 1.
2. The Berger Sphere
We recall some facts on the Berger spheres.
2.1. S3 as a special unitary group
Let us identify the unit sphere S3 = {(z, w) ∈ C2 : |z|2 + |w |2 = 1} and the special unitary group
SU(2) by the map
(z, w) 7→
[
z w
−w z
]
.
Heayong Shin, Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee and Seong-Deog Yang 259
The Lie algebra su(2) is spanned by
X1 =
[
0 1
−1 0
]
, X2 =
[
0 i
i 0
]
, X3 =
[
i 0
0 −i
]
which generate the left-invariant vector fields:
X1(z, w) =
[−w z
−z −w
]
, X2(z, w) =
[
i w i z
i z −i w
]
, X3(z, w) =
[
i z −i w
−i w −i z
]
.
Viewed as tangent vector fields on S3,
X1(z, w) = (−w, z), X2(z, w) = (i w, i z), X3(z, w) = (i z,−i w).
2.2. A circle action
The orbits of the right circle action on S3[
z w
−w z
]
7→
[
z w
−w z
][
e iθ 0
0 e−iθ
]
or (z, w) 7→ (e iθz,e−iθw) (1)
are the fibers of the Hopf fibration H :S3 →S2(1/2) given by
H(z, w) =
(
zw,
1
2
(|z|2 −|w |2)
)
.
The fiber over the general point (z, w) is the circle {(e iθz,e−iθw)} to which the vector field
X3(z, w) = (i z,−i w) is tangent.
2.3. The Berger metric
Now consider the left invariant Riemannian metrics gδ on S
3 = SU(2) given in terms of the left
invariant vector fields X1, X2, X3 by
Xi ·X j = 0, i 6= j ,
X1 ·X1 = X2 ·X2 = δ2, X3 ·X3 = 1.
The Berger sphere is the Riemannian manifold (S3, gδ). When δ = 1, the Berger sphere is the
standard round sphere S3.
2.4. Isometries
Since it is a left-invariant metric, one can see that the left multiplication of SU(2)[
z w
−w z
]
7→
[
a b
−b a
][
z w
−w z
]
, aa +bb = 1,
that is,
(z, w) 7→ (az −bw ,bz +aw)
is an isometry. Note that the circle action (1) is also an isometry. Then the maps
(z, w) 7→ (e iθz, w), (z, w) 7→ (z,e iθw) (2)
are isometries as well.
260 Heayong Shin, Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee and Seong-Deog Yang
2.5. Geodesics and ruled minimal surfaces
A geodesic in the Berger sphere is called horizontal if it is orthogonal to the Hopf fibers every-
where and is called vertical if it is tangent to the Hopf fibers everywhere. One can see that the
curve
t 7→ (e i t z,e−i t w)
is a vertical geodesic passing through the point (z, w) whose whole image is the Hopf fiber
through (z, w) and that the curve
t 7→ (z cos t −e−iθw sin t ,e iθz sin t +w cos t )
which is the integral curve of the vector field cosθX1 +sinθX2 passing through the point (z, w) is
a horizontal geodesic.
The following was shown in [3]:
Proposition 1. A surface in the Berger sphere is a ruled minimal surface if and only if it is
congruent to the parametric surface
X (s, t ) = (e iαs cos t ,e i s sin t ), α ∈R.
3. A parametrization
Let P = {(s, t ,θ) : 0 ≤ s ≤ 2π,0 ≤ t ≤π/2,0 < θ ≤ 2π} and consider the mapΦ : P →S3
Φ(s, t ,θ) = (e i s cos t ,e i (θ−s) sin t ),
which is one-to-one on P ′, the interior of P , and yields coordinates onΦ(P ′).
In [4], we used the parametrization
Ψ : P →S3, Ψ(s, t ,θ) = (e i s cos t ,e i (s−θ) sin t ).
Switching the parametrization fromΨ toΦ enables us to adopt most of the formulae and figures
in [4] without modification and the same construction procedure. However, we change the
construction procedure following that of [5], which we think is a little simpler.
Since
Φ(s,0,θ) = (e i s ,0), Φ(s,π/2,θ) = (0,e i (θ−s)),
we have the following:
Proposition 2. For any θ, the image of the curve s 7→Φ(s,0,θ) is the Hopf fiber over (1,0) and the
image of the curve s 7→Φ(s,π/2,θ) is the Hopf fiber over (0,1).
Note also that, when t = π/2, since Φ(s,π/2,θ) = Φ(s′,π/2,θ′) if and only if s + θ = s′ + θ′
mod 2π, those points s + θ = s′ + θ′ mod 2π correspond to a single point on the fiber {(0,e i s )}
over (0,1).
3.1. The Berger metric in terms of the parametrizationΦ
Now let
∂s := ∂Φ
∂s
(s, t ,θ), ∂t := ∂Φ
∂t
(s, t ,θ), ∂θ :=
∂Φ
∂θ
(s, t ,θ).
Computations give
∂s = X3,
∂t = cos(θ−2s)X1 + sin(θ−2s)X2,
tan t ∂s + (1+ tan t )∂θ =−sin(θ−2s)X1 +cos(θ−2s)X2.
Then we have the following:
Heayong Shin, Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee and Seong-Deog Yang 261
Proposition 3. For any s0,θ0,
(i) the t-curve Hs0,θ0 (t ) :=Φ(s0, t ,θ0) is a horizontal geodesic.
(ii) the s-curve Vθ0 (s) :=Φ(s,π/4,θ0) is a vertical geodesic.
It was shown in [6] that the reflection across a horizontal or a vertical geodesic is an isometry
of the Berger sphere. In particular, we have the following:
Proposition 4. The following two maps
Φ(s, t ,θ) 7→Φ(2s0 − s, t ,2θ0 −θ),
Φ(s, t ,θ) 7→Φ(s −θ+θ0,π/2− t ,2θ0 −θ)
which represent reflections along the geodesics Hs0,θ0 (t ) and Vθ0 (s), respectively, are isometries.
One can also see that the rotation
Rotθ0 :Φ(s, t ,θ) 7→Φ(s, t ,θ+θ0)
with respect to the Hopf fiber over the point (1,0) and the translation
Ts0 :Φ(s, t ,θ) 7→Φ(s + s0, t ,θ)
along the Hopf fiber are also isometries. Note that Ts0 is the circle action in Section 2.2.
Then Proposition 1 and (1), (2) give the following:
Proposition 5. The following surfaces are ruled minimal surfaces.
(t ,θ) 7→ Φ(s0, t ,θ) = (e i s0 cos t ,e i (θ−s0) sin t ),
(s, t ) 7→ Φ(s, t ,θ0) = (e i s cos t ,e i (θ0−s) sin t ),
(s,θ) 7→Φ(s,π/4,θ) = 1p
2
(e i s ,e i (θ−s)).
4. Construction
Note that the surface
T :=Φ(s,π/4,θ) = 1p
2
(e i s ,e i (θ−s))
divides the Berger sphere into two regions {Φ(s, t ,θ) : 0 ≤ t ≤ π/4)} and {Φ(s, t ,θ) : π/4 ≤ t ≤ π/2}
and that the two regions are congruent; in fact, the reflection along any vertical geodesic Vθ0 gives
the congruence.
We first construct a minimal surface D− embedded in {Φ(s, t ,θ) : 0 ≤ t ≤π/4} whose boundary
components consist of vertical geodesics, and then reflect the surface D− with respect to a vertical
geodesic in the boundary to get the minimal surface D+ embedded in {Φ(s, t ,θ) : π/4 ≤ t ≤ π/2}.
Then we show that the embedded minimal surface D :=D−∪D+ is smooth without boundary.
Let m and l be positive integers and
k = 2ml .
As the basic block of our construction, we consider the pentahedral region
P0 := {Φ(s, t ,θ) : 0 ≤ s ≤π/k, 0 ≤ t ≤π/4, 0 ≤ θ ≤π/m}
bounded by five ruled minimal surfaces
H0 :=Φ(0, t ,θ), H1 :=Φ(π/k, t ,θ), V0 :=Φ(s, t ,0), V1 :=Φ(s, t ,π/m)
and T , see Figure 2.
For notational convenience, let
Hp,q := Hπp/k,πq/m , Vq :=Vπq/m
262 Heayong Shin, Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee and Seong-Deog Yang
ℋ1
V0
V1
ℋ0
T
Figure 2. P0
and let Rp,q denote the reflection across the horizontal geodesic Hp,q and let Rq denote the
reflection across the vertical geodesic Vq .
Let Γ ⊂ ∂P0 be the piecewise geodesic polygon of six segments of horizontal geodesics
H0,0, H1,0, H1,1, H0,1 and vertical geodesics V0,V1, which is a subset of the 1-skeleton of P0, see
Fig.3.
𝐻0,0
𝑉1𝑉0
𝐻1,0
𝐻1,1
𝐻0,1
Figure 3. Γ
Then, since the region P0 is mean-convex by [2], Γ spans an embedded minimal disk D0 which
lies inside of P0. Now D0 can be analytically extended across the boundary segments H1,1 by
reflection to get a minimal surface
D0,0 := D0 ∪R1,1(D0)
which is embedded in the region
P0,0 := {Φ(s, t ,θ) : 0 ≤ s ≤ 2π/k, 0 ≤ t ≤π/4, 0 ≤ θ ≤ 2π/m}.
The boundary of D0,0 consists of the geodesic segments
H0,0(t ), H1,0(t ), H0,1(t ), H1,2(t ), H2,1(t ), H2,2(t ) ; 0 ≤ t ≤π/4
Heayong Shin, Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee and Seong-Deog Yang 263
and
V0(s); 0 ≤ s ≤π/k, V1(s); 0 ≤ s ≤ 2π/k, V2(s); π/k ≤ s ≤ 2π/k.
It is clear that
R1,1(D0.0) = D0,0.
For each p = 0,1, . . . ,k −1 and q = 0,1, . . . ,m −1, we define
Dp,q := (T2πp/k ◦Rot2πq/m)(D0,0).
The surface Dp,q is embedded minimal surface in the region
Pp,q := {Φ(s, t ,θ) : 2πp/k ≤ s ≤ 2π(p +1)/k, 0 ≤ t ≤π/4, 2πq/m ≤ θ ≤ 2π(q +1)/m}
with the boundary consisting of the geodesic segments
H2p,2q (t ), H2p+1,2q (t ), H2p,2q+1(t ), H2p+1,2q+2(t ), H2p+2,2q+1(t ), H2p+2,2q+2(t ) ; 0 ≤ t ≤π/4
and
V2q (s), 2pπ/k ≤ s ≤ (2p +1)π/k,
V2q+1(s), 2pπ/k ≤ s ≤ (2p +2)π/k,
V2q+2(s), (2p +1)π/k ≤ s ≤ (2p +2)π/k.
One can see that the surfaces Dp,q are disjoint with each other except along the common bound-
ary segments. More specifically, Dp,q has common boundary H2p,2q with Dp−1,q−1, H2p+1,2q with
Dp,q−1, H2p,2q+1 with Dp−1,q , H2p+2,2q+1 with Dp+1,q , H2p+1,2q+2 with Dp,q+1 and H2p+2,2q+2 with
Dp+1,q+1.
Now, let us consider the surface
Dp,q ∪Dp−1,q−1
joined along H2p,2q . Since R1,1(D0,0) = D0,0 and since direct computation shows that
Rp,2q ◦T2πp/k ◦Rot2πq/m ◦R1,1 = T2π(p−1)/k ◦Rot2π(q−1)/m
one can see that
R2p,2q (Dp,q ) = (R2p,2q ◦T2πp/k ◦Rot2πq/m)(D0,0)
= (R2p,2q ◦T2πp/k ◦Rot2πq/m ◦R1,1)(D0,0)
= (T2π(p−1)/k ◦Rot2π(q−1)/m)(D0,0)
= Dp−1,q−1.
Therefore, Dp,q and Dp−1,q−1 are joined smoothly across the common boundary H2p,2q .
This argument shows that the surface
D− :=∪p,q Dp,q
is a smooth embedded minimal surface in the region {Φ(s, t ,θ) : 0 ≤ t ≤ π/4}. By construction,
one can see that the surface D− is invariant under the transformation T2π/k and Rot2π/m and that
∂D−, the boundary of D−, is
∂D− =
2m−1⋃
q=0
Vq .
Now let
D+ := R0(D−)
then D+ is a smooth minimal surface embedded in {Φ(s, t ,θ) : π/4 ≤ t ≤ π/2}. Since R0(Vq ) =
V2m−q , the boundary of D+ also consists of vertical geodesics V0,V1, . . . ,V2m−1. Since direct
computation gives that
R i ◦R0 = Tπi /m ◦Rot2πi /m
264 Heayong Shin, Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee and Seong-Deog Yang
and since k = 2ml , we have
R i (D+) = (R i ◦R0)(D−) = (Tπi /m ◦Rot2πi /m)(D−) = (T2πi l/k ◦Rot2πi /m)(D−) =D−.
This implies that the surface D+ is the reflection of D− along each common boundary Vi .
Hence the surface
D :=D+−∪D−
is a smooth minimal surface without boundary embedded in the Berger sphere. This completes
the construction.
References
[1] https://minimalsurfaces.blog/home/repository/singly-periodic/helicoidal-karcher-scherk-surfaces/.
[2] I. Meeks, William H., S.-T. Yau, “The existence of embedded minimal surfaces and the problem of uniqueness”, Math.
Z. 179 (1982), p. 151-168.
[3] H. Shin, Y. W. Kim, S.-E. Koh, H. Y. Lee, S.-D. Yang, “Ruled minimal surfaces in the Berger sphere”, Differ. Geom. Appl.
40 (2015), p. 209-222.
[4] ——— , “Compact embedded minimal surfaces in the Berger sphere”, C. R. Acad. Sci. Paris 356 (2018), no. 3, p. 333-
339.
[5] ——— , “Schwarz’ CLP-surfaces in Nil3”, Proc. Am. Math. Soc. 147 (2019), no. 4, p. 1677-1685.
[6] F. Torralbo, “Compact minimal surfaces in the Berger spheres”, Ann. Global Anal. Geom. 41 (2012), no. 4, p. 391-405.


Addendum to the paper: Compact embedded minimal surfaces in the Berger sphere

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Addendum to the paper: Compact embedded minimal surfaces in the Berger sphere

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