Comptes Rendus Mathématique

Golla Ramesh

Shanola S. Sequeira

Comptes Rendus
Mathématique
Golla Ramesh and Shanola S. Sequeira
Absolutely minimum attaining Toeplitz and absolutely norm attaining Hankel
operators
Volume 361 (2023), p. 973-977
Published online: 7 September 2023
https://doi.org/10.5802/crmath.457
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. 973-977
https://doi.org/10.5802/crmath.457
Operator theory / Théorie des opérateurs
Absolutely minimum attaining Toeplitz and
absolutely norm attaining Hankel operators
Golla Ramesh∗, a and Shanola S. Sequeiraa
a Department of Mathematics, IIT Hyderabad, Kandi, Sangareddy, Telangana-502284, India
E-mails: rameshg@math.iith.ac.in, ma18resch11001@iith.ac.in
Abstract. In this article, we completely characterize absolutely norm attaining Hankel operators and abso-
lutely minimum attaining Toeplitz operators. We also improve [19, Theorem 2.1], by characterizing the abso-
lutely norm attaining Toeplitz operator Tϕ in terms of the symbol ϕ ∈ L∞.
Mathematical subject classification (2010). 47B35, 47A10, 47B07.
Funding. The first author is supported by SERB Grant No. MTR/2019/001307, Govt. of India. The sec-
ond author is supported by the Department of Science and Technology- INSPIRE Fellowship (Grant No.
DST/INSPIRE FELLOWSHIP/2018/IF180107).
Manuscript received 13 October 2022, revised 4 November 2022, accepted 18 December 2022.
1. Introduction
Let L2 denote the Lebesgue space of all square integrable functions on the unit circle T in the
complex plane with respect to the normalized Lebesgue measure µ and L∞ be the Banach space
of all complex valued essentially boundedµ-measurable functions onT. Let H 2 denote the closed
subspace of L2 consisting of all those functions whose negative Fourier co-efficients vanish.
For any ϕ ∈ L∞, the Toeplitz operator Tϕ : H 2 → H 2 is defined by
Tϕ f = P (ϕ f ) for all f ∈ H 2,
and the Hankel operator Hϕ : H 2 → H 2 is defined by
Hϕ f = J (I −P )ϕ f for all f ∈ H 2,
where (ϕ f )(z) =ϕ(z) f (z) for all z ∈T, P is the orthogonal projection of L2 onto H 2 and J (z−n) =
zn−1,n = 0,±1,±2, . . . is the unitary operator on L2.
Toeplitz and Hankel operators are well studied in matrix theory, operator theory, function
theory etc., as they find applications in many fields. The initial study of Toeplitz operators was
done by Otto Toeplitz and thereafter Brown and Halmos in [4] discussed some of the important
properties of them. Further, H. Widom, R. G. Douglas, D. Sarason, etc., also worked on Toeplitz
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
974 Golla Ramesh and Shanola S. Sequeira
operators and most of their results can be found in [7]. We refer to [13,17] for the results on Hankel
operators.
One of the important properties of any bounded linear operator is it’s norm attaining prop-
erty. Such a property is vastly studied on Banach spaces (for example, see [1, 2, 12]). A few de-
velopments of norm attaining operators on Hilbert spaces can be found in [5, 8]. For more de-
tails on norm attaining Toeplitz and Hankel operators we refer to [3, 21]. It is important to note
that the set of all compact operators and isometries defined between the Hilbert spaces are al-
ways norm attaining. Moreover, on restriction of such operators to any non-zero closed subspace
of the Hilbert space is also norm attaining. This observation initiated the study of new class of
operators by Carvajal and Neves [5] and called them the absolutely norm attaining operators or
A N -operators.
A natural counterpart of the norm attaining and the absolutely norm attaining operators are
the minimum attaining and the absolutely minimum attaining operators or A M -operators,
which were also introduced by Carvajal and Neves in [6].
Due to the lack of non-zero compact Toeplitz operators on H 2, the authors in [19] were inter-
ested in looking at A N -Toeplitz operators, since this set is non-empty as it contains isometric
Toeplitz operators. The authors also characterized absolutely minimum attaining Hankel opera-
tors in the same article.
In this article, we address the following question which is posed in [19].
Question 1. Characterize all the A M -Toeplitz operators and all the A N -Hankel operators.
In [19, Theorem 2.1], a characterization of an A N -Toeplitz operator Tϕ,ϕ ∈ L∞ is given as
follows:
Theorem 2. Let ϕ ∈ L∞. Then Tϕ is an A N -operator if and only if ∥ϕ∥2∞I −T ∗ϕTϕ is a finite rank
operator.
Here we characterize such an operator in terms of the symbol ϕ itself, which improves the
above result.
We also characterize A M -Toeplitz operators and further prove that there are no non-compact
A N -Hankel operators. All these results are found in Section 2.
In the remaining part of this section, we give a few notations and definitions which are
necessary for developing the article.
1.1. Preliminaries
Let H1, H2 be infinite dimensional complex Hilbert spaces and B(H1, H2) denote the Banach
space of all bounded linear operators from H1 to H2. If T ∈ B(H1, H2), then T is said to be finite
rank if its range space is finite dimensional and compact if T maps any bounded set in H1 to a
pre-compact set in H2.
Definition 3 ([5, Definitions 1.1, 1.2]). An operator T ∈ B(H1, H2) is called norm attaining if
there exists a unit vector x ∈ H1 such that ∥T x∥ = ∥T ∥. If for every non-zero closed subspace M
of H1, T |M : M → H2 is norm attaining, then T is called absolutely norm attaining or an A N -
operator.
Definition 4 ([6, Definitions 1.1, 1.4]). An operator T ∈ B(H1, H2) is said to be minimum
attaining if there exists a unit vector x ∈ H1 such that m(T ) = ∥T x∥, where
m(T ) := inf{∥T x∥ : x ∈ H1,∥x∥ = 1}.
If for every non-zero closed subspace M of H1, T |M : M → H2 is minimum attaining, then T is
said to be absolutely minimum attaining or an A M -operator.
Golla Ramesh and Shanola S. Sequeira 975
For more details on A N -operators we refer to [5,14,16,18] and for that on A M -operators we
refer to [6, 9, 15].
Let B(H 2) be the Banach space of all bounded linear operators on H 2 and let F (H 2) and
K (H 2) denote the set of all finite rank and compact operators on H 2, respectively. The set of
all positive elements (operators) in F (H 2) and K (H 2) are denoted by F (H 2)+ and K (H 2)+, re-
spectively. The set of all A N and A M -operators on H 2 are denoted by A N (H 2) and A M (H 2),
respectively.
Let C (T) denote the set of all complex valued continuous functions on T. The Sarason algebra
H∞+C (T) is a closed subalgebra of L∞, where H∞ = H 2 ∩L∞. The notations and terminologies
used in [19] will be adhered throughout this article.
2. Main Results
We start this section by recalling a result which connects both Toeplitz and Hankel operators.
Proposition 5 ([21, Proposition 8]). For any ϕ,ψ ∈ L∞, we have H∗ψHϕ = Tψϕ−T ∗ψTϕ.
The following results characterize all finite rank and compact Hankel operators.
Theorem 6 ([17, Corollary 3.2, p. 21]). Let ϕ ∈ L∞. Then Hϕ ∈ F (H 2) if and only if (I −P )ϕ is a
rational function.
Theorem 7 ([13, Corollary 4.3.3, p. 145]). Let ϕ ∈ L∞. Then Hϕ ∈ K (H 2) if and only if ϕ ∈
H∞+C (T).
Now we characterize an A N -Toeplitz operator Tϕ in terms of ϕ.
Theorem 8. Let ϕ ∈ L∞. Then Tϕ ∈ A N (H 2) if and only if |ϕ| = ∥ϕ∥∞ a.e.(µ) and (I −P )ϕ is a
rational function.
Proof. By Theorem 2 and [19, Corollary 2.3], we have ∥ϕ∥2∞I −T ∗ϕTϕ = F , where F ∈F (H 2)+ and
|ϕ| = ∥ϕ∥∞ a.e.(µ). Hence by Proposition 5, we get H∗ϕHϕ = F ∈F (H 2)+. Therefore by Theorem 6,
it follows that (I −P )ϕ is a rational function.
Conversely, if |ϕ| = ∥ϕ∥∞ a.e.(µ) and (I −P )ϕ is a rational function, then H∗ϕHϕ = F , where
F ∈ F (H 2)+. Again by Proposition 5, we get ∥ϕ∥2∞I − T ∗ϕTϕ = F . Hence by Theorem 2, Tϕ ∈
A N (H 2). □
Corollary 9. Letϕ ∈ L∞. Then Tϕ ∈A N (H 2) if and only if |ϕ| = ∥ϕ∥∞ a.e.(µ) and Hϕ ∈A M (H 2).
Proof. By [19, Theorem 3.3], we have Hϕ ∈ A M (H 2) if and only if Hϕ ∈ F (H 2). Now, from
Theorems 6 and 8, we get the required conclusion. □
Remark 10. If Tϕ ∈A N (H 2), then it satisfies T ∗ϕTϕ+H∗ϕHϕ = ∥ϕ∥2∞I .
Next we characterize all A M -Toeplitz operators. For this, we need the following definition.
Definition 11. For T ∈B(H 2),
(1) the spectrum of T is defined by
σ(T ) = {λ ∈C : T −λI is not invertible in B(H 2)},
(2) the essential spectrum of T is defined by σess(T ) = σ(π(T )), where π : B(H 2) →
B(H 2)/K (H 2) is the canonical quotient map.
Theorem 12. Let ϕ ∈ L∞. Then Tϕ ∈ A M (H 2) if and only if |ϕ| = α a.e.(µ) and ϕ ∈ H∞+C (T),
where α ∈σess(|Tϕ|) and |Tϕ| is the positive square root of T ∗ϕTϕ.
976 Golla Ramesh and Shanola S. Sequeira
Proof. If Tϕ ∈A M (H 2), then by [9, Theorem 5.14], T ∗ϕTϕ ∈A M (H 2). From [9, Theorem 5.8], we
have TϕTϕ =α2I−K+F , whereα2 ∈σess(TϕTϕ),α≥ 0,K ∈K (H 2)+,F ∈F (H 2)+ with K ≤α2I and
K F = 0. Hence by [19, Propsition 2.18], we get |ϕ| =α a.e.(µ) and by [11, Lemma 3], α ∈σess(|Tϕ|).
By Proposition 5, we get H∗ϕHϕ = K − F ∈ K (H 2). Therefore Hϕ ∈ K (H 2) and by Theorem 7,
ϕ ∈ H∞+C (T).
Conversely, assume that |ϕ| = α a.e.(µ) and ϕ ∈ H∞ +C (T). Then H∗ϕHϕ = K , where K ∈
K (H 2)+. By Proposition 5, we get TϕTϕ =α2I−K . Then by [19, Proposition 3.5], TϕTϕ ∈A M (H 2)
and hence Tϕ ∈A M (H 2) by [9, Theorem 5.14]. □
In general, the set of all A N -operators (or A M -operators) is not closed. As {ϕ ∈ L∞ :
(I − P )ϕ is a rational function} is not a closed set, the set of all A N -Toeplitz operators is not
closed. But the set of all A M -Toeplitz operators is closed.
Let T∞(H 2) denote the set of all Toeplitz operators on H 2 induced by the symbols from L∞.
Proposition 13. The set of all A M - Toeplitz operators is closed in T∞(H 2).
Proof. Let Tψ be in the closure of set of all A M -Toeplitz operators. Then there exists a sequence
of A M - Toeplitz operators {Tϕn } such that Tϕn converges to Tψ in the operator norm. So by
Theorem 12, we have ϕn ∈ H∞ +C (T) and |ϕn | = αn , where αn ∈ σess(|Tϕn |) which further
implies α2n ∈ σess(T ∗ϕn Tϕn ) for all n ∈ N ([11, Lemma 3]). Since H∞+C (T) is a closed subalgebra
of L∞, we get ψ ∈ H∞ +C (T). Now by [9, Theorem 5.14], we have T ∗ϕn Tϕn ∈ A M (H 2) and
by [9, Theorem 5.8], we write T ∗ϕn Tϕn = β2n I − Kn + Fn , where β2n ∈ σess(T ∗ϕn Tϕn ),βn ≥ 0,Kn ∈
K (H 2)+,Fn ∈ F (H 2)+ with Kn ≤ β2n I and KnFn = 0 for all n ∈ N. From [15, Theorem 3.10], we
have σess(T ∗ϕn Tϕn ) = {β2n} = {α2n}, so αn = βn for all n ∈N. As α2n ≤ ∥T ∗ϕn Tϕn∥ ≤ supn∈N ∥Tϕn∥2, we
have {α2n} is bounded. Then there exists a subsequence {αnk } of {αn} such that α
2
nk converges
to α2 for some α ≥ 0. Hence T ∗ϕnk Tϕnk − α
2
nk I = Knk − Fnk converges to some K ∈ K (H 2).
Therefore T ∗ψTψ = α2I +K . Now by [7, Exercise 7.7], we get |ψ| = α, where α ∈ σess(|Tψ|). Hence
by Theorem 12, Tψ ∈A M (H 2). □
Remark 14. In [20], it was proved that the operator norm closure of A M -operators is the same
as the operator norm closure of A N -operators. Here by Theorems 8 and 12, we observe that
the set of all A N -Toeplitz operators is contained in the set of all A M -Toeplitz operators. So by
Proposition 13, we conclude that the operator norm closure of the set of A N -Toeplitz operators
is the set of A M -Toeplitz operators. In other words, set of all A N - Toeplitz operators is dense
in the set of all A M - Toeplitz operators with respect to the operator norm.
The next theorem shows that there is no non-compact absolutely norm attaining Hankel
operator. For a detailed study on compact Hankel operators we refer to [10].
Theorem 15. Let ϕ ∈ L∞. Then Hϕ ∈A N (H 2) if and only if Hϕ ∈K (H 2).
Proof. Clearly if Hϕ ∈K (H 2), then Hϕ ∈A N (H 2).
On the other hand, if Hϕ ∈ A N (H 2), then by [14, Corollary 2.11], H∗ϕHϕ ∈ A N (H 2). Hence
by [14, Theorem 2.5], H∗ϕHϕ = αI −F +K for some α ≥ 0, F ∈ F (H 2)+, K ∈ K (H 2)+ with F ≤ αI
and K F = 0. By Proposition 5, we getαI −F +K = T|ϕ|2 −T ∗ϕTϕ or T|ϕ|2−α−TϕTϕ = K −F ∈K (H 2).
So by [7, Exercise 7.7], |ϕ|2 −α=ϕϕ, which implies α= 0. Hence H∗ϕHϕ ∈K (H 2)+ which in turn
results in Hϕ ∈K (H 2). □
Corollary 16. Let ϕ ∈ L∞. Then Hϕ ∈A N (H 2) if and only if ϕ ∈ H∞+C (T).
Proof. The proof follows by Theorems 7 and 15. □
Golla Ramesh and Shanola S. Sequeira 977
Remark 17.
(1) By Theorem 15, we have that the set of all A N -Hankel operators is a closed linear
subspace of the space of all Hankel operators on H 2.
(2) By [19, Theorem 3.3], we have that the set of all A M -Hankel operators are exactly the
finite rank Hankel operators. Hence by Theorem 15, we conclude that the set of all A M -
Hankel operators is a dense subspace of the set of all A N -Hankel operators.
We summarize the results of this paper.
For any ϕ ∈ L∞,
(1) Tϕ ∈A N (H 2) if and only if |ϕ| = ∥ϕ∥∞ a.e.(µ) and (I −P )ϕ is a rational function.
(2) Tϕ ∈A M (H 2) if and only if |ϕ| =α a.e.(µ) and ϕ ∈ H∞+C (T), where α ∈σess(|Tϕ|).
(3) Hϕ ∈A N (H 2) if and only if ϕ ∈ H∞+C (T).
Acknowledgements
We thank the referee for valuable suggestions and the handling editor for the help during the
editorial process.
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Comptes Rendus
Mathématique
Dawei Chen
Complete curves in the strata of differentials
Volume 361 (2023), p. 371-373
https://doi.org/10.5802/crmath.416
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. 371-373
https://doi.org/10.5802/crmath.416
Complex analysis and geometry / Analyse et géométrie complexes
Complete curves in the strata of differentials
Dawei Chen∗, a
a Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA
E-mail: dawei.chen@bc.edu
Abstract. Gendron proved that the strata of holomorphic differentials with prescribed orders of zeros do not
contain complete algebraic curves by applying the maximum modulus principle to saddle connections. Here
we provide an alternative proof for this result by using positivity of divisor classes on moduli spaces of curves.
Funding. Research partially supported by National Science Foundation Grant DMS-2001040 and Simons
Foundation Collaboration Grant 635235.
Manuscript received 15 May 2022, revised 19 August 2022, accepted 1 September 2022.
For µ = (m1, . . . , mn) with ∑ni=1 mi = 2g − 2 and mi ≥ 0 for all i , let Hg (µ) be the stratum of
holomorphic differentials ω of zero type µ on smooth and connected complex genus g curves X ,
i.e., the underlying canonical divisor of ω is of type
∑n
i=1 mi pi for distinct points p1, . . . , pn in X .
The study of differentials is significant in surface dynamics and moduli theory. We simply
refer to [3, 10, 11] for an introduction to this fascinating subject. An important question for
understanding the geometry of a (non-compact) moduli space is to study complete subvarieties
contained in it. For instance for the moduli space of curves Mg , this amounts to finding families
of smooth curves that have non-constant complex structures and do not degenerate when
approaching to the boundary. While there exist complete subvarieties of positive dimension in
Mg , the sharp upper bound for their dimensions is largely unknown (see [8] for an upper bound),
e.g., it is unknown whether M4 contains a complete (algebraic) surface.
One can similarly ask this question for the strata of holomorphic differentials. In [9] Gendron
proved the following nice result.
Theorem. For µ= (m1, . . . , mn) with mi ≥ 0 for all i , the stratum Hg (µ) of holomorphic differen-
tials of type µ does not contain complete curves.
Gendron’s argument relies on flat geometry. It applies the maximum modulus principle to sad-
dle connections (i.e., geodesics joining the zeros of ω under the induced metric), which essen-
tially forces a complete family of differentials to be trivial. Below we provide a short algebraic
proof for this result, which uses positivity of divisor classes on moduli spaces of curves.
We first review some related moduli spaces and divisor classes. Denote by Pg (µ) =Hg (µ)/C∗
the projectivized stratum parameterizing canonical divisors of type µ; then Hg (µ) can be identi-
fied with the tautological bundle O (−1) over Pg (µ) with the zero section removed. We denote by
η= c1(O (−1)) the first Chern class of the tautological bundle. Consider the zeros of the canonical
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
372 Dawei Chen
divisors of type µ as marked points in the underlying curves; then Pg (µ) can be embedded in
Mg ,n as a subvariety. Let κ= f∗(c1(ω f )2) where ω f is the relative dualizing sheaf on the universal
curve f : X →Mg ,n . Let ψi be the first Chern class of the bundle of cotangent lines with respect
to the i th marked point. We also use the same notations for the restrictions of these divisor classes
from Mg ,n to Pg (µ). It is known that the divisor classes ψi and κ are proportional to η on Pg (µ)
(see [5, Proposition 2.1]):
η= (mi +1)ψi , κ= κµη
where κµ = 2g −2+n −∑ni=1 1/(mi +1).
Proof of the Theorem. Suppose C is a complete (irreducible) curve contained in Hg (µ). Let
C0 ⊂ Pg (µ) be the image of C under the projection Hg (µ) → Pg (µ). Since each fiber of Hg (µ)
over Pg (µ) is C∗, the complete curve C cannot be contained in a fiber. Therefore, C corresponds
to a multi-section of O (−1) over C0. Let s be the degree of C over C0. Then C gives a nowhere
vanishing section of O (−s) over C0, which implies that sη and hence ψi and κ are all trivial
restricted to C0 by the above divisor class relations. However, κ+∑ni=1ψi is an ample divisor class
on M g ,n (see [7, Theorem (2.2)] and [1, Chapter XIV, Theorem (5.1)])1 which thus has positive
degree on every complete curve contained in it and cannot be trivial on C0. This leads to the
desired contradiction. 
Remark 1. For the strata of meromorphic differentials, i.e., when µ contains some negative
entries, in [4] it was shown that both Hg (µ) and Pg (µ) do not contain complete curves. The
proof relies on the sign change of mi +1 at a zero and at a pole. However for the case of all mi ≥ 0,
a complete curve C0 in Pg (µ), if it exists, would not necessarily lift to a complete curve in Hg (µ),
e.g., the lift might cross the zero section of O (−1). Hence it remains open to determine whether
Pg (µ) contains a complete curve when all the entries of µ are nonnegative.
Note that if one adds the boundary divisor ∆0 parameterizing differentials on nodal curves
with non-separating nodes only, then for every signature µ with nonnegative entries the partial
compactification Pg (µ)∪∆0 contains infinitely many complete curves arising from Teichmüller
curves (see [6, Corollary 3.2]).
Finally one can study the strata of k-differentials H kg (µ) and k-canonical divisors P
k
g (µ) where
µ is a partition of k(2g −2). Via the canonical k-cover π, a k-differential ξ can lift to an abelian
differential ω (up to kth roots of unity), i.e., π∗ξ=ωk , where ω is holomorphic if and only if ξ has
no pole of order ≤ −k (see e.g., [2, Section 2.1]). Therefore, H kg (µ) contains no complete curves
for any µ and P kg (µ) contains no complete curves if µ has an entry ≤ −k. It remains open to
determine whether P kg (µ) contains a complete curve when all entries of µ are bigger than −k.
Acknowledgements
The author thanks the referees for carefully reading the paper and very helpful comments.
References
[1] E. Arbarello, M. Cornalba, P. Griffiths, Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen
Wissenschaften, vol. 268, Springer, 2011, with a contribution by Joseph Daniel Harris.
1This ample divisor class can also be written as 12λ−δ+∑ni=1ψi by using the well-known relation 12λ= κ+δ where
λ is the first Chern class of the Hodge bundle and δ is the total boundary divisor class. Note that our κ class differs from
the one in [1] by the totalψ class, as we use the ordinary relative dualizing sheaf instead of the log version in the definition
of κ.
Dawei Chen 373
[2] M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, M. Möller, “Strata of k-differentials”, Algebr. Geom. 6 (2019),
no. 2, p. 196-233.
[3] D. Chen, “Teichmüller dynamics in the eyes of an algebraic geometer”, in Surveys on recent developments in algebraic
geometry, Proceedings of Symposia in Pure Mathematics, vol. 95, American Mathematical Society, 2017, p. 171-197.
[4] ——— , “Affine geometry of strata of differentials”, J. Inst. Math. Jussieu 18 (2019), no. 6, p. 1331-1340.
[5] ——— , “Tautological ring of strata of differentials”, Manuscr. Math. 158 (2019), no. 3-4, p. 345-351.
[6] D. Chen, M. Möller, “Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus”, Geom. Topol.
16 (2012), no. 4, p. 2427-2479.
[7] M. D. T. Cornalba, “On the projectivity of the moduli spaces of curves”, J. Reine Angew. Math. 443 (1993), p. 11-20.
[8] S. Diaz, “A bound on the dimensions of complete subvarieties of Mg ”, Duke Math. J. 51 (1984), no. 2, p. 405-408.
[9] Q. Gendron, “Les strates ne possèdent pas de variétés complètes”, C. R. Math. Acad. Sci. Paris 358 (2020), no. 2,
p. 197-200.
[10] A. Wright, “Translation surfaces and their orbit closures: an introduction for a broad audience”, EMS Surv. Math. Sci.
2 (2015), no. 1, p. 63-108.
[11] A. Zorich, “Flat surfaces”, in Frontiers in number theory, physics, and geometry. I, Springer, 2nd printing ed., 2006,
p. 437-583.


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