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Comptes Rendus
Mathématique
Sergio A. Carrillo
A quick proof of the regularity of the flow of analytic vector fields
Volume 359, issue 9 (2021), p. 1155-1159
<https://doi.org/10.5802/crmath.271>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
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
Comptes Rendus
Mathématique
2021, 359, n 9, p. 1155-1159
https://doi.org/10.5802/crmath.271
Complex analysis and geometry, Dynamical systems / Analyse et géométrie complexes,
Systèmes dynamiques
A quick proof of the regularity of the flow of
analytic vector fields
Sergio A. Carrilloa
a Programa de matemáticas, Universidad Sergio Arboleda, Calle 74 # 14-14, Bogotá,
Colombia.
E-mail: sergio.carrillo@usa.edu.co
Abstract. We offer a new and elementary proof of the convergence of the Lie series giving the flow of an
analytic vector field as well as a natural deduction of such series.
2020 Mathematics Subject Classification. 32M25.
Funding. The author is supported by Ministerio de Economía y Competitividad from Spain, under the
Project “Métodos asintóticos, algebraicos y geométricos en foliaciones singulares y sistemas dinámicos” (Ref.:
PID2019-105621GB-I00) and Univ. Sergio Arboleda project IN.BG.086.20.002.
Manuscript received 29th July 2021, accepted 8th September 2021.
1. Introduction
Consider a germ X of an analytic vector field defined at a point p on a n-complex analytic
manifold M . In local coordinates z = (z1, . . . , zn) such that z(p) = 0 ∈ Cn , X is a vector field in
(Cn ,0), i.e., a n-tuple of convergent power series
X = (X1, . . . , Xn) ∈C{z}n ,
or the derivation
X ( f )(z) =
n∑
j=1
X j (z)
∂ f
∂z j
(z), where f ∈C{z}.
The dynamics of X is determined through its flowφX (t , z), which is the solution of the differential
equation
∂tφX (t , z) = X (φX (t , z)), φX (0, z) = z. (1)
Geometrically, for z0 ∈ Cn near 0, the map t ∈ (C,0) 7→ φX (t , z0) is a curve passing through z0 at
t = 0, having at the point φX (t , z0) the tangent vector X (φX (t , z0)).
A classical result in the general theory of ordinary differential equations is that (1) has a unique
solution which is as regular as X is. In our case, if X is analytic, the flow will be analytic in both
variables (t , z). More specifically, we have:
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1156 Sergio A. Carrillo
Theorem 1. Given a germ X of an analytic vector field at 0 ∈ Cn , the differential equation (1)
admits a unique analytic solution φX (t , z) ∈C{t , z}n which is given by the Lie series
φX (t , z) =
∞∑
m=0
(X m(z1), . . . , X
m(zn))
t m
m!
. (2)
Here X m denotes the iterations of X as a derivation, i.e.,
X 0( f ) = f , X m+1( f ) := X (X m( f )), m ≥ 0, for f ∈C{z}. (3)
The proof of the analyticity of φX is usually done using the Cauchy mayorant series tech-
nique [3, Chapter VII] or by the contraction map principle [5, Chapter I.1] which extends to less
regular systems. Then, the expansion (2) is obtained as a consequence of analyticity writing φX
as its Taylor series at t = 0. In particular, once we have an analytic solution of (1) at (t , z) = (0,0),
it must be series (2). On the other hand, the series (2) can be shown to be convergent employing
mayorizing series as explained in Gröbner’s book [4, Satz 2]. Due to his contribution this series is
also known as a Gröbner–Lie series, having applications in the inversion of a mapping of power
series [9, Section 2], in the resolution of systems of autonomous and non-autonomous differen-
tial equations [4, 8], and in the classification of tangent to the identity biholomorphisms [1, 6].
We will prove Theorem 1 finding the series (2) as a solution to an auxiliary linear partial dif-
ferential equation, proving its convergence with the aid of Nagumo norms, and finally checking
that (2) solves (1) using the analyticity of this map. This proof is even more elementary than previ-
ous approaches due to the linear character of the auxiliary problem. In fact, it only uses Cauchy’s
formulas and Taylor expansions.
2. The proof of the theorem
We need some preliminaries. First, we extend (3) to maps F = ( f1, . . . , fn) ∈C{z}n by
X m(F ) := (X m( f1), . . . , X m( fn)), m ≥ 0.
In particular, X m(id) = (X m(z1), . . . , X m(zn)) where id(z) = z. Moreover,
X m+1(F )(z) = Dz [X m(F )](z) ·X (z), (4)
where Dz denotes the Jacobian matrix with respect to z and · stands for the usual product of a
matrix and a vector.
Second, ifφX (s,φX (τ, z0)) andφX (τ+s, z0) are defined, the 1-parametric group law asserts that
φX (s,φX (τ, z0)) =φX (τ+ s, z0),
because both sides of the equality are solutions of (1) at the point φX (τ, z0), but such solution is
unique. In particular, φX (t ,φX (−t , z0)) = z0, for small z0 and t . If we differentiate this equation
with respect to t , we find that
∂tφX (t ,φX (−t , z0))+DzφX (t ,φX (−t , z0))∂tφX (−t , z0)(−1) = 0.
Using (1) at −t and setting z =φX (−t , z0) it turns out that w =φX satisfies
∂t w(t , z) = Dz w(t , z) ·X (z), (5)
see [1, p. 546]. This is a linear PDE in w ∈Cn that will lead to the series (2), as we shall see.
Finally, to establish the convergence of (2) we use the Nagumo norms. These were introduced
by M. Nagumo in his work [7] on power series solutions of analytic PDEs. These norms have
numerous applications, for instance, in singularly perturbed problems, see [2].
Let Dnr = {z ∈ Cn : |z j | < r, j = 1, . . . ,n} be the polydisc centered at the origin with polyradius
(r, . . . ,r ), for a common r > 0. We also write O (Dnr ) for the set of complex-valued holomorphic
functions on Dnr .
C. R. Mathématique — 2021, 359, n 9, 1155-1159
Sergio A. Carrillo 1157
Definition 2. If r > 0, f ∈O (Dnr ), and m ∈N the mth Nagumo norm of f is
‖ f ‖m := sup
z∈Dnr
| f (z)|(r −|z1|)m · · · (r −|zn |)m .
If m = 0 this is just the supremum norm which is finite in case f is bounded on Dnr . In this
situation ‖ f ‖m ≤ r mn‖ f ‖0. Moreover, we have:
Lemma 3. If f , g ∈O (Dnr ) and m,k ∈N, then
(1) ‖ f + g‖m ≤ ‖ f ‖m +‖g‖m and ‖ f g‖m+k ≤ ‖ f ‖m‖g‖k .
(2) ‖ ∂ f∂z j ‖m+1 ≤ e(m +1)r
n−1‖ f ‖m , j = 1, . . . ,n.
For the sake of completeness we include the proof at the end of the paper. Since we work
with vector- and matrix-valued maps, we extend these norms for F = ( f1, . . . , fn) ∈ O (Dnr )n and
A = (Ai j )i , j=1,...,n ∈O (Dnr )n×n by the rules
‖F‖m := max
1≤ j≤n
‖ f j ‖m and ‖A‖m := max
1≤i≤n
n∑
j=1
‖Ai j ‖m .
Then Lemma 3 shows that
‖A ·F‖m+k ≤ ‖A‖m‖F‖k , ‖Dz F‖m+1 ≤ ner n−1(m +1)‖F‖m . (6)
We are now ready to prove the result.
Proof of Theorem 1. We assume X is not identically zero. Let us fix a closed polydisc Dnr where
each X j is bounded and holomorphic in a neighborhood of this domain. Taking into account
equation (5), we search for a solution w(t , z) of the initial-value problem
∂t w(t , z) = Dz w(t , z) ·X (z), w(0, z) = z. (7)
Setting w(t , z) =∑∞m=0ϕm(z)t m/m!, after equating coefficients in common powers of t m , we find
ϕ0(z) = w(0, z) = z, ϕm+1(z) = Dzϕm(z) ·X (z), m ≥ 0. (8)
Due to (4), ϕm(z) = X m(id)(z), and thus ϕm is holomorphic and bounded on Dnr , for all m ≥ 0. To
prove that w(t , z) converges we study the series
W (τ) :=
∞∑
m=0
‖ϕm‖m τ
m
m!
,
where ‖·‖m denotes the vector mth Nagumo norm on the polydisc Dnr . First, since X is bounded,
then ‖X ‖0 > 0 is finite. Now, thanks to (6) we find that
‖ϕm+1‖m+1
(m +1)! ≤
‖Dzϕm‖m+1
(m +1)! ‖X ‖0 ≤ ner
n−1‖X ‖0 ‖ϕm‖m
m!
.
Since ‖ϕ0‖0 = ‖z‖0 = r , the previous inequality shows that
‖ϕm‖m
m!
≤ rαm , α := ner n−1‖X ‖0.
This proves that W (τ) is convergent. Finally, if |z1|, . . . , |zn | ≤ ρ < r , then
max
1≤ j≤n
|X m(z j )|
m!
≤ ‖ϕm‖m/m!
(r −|z1|)m · · · (r −|zn |)m
≤ rαm(r −ρ)−mn ,
showing that w(t , z) converges in the polydisc D(r−ρ)n /α×Dnρ .
To conclude we prove that w(t , z) solves (1). For this, since X ◦w is analytic, it is equal to its
Taylor series around t = 0:
X (w(t , z)) =
∞∑
m=0
∂mt (X ◦w)(0, z)
t m
m!
.
Therefore, we need to check that
∂mt (X ◦w) (0, z) =ϕm+1(z), for all m ≥ 0, (9)
C. R. Mathématique — 2021, 359, n 9, 1155-1159
1158 Sergio A. Carrillo
since then
X (w(t , z)) =
∞∑
m=0
ϕm+1(z)
t m
m!
= ∂t
( ∞∑
m=0
ϕm(z)
t m
m!
)
= ∂t w(t , z),
as required. We proceed by induction on m. If m = 0, X (w(0, z)) = X (z) =ϕ1(z) and (9) is valid. If
we assume it holds for some m, note first that
∂m+1t (X ◦w) = ∂mt ∂t (X ◦w) = ∂mt ((Dz X ◦w) ·∂t w) = ∂mt ((Dz X ◦w) ·Dz w ·X )
= ∂mt Dz (X ◦w) ·X = Dz∂mt (X ◦w) ·X .
Setting t = 0 and using the induction hypothesis we conclude that
∂m+1t (X ◦w)(0, z) = Dz∂mt (X ◦w)(0, z) ·X (z)
= Dz
(
∂mt (X ◦w)(0, z)
) ·X (z) = Dzϕm+1(z) ·X (z) =ϕm+2(z),
as desired. 
Theorem 1 gives an alternative to establish some elementary observations:
(i). If we change the initial condition in (7) and consider instead
∂t w(t , z) = Dz w(t , z) ·X (z), w(0, z) = F (z),
for F ∈ C{z}n , this system has as unique solution Φ(t , z) = ∑∞m=0 X m(F )(z) t mm! , whose form and
convergence are determined in exactly the same way as before. But Ψ(t , z) = F (φX (t , z)) also
satisfies this system sinceΨ(0, z) = F (z) and
∂tΨ(t , z) = Dz [F ](φX (t , z))∂tφX (t , z)
= Dz [F ](φX (t , z))Dz [φX ](t , z)X (z) = DzΨ(t , z)X (z),
where we used (5). In conclusion, we recover the classical formula
F (φX (t , z)) =
∞∑
m=0
X m(F )(z)
t m
m!
,
for the Lie series associated to X and F .
(ii). If X (0) = 0 and X (z) = Az +·· · , where A = Dz X (0), then
Dz [X
m(id)](0) = Am .
In fact, differentiating the second equation in (8) and taking z = 0 shows that Dzϕm+1(0) =
Dzϕm(0)·A, which proves the claim. WritingφX (t , z) as a power series in z with coefficients series
in t we get
φX (t , z) = e At z +·· · .
In particular, φX (t , · ) : (Cn ,0) → (Cn ,0) is a biholomorphism. Moreover, if A = 0, then φX (t , · ) is
tangent to the identity.
We conclude this note with the proof of Lemma 3.
Proof of Lemma 3. We prove (2) for m > 0 and j = 1. The remaining statements are easily
supplied. Fix z = (z1, z ′) ∈ Dnr , z ′ ∈Cn−1, and take 0 < ρ < r −|z1| so that D = {y ∈C : |y − z1| ≤ ρ} is
contained in Dr . Cauchy’s formulas show that∣∣∣∣ ∂ f∂z1 (z)
∣∣∣∣≤ 1ρ sup|y−z1|=ρ | f (y, z ′)|, (10)
By Definition 2 we have that
| f (y, z ′)| ≤ ‖ f ‖m
(r −|y |)mR , where R =
n∏
j=2
(r −|z j |)m ,
C. R. Mathématique — 2021, 359, n 9, 1155-1159
Sergio A. Carrillo 1159
and taking the supremum for y on the boundary of the disc D , we obtain
sup
|y−z1|=ρ
| f (y, z ′)| ≤ ‖ f ‖m
(r − (ρ+|z1|))mR
. (11)
Putting together (10) and (11) we find∣∣∣∣∣ ∂ f∂z1 (z)
n∏
j=1
(r −|z j |)m+1
∣∣∣∣∣≤ r n−1 ‖ f ‖mρ (r −|z1|)m+1(r −|z1|−ρ)m .
The result follows taking ρ = r−|z1|m+1 and using the inequality
(
1− 1m+1
)−m < e. 
References
[1] P. Ahern, J.-P. Rosay, “Entire functions, in the classification of differentiable germs tangent to the identity, in one or
two variables”, Trans. Am. Math. Soc. 347 (1995), no. 2, p. 543-572.
[2] M. Canalis-Durand, J.-P. Ramis, R. Schäfke, Y. Sibuya, “Gevrey solutions of singularly perturbed differential equations”,
J. Reine Angew. Math. 518 (2000), no. 2, p. 95-129.
[3] H. Cartan, Elementary theory of analytic functions of one or several complex variables, Hermann, 1973.
[4] W. Gröbner, Die Lie-Reihen und ihre Anwendungen, VEB Deutscher Verlag der Wissenschaften, 1960.
[5] Y. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86,
American Mathematical Society, 2008.
[6] J. Martinet, J.-P. Ramis, “Classification analytique des équations différentielles non linéaires résonnantes du premier
ordre”, Ann. Sci. Éc. Norm. Supér. 16 (1983), no. 4, p. 571-621.
[7] M. Nagumo, “Über das Anfangswertproblem partieller Differentialgleichunge”, Jap. J. Math. 18 (1942), p. 41-47.
[8] S. Steinberg, “Lie series and nonlinear ordinary differential equations”, J. Math. Anal. Appl. 101 (1984), p. 39-63.
[9] R. Winkel, “An exponential formula for polynomial vector fields: II. Lie series, exponential substituition, and rooted
trees”, Adv. Math. 147 (1999), p. 260-303.
C. R. Mathématique — 2021, 359, n 9, 1155-1159


A quick proof of the regularity of the flow of analytic vector fields

Fabiano G. B. Brito

Jackeline Conrado

Icaro Gonçalves

Adriana V. Nicoli

Comptes Rendus
Mathématique
Fabiano G. B. Brito, Jackeline Conrado, Icaro Gonçalves and Adriana
V. Nicoli
Area minimizing unit vector fields on antipodally punctured unit
2-sphere
Volume 359, issue 10 (2021), p. 1225-1232
<https://doi.org/10.5802/crmath.258>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
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
Comptes Rendus
Mathématique
2021, 359, n 10, p. 1225-1232
https://doi.org/10.5802/crmath.258
Geometry and Topology / Géométrie et Topologie
Area minimizing unit vector fields on
antipodally punctured unit 2-sphere
Fabiano G. B. Britoa, Jackeline Conradob, Icaro Gonçalvesa
and Adriana V. Nicolib
a Centro de Matemática, Computação e Cognição, Universidade Federal do ABC,
Santo André, 09210-170, Brazil
b Dpto. de Matemática, Instituto de Matemática e Estatística, Universidade de São
Paulo, R. do Matão 1010, São Paulo-SP, 05508-900, Brazil
E-mails: fabiano.brito@ufabc.edu.br, avnicoli@ime.usp.br,
icaro.goncalves@ufabc.edu.br, avnicoli@ime.usp.br
In memory of Amine Fawaz
Abstract. We provide a lower value for the volume of a unit vector field tangent to an antipodally punctured
Euclidean sphere S2 depending on the length of an ellipse determined by the indexes of its singularities. We
also exhibit minimizing vector fields ~vk within each index class and show that they are the only ones that are
sharp for the volume. These fields have areas given essentially by the length of ellipses depending just on the
indexes in N and S.
Funding. The second and fourth authors was financed by Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior- Brasil (CAPES) - Finance Code 001.
Manuscript received 18th February 2021, revised 20th May 2021 and 4th August 2021, accepted 11th Au-
gust 2021.
1. Introduction and main results
For a compact Riemannian manifold (M , g ), the volume of a smooth vector field ~v : M → T M is
the volume of its image~v(M) 7→ (T M ; g Sas ), where g Sas is called Sasaki metric and it is defined by
declaring the orthogonal complement of the vertical distribution to be the horizontal distribution
given by the Levi-Civita connection ∇. In terms of ∇ and g ,
vol(~v) =
∫
M
√
det(I + (∇~v)(∇~v)∗)ν, (1)
where I is the identity, (∇~v)∗ is the adjoint operator an ν is the volume form of M .
Back in 1986, Gluck and Ziller ( [5]) proved that Hopf flows are the unit vector fields of
minimum volume in M =S3. The theorem reads
Theorem 1 (Gluck and Ziller). Hopf unit vector fields (~vH ) are the minimum for the volume on
S3 and no others.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1226 Fabiano G. B. Brito, Jackeline Conrado, Icaro Gonçalves and Adriana V. Nicoli
However, in 1988 Johnson proved that Hopf vector fields are unstable in higher dimensions
(see [6]):
Theorem 2 (Johnson). The Hopf vector fields on S2n+1 are unstable for n > 1.
Later, in 2008 Brito, Chacón and Johnson ( [2]) established a relationship between the volume
of unit vector fields and their indexes around isolated singularities. More precisely:
Theorem 3 (Brito, Chacón and Johnson). Let S2 or S3 be the standard Euclidean sphere with
two antipodal points N and S are removed. Let ~v be a unit smooth vector field defined in those
manifolds and I~v (P ) the Poincaré index of ~v around P. Then,
• for S2\{N ,S}, vol(~v) ≥ 12 (π+|I~v (N )|+ |I~v (S)|−2)vol(S2)• for S3\{N ,S}, vol(~v) ≥ (|I~v (N )|+ |I~v (S)|)vol(S3)
Let~vR be the north-south field, then it will achieve both equalities in the theorem. In this case,
for S2\{N ,S}, vol(~vR ) = 12π vol(S2) and in S3\{N ,S}, vol(~vR ) = 2 vol(S3) = vol(~vH )
In terms of foliations on S2, Fawaz (see [4]) studied the minimal value of meromorphic
foliations. The minimum value is achieved by taking the foliation of the sphere S2 by parallels.
Formally:
Theorem 4 (Fawaz). Let F be a foliation on the Riemann sphere S2 given by the real part of a
meromorphic or holomorphic vector field, then vol(F ) ≥ 2π2.
In 2010, Borrelli and Gil-Medrano proved (see [1]) that the Pontryagin fields are area-
minimizing of the unit 2-sphere. Pontryagin fields of Sn are any unit vector field ~vP tr y defined
in a dense open subset U such that the closure of~vP tr y (U ) is the n-dimensional generalized Pon-
tryagin cycle of the unit vector bundle of the n-sphere (T 1Sn).
Theorem 5 (Borrelli and Gil-Medrano). Among unit vector fields without boundary of S2(1)\{P }
those of least area are Pontryagin fields (~vP tr y ) and no others.
Recently, in 2019, Theorem 3 was extended to odd dimensional spheres S2n+1\{±P }, see [3].
Theorem 6 (Brito, Gomes and Gonçalves). If ~v is a unit vector field on S2n+1\{±P }, then
vol(~v) ≥ π
4
(|I~v (P )|+ |I~v (−P )|)vol(S2n).
In this article, we establish sharp lower bounds for the total area of unit vector fields on
antipodally punctured Euclidean sphere S2, and these values depend on the indexes of their
singularities. We show that this lower bound is sharp and describe the vector fields~vk that achieve
the minimum for each positive index k, we also show that these unit vector fields are the only ones
with this property. In fact, given an index k, we informally declare a unit vector field ~vk with one
or two singularities satisfying:
(1) ~vk is parallel along meridians
(2) ~vk turns k −1 times along each parallel at a constant angle speed
(3) we establish an “initial meridian” along which ~vk makes an angle θ with each parallel
Theorem. Let ~v be a unit vector field defined on M =S2\{N ,S}. If k = max{I~v (N ), I~v (S)}, then
vol(~v) ≥πL(εk ),
where L(εk ) is the length of the ellipse
x2
k2
+ y2
(k−2)2 = 1, with a positive index k and I~v (P ) stands for
the Poincaré index of ~v around P.
This is a natural extension of Theorems 3 and 5 aforementioned. We also exhibit vector fields
~vk achieving the minimum volume for each index k, i.e. the lower bound is sharp. These results,
as long as we know, completely solve the Gluck-Ziller problem for the antipodally punctured unit
2-sphere. Borreli and Gil-Medrano solved the case where k = 2, [1].
C. R. Mathématique — 2021, 359, n 10, 1225-1232
Fabiano G. B. Brito, Jackeline Conrado, Icaro Gonçalves and Adriana V. Nicoli 1227
2. Existence and uniqueness of vector fields with total area πL(ε)
Let M =S2\{N ,S} be the Euclidean sphere in which two antipodal points N and S are removed.
Denote by g the usual metric of S2 induced from R3, and by ∇ the Levi-Civita connection
associated to g . Consider the oriented orthonormal local frame {e1,e2} on M , where e1 is tangent
to the parallels and e2 to the meridians. Let ~v be a unit vector field tangent to M and consider
another oriented orthonormal local frame
{
~v ,~v⊥
}
on M and its dual basis {ω1,ω2} compatible
with the orientation of {e1,e2}.
In dimension 2, the volume of ~v given in equation (1) reduces to
vol(~v) =
∫
S2
√
1+γ2 +δ2ν, (2)
where γ = g (∇~v~v ,~v⊥) and δ = g (∇v⊥~v⊥,~v) are the geodesic curvatures associated to ~v and ~v⊥,
respectively, and ν is the volume form.
Let S1α be the parallel of S
2 at latitude α ∈ (−π2 , π2 ) and S1β be the meridian of S2 at longitude
β ∈ (0,2π).
We formally define a family of unit vector fields achieving the volume given by the Theorem.
Definition 7. Let k be a positive integer and define:
(1) ~v1(p) =~e2(p), if k = 1;
(2) ~vk (p) = cosθ(p)~e1(p) + sinθ(p)~e2(p), if k > 1, where θ : S2\{N ,S} → R is given by
θ(α,β) = (k −1)β+θ0, (where θ0 is a constant), in that way,
θ1(p) = k −1√
x2 + y2
and θ2(p) = 0,
where θ1 = dθ(e1) and θ2 = dθ(e2) are the derivatives of θ on e1 and e2, respectively.
Notice that θ has constant variation along the parallel x2 + y2 = cosα, with α ∈ (−π2 , π2 )
constant, and this includes the case where k = 1. Remember that the case k = 2 has one
singularity (see [1]).
If we use spherical coordinates (α,β) so that p = (cosαcosβ,cosαsinβ, sinα), we can say that
the vector~vk spins around a point P at a constant speed of rotation along the parallelα. Moreover,
~vk gives exactly k −1 turns when it passes the α parallel, with respect to the referential
{
~e1,~e2
}
,
and it gives k turns with respect to a fixed polar referential, in this case, θ1(p) = k−1cosα . In Figure 1
is given a visual representation about the behavior of ~vk s and in Figure 2 we have a unit vector
field with k = 4.
Lemma. Let θ ∈ [0,π/2] be the oriented angle from e1 to ~v. If ~v = (cosθ)e1 + (sinθ)e2 and
~v⊥ = (−sinθ)e1 + (cosθ)e2, then
1+γ2 +δ2 = 1+ (tanα+θ1)2 +θ22 .
Proof. We write γ and δ as the following sums
γ= A+B +C +D and δ= A′+B ′+C ′+D ′,
with
A = g (∇(cosθ)e1 (cosθ)e1,~v⊥) , B = g (∇(sinθ)e2 (cosθ)e1,~v⊥) ,
C = g (∇(cosθ)e1 (sinθ)e2,~v⊥) , D = g (∇(sinθ)e2 (sinθ)e2,~v⊥)
and
A′ = g (∇(−sinθ)e1 (−sinθ)e1,~v) , B ′ = g (∇(cosθ)e2 (−sinθ)e1,~v)
C ′ = g (∇(−sinθ)e1 (cosθ)e2,~v) , D ′ = g (∇(cosθ)e2 (cosθ)e2,~v) .
C. R. Mathématique — 2021, 359, n 10, 1225-1232
1228 Fabiano G. B. Brito, Jackeline Conrado, Icaro Gonçalves and Adriana V. Nicoli
Figure 1. Visual representation of ~vk s: The angle θ, between the parallels and the field
~vk , changes as k changes. For example, the angle θ between ~v3 and parallels is twice the
angle θ between parallels and ~v2. This is to be expected, since ~v3 has a singularity with an
index equal to 3 (with four “petals”) and ~v2 has a singularity with an index equal to 2 (with
two “petals”). Also, ~v1 is the south-north vector field, which forms an angle θ = π2 with the
parallels. See [1] and [7] for more details.
Observe that tanα = g (∇e1 e1,e2) and ∇e2 e2 = 0. Indeed, define ψ : U → R3 as ψ(α,β) =
(cosαsinβ,cosαcosβ, sinα), where
(− π2 , π2 )× (0,2π) := U ⊂ R2. In this case, ψα = e2, ψβ =
e1 cos(α) and ψαα = ∇e2 e2. Therefore, tanα = g
(∇e1 e1,e2) and since the e2 is tangent to the
meridians, ∇e2 e2 = 0 in S2.
By a straightforward computation,
γ= cosθ(tanα+θ1)+ (sinθ)θ2, (3)
δ= sinθ(tanα+θ1)− (cosθ)θ2. (4)
From equations (3) and (4), we have
1+γ2 +δ2 = 1+ (cosθ(tanα+θ1)+ (sinθ)θ2)2 + (sinθ(tanα+θ1)− (cosθ)θ2)2
= 1+cos2θ(tanα+θ1)2 + (sinθ)2θ22 + sin2θ(tanα+θ1)2 + (cos2θ)θ22
= 1+ (tanα+θ1)2 +θ22 .
Finally,
1+γ2 +δ2 = 1+ (tanα+θ1)2 +θ22 . 
C. R. Mathématique — 2021, 359, n 10, 1225-1232
Fabiano G. B. Brito, Jackeline Conrado, Icaro Gonçalves and Adriana V. Nicoli 1229
This Lemma allows us to rewrite the volume functional as an integral depending on the
latitude α and the derivatives of θ
vol(~v) =
∫
M
√
1+ (tanα+θ1)2 +θ22 ν. (5)
Proposition 8. Let ~vk be unit vector field on M =S2\{N ,S}. Then,
vol(~vk ) =πL(εk ),
if and only if ~vk satisfies the aforementioned definition.
Proof. Using the Lemma we have
vol(~vk ) =
∫
M
√
1+ (tanα+θ1)2 +θ22 ν.
Assuming θ1 = k−1cosα and θ2 = 0, we obtain
vol(~vk ) =
∫
M
√
1+
(
tanα+ k −1
cosα
)2
ν
=
∫
M
√
1+
(
sinα+k −1
cosα
)2
ν
=
∫
M
√
1+ (k −1)2 +2(k −1)sinα
cosα
ν
= lim
α0→− π2
∫ π
2
α0
∫ 2π
0
√
1+ (k −1)2 +2(k −1)sinα
cosα
cosαdβdα
= 2π
∫ π
2
− π2
√
1+ (k −1)2 +2(k −1)sinαdα.
Taking t = α2 + π4 we have
vol(~vk ) = 4π
∫ π
2
0
√
(k −2)2 +4(k −1)sin2 t dt
=πL(εk ).
On the other hand, if vol(~vk ) =πL(εk ),
vol(~vk ) =
∫ √
1+ (tanα+θ1)2 +θ22 ν
≥
∫ √
1+ (tanα+θ1)2ν
≥
∫
|cosϕ+ sinϕ(tanα+θ1)|ν
=πL(εk ),
then θ2 = 0 and cosϕ(tanα+θ1) = sinϕ, where ϕ ∈ R. We conclude that θ1 = tanϕ− tanα and
ϕ=ϕ(α) = arctan
(
tanα+ k−1cosα
)
, which implies θ1 = k−1cosα . 
Example (Poincaré Index 4 and −2). Let~v be a unit vector field inS2 with two singularities with
indexes 4 and −2. Thus the volume of ~v is bounded by
vol(~v) ≥ 4π
∫ 2π
0
√
4+12sin2 t dt
Figures 2 and 3 provide a visual representation for a unit vector field with two singularities with
Poincaré indexes 4 and −2.
C. R. Mathématique — 2021, 359, n 10, 1225-1232
1230 Fabiano G. B. Brito, Jackeline Conrado, Icaro Gonçalves and Adriana V. Nicoli
Figure 2. Singularity of index 4 in north
pole
Figure 3. Singularity of index −2 in
south pole
3. Proof of the Theorem
We will follow most of the arguments found in the proof of the main Theorem of [2].
Proof of the Theorem. Given a, b, ϕ ∈ R we have a general inequality
p
a2 +b2 ≥ |a cosϕ+
b sinϕ|, which implies √
1+ (tanα+θ1)2 ≥
∣∣cosϕ+ sinϕ(tan(α)+θ1)∣∣ .
Therefore,
vol(~v) ≥
∫
M
(
cosϕ+ sinϕ| tanα+θ1|
)
ν. (6)
This inequality is valid for all ϕ such that 0 ≤ ϕ ≤ 2π, therefore we may use ϕ obtained in the
previous Proposition:
ϕk (α) = arctan
(
tanα+ k −1
cosα
)
; where α ∈
(
−π
2
,
π
2
)
. (7)
Replacing this condition in equation (6) we find
vol(~v) ≥
∫
M
(
cos
(
ϕk (α)
)+ sin(ϕk (α)) |tanα+θ1|)ν. (8)
On the other hand, 7 provides that
cos
(
ϕk (α)
)= cosα√
1+ (k −1)2 +2(k −1)sinα
and
sin
(
ϕk (α)
)= k −1+ sinα√
1+ (k −1)2 +2(k −1)sinα
.
Thus, the second part of the inequality (8) is equal to
lim
α0→− π2
∫ π
2
α0
∫ 2π
0
(
cosα√
1+ (k −1)2 +2(k −1)sinα
+ k −1+ sinα√
1+ (k −1)2 +2(k −1)sinα
|tanα+θ1|
)
cosαdβdα. (9)
C. R. Mathématique — 2021, 359, n 10, 1225-1232
Fabiano G. B. Brito, Jackeline Conrado, Icaro Gonçalves and Adriana V. Nicoli 1231
Remember that Cartan’s connection form ω12 is given by
ω12 = δω1 −γω2,
where {ω1,ω2} is dual basis of {~v ,~v⊥}. If i :S1α ,→S2 is the inclusion map, and e1 = sinθ~v⊥+cosθ~v ,
then
i∗(ω12)(e1) = δsinθ−γcosθ,
where i∗ is the pullback of i .
From equations (3) and (4), we have
i∗(ω12)(e1) = sinθ
[
sinθ
(
tanα+θ1
)−cosθ(θ2)]−cosθ [cosθ( tanα+θ1)+ sinθ(θ2)]
= tanα+θ1.
Thus, from (9)
vol(~v) ≥ lim
α0→− π2
∫ π
2
α0
∫ 2π
0
(
cosα+ ((k −1)+ sinα)i∗(ω12)(e1)√
1+ (k −1)2 +2(k −1)sinα
)
cosαdβdα. (10)
In order to compute the integral of i∗ω12 over the parallel of S2 at constant latitude α, we follow
the same arguments in the proof Theorem 1.1 of [2].
S2α = {(x, y, z) ∈R3; z ≥ sinα}, α0 ≤α≤
π
2
.
The 2-form dω12 is given by
dω12 =ω1 ∧ω2.
A simple application of Stokes’ theorem implies that∫
S2α
dω12 = 2π
(
IN (~v)
)−∫
S1α
i∗ω12.
Suppose that IN (~v) = sup{IN (~v), IS (~v)} = k. We obtain∫
S1α
i∗ω12 = 2πk −Area
(
S2α
)= 2πk −2π(1− sinα)= 2π(k −1+ sinα). (11)
From inequality (10),
vol(~v) ≥ lim
α0→− π2
∫ π
2
α0
∫ 2π
0
(
cosα+ ((k −1)+ sinα)i∗(ω12)(e1)√
1+ (k −1)2 +2(k −1)sinα
)
cosαdβdα
= lim
α0→− π2
∫ π
2
α0
(
cosα√
1+ (k −1)2 +2(k −1)sinα
∫ 2π
0
cosαdβ
+
(
(k −1)+ sinα)√
1+ (k −1)2 +2(k −1)sinα
∫
S1α
i∗ω12
)
dα
= lim
α0→− π2
∫ π
2
α0
(
2πcos2α√
1+ (k −1)2 +2(k −1)sinα
+ 2π
(
(k −1)+ sinα)2√
1+ (k −1)2 +2(k −1)sinα
)
dα,
where the last inequality is obtained from (11). Therefore,
vol(~v) ≥ 2π lim
α0→− π2
∫ π
2
α0
(
cos2α+ ((k −1)+ sinα)2√
1+ (k −1)2 +2(k −1)sinα
)
dα.
Analogously,
vol(~v) ≥ 2π lim
α0→− π2
∫ π
2
α0
(√
1+ (k −1)2 +2(k −1)sinα
)
dα.
A trigonometrical identity give us
vol(~v) ≥ 2π
∫ π
2
− π2
√
(k −2)2 +4(k −1)sin2
(α
2
+ π
4
)
dα.
C. R. Mathématique — 2021, 359, n 10, 1225-1232
1232 Fabiano G. B. Brito, Jackeline Conrado, Icaro Gonçalves and Adriana V. Nicoli
Assume that t = α2 + π4 , then
vol(~v) ≥ 4π
∫ π
2
0
√
(k −2)2 +4(k −1)sin2 tdt . (12)
Consider k > 2 and an ellipse εk given by
x2
k2
+ y
2
(k −2)2 = 1.
Let µ be a parametrization for εk defined by µ(t ) = (k cos t , (k −2)sin t ). Its length is
L(εk ) = 4
∫ 2π
0
(√
(k −2)2 +4(k −1)sin2 t
)
dt . (13)
Therefore,
vol(~v) ≥πL(εk ). 
Acknowledgment
We would like to thank professor Giovanni Nunes for carefully reading and commenting a
previous version of this manuscript and for the referees’ great suggestions and corrections.
References
[1] V. Borrelli, O. Gil-Medrano, “Area minimizing vector fields on round 2-spheres”, J. Reine Angew. Math. 640 (2010),
p. 85-99.
[2] F. G. B. Brito, P. M. Chacón, D. L. Johnson, “Unit vector fields on antipodally punctured spheres: big index, big volume”,
Bull. Soc. Math. Fr. 136 (2008), no. 1, p. 147-157.
[3] F. G. B. Brito, A. O. Gomes, I. Gonçalves, “Poincaré index and the volume functional of unit vector fields on punctured
spheres”, Manuscr. Math. 162 (2019), no. 3-4, p. 487-500.
[4] A. Fawaz, “Total curvature and volume of foliations on the sphere S2”, Cent. Eur. J. Math. 7 (2009), no. 4, p. 660-669.
[5] H. Gluck, W. Ziller, “On the volume of a unit vector field on the three-sphere”, Comment. Math. Helv. 61 (1986), p. 177-
192.
[6] D. L. Johnson, “Volumes of flows”, Proc. Am. Math. Soc. 104 (1988), no. 3, p. 923-932.
[7] S. L. Pedersen, “Volumes of vector fields on spheres”, Trans. Am. Math. Soc. 336 (1993), no. 1, p. 69-78.
C. R. Mathématique — 2021, 359, n 10, 1225-1232


Area minimizing unit vector fields on antipodally punctured unit 2-sphere

Dimitris Vardakis

Alexander Volberg

Comptes Rendus
Mathématique
Dimitris Vardakis and Alexander Volberg
Free boundary problems in the spirit of Sakai’s theorem
Volume 359, issue 10 (2021), p. 1233-1238
<https://doi.org/10.5802/crmath.259>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
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
Comptes Rendus
Mathématique
2021, 359, n 10, p. 1233-1238
https://doi.org/10.5802/crmath.259
Complex analysis and geometry, Functional analysis / Analyse et géométrie complexes,
Analyse fonctionnelle
Free boundary problems in the spirit of Sakai’s
theorem
Dimitris Vardakisa and Alexander Volberga, b
a Department of Mathematics, Michigan State University, East Lansing MI. 48823,
USA
b Hausdorff Center for Mathematics, Universität Bonn, Bonn, Germany
E-mails: jimvardakis@gmail.com, volberg@msu.edu
Abstract. A Schwarz function on an open domainΩ is a holomorphic function satisfying S(ζ) = ζ on Γ, which
is part of the boundary of Ω. Sakai in 1991 gave a complete characterization of the boundary of a domain
admitting a Schwarz function. In fact, ifΩ is simply connected and Γ= ∂Ω∩D(ζ0,r ), then Γ has to be regular
real analytic (with possible cusps). Sakai’s result has natural applications to 1) quadrature domains, 2) free
boundary problem for∆u = 1 equation. In our scenarios Γ can be, respectively, from real-analytic to just C∞,
regular except for a harmonic-measure-zero set, or regular except finitely many points.
Résumé. Dans le présent article, nous considérons la pléthore de résultats dans l’esprit de théorème de Sa-
kai concernant les fonctions de Schwarz, c’est-à-dire les fonctions holomorphes dans un domaine ouvert Ω
satisfaisant S(ζ) = ζ sur Γ, qui fait partie de la frontière deΩ. Sakai en 1991 a donné une caractérisation com-
plète de la frontière d’un domaine admettant une fonction de Schwarz. Les résultats ci-dessous concernent
trois scénarios de généralisation du résultat de Sakai, motivés plutôt par l’application au problème de dy-
namique complexe étudié dans [13]. À la fin de cette note, nous mentionnons quelques problèmes encore
ouverts.
Funding. AV is supported by NSF grant DMS 1900286 and Alexander von Humboldt Foundation.
Manuscript received 7th June 2021, revised 25th July 2021, accepted 15th August 2021.
Version française abrégée
Le résultat principal de cet article consiste en des généralisations du résultat de Sakai ayant
été motivées par une application à la dynamique complexe. Ici, nous nous intéressons à trois
scénarios différents sur un domaine simplement connexeΩ :
(1) lorsque f1(ζ) = ζ f2(ζ) sur Γ, avec f1, f2 holomorphes et continues jusqu’au bord,
(2) lorsque U /V est égal à certaine fonction analytique réelle sur Γ avec U ,V positives et
harmoniques surΩ et s’annulant sur Γ, et
(3) lorsque S(ζ) =Φ(ζ,ζ) sur Γ, avecΦ une fonction holomorphe de deux variables.
Il s’avère que la partie Γ de la frontière peut être, respectivement et indifféremment, une partie
analytique réelle ou seulement C∞, régulière à l’exception d’un nombre fini de points, ou même
régulière à l’exception d’un ensemble de mesure harmonique zéro.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1234 Dimitris Vardakis and Alexander Volberg
Introduction
In 1991, Sakai proved a beautiful theorem [12, Theorem 5.2] which fully characterizes the bound-
aries of sets admitting a Schwarz function. Before we dwell on his result and its applications, we
introduce some necessary notation: The disk of radius r > 0 and centre ζ0 will be denoted by
D(ζ0,r ), while the unit disk and the unit circle by D and T, respectively. In general, D will be a
(small) open domain.Ωwill stand for a bounded open set on the complex plane and ζ0 ∈ ∂Ω for a
non-isolated point of its boundary Γ= ∂Ω∩D (for some D). For convenience, we will sometimes
work with ζ0 = 0.
Now, suppose there is a function S holomorphic onΩ, continuous onΩ∪Γ, and which satisfies
that S(ζ) = ζ for all ζ ∈ Γ, where ζ0 ∈ Γ= ∂Ω∩D(ζ0,r ). The function S is called a Schwarz function
onΩ∪Γ at ζ0. Sakai proved the following theorem:
Theorem 1. LetΩ be an arbitrary bounded open set and S a Schwarz function onΩ∪Γ at 0. Then,
there exists a small neighbourhood of 0, D, so that exactly one of the following must be true:
(1) Ω∩D is simply connected and Γ∩D is a regular real analytic simple arc through 0.
(2a) Γ∩D determines uniquely a regular real analytic arc through 0. The set Γ∩D is either
an infinite proper subset of this arc with 0 as an accumulation point or equal to it. Also,
Ω∩D = D \Γ.
(2b) Ω∩D = Ω1 ∪Ω2 where Ω1 and Ω2 are simply connected and ∂Ω1 ∩D and ∂Ω2 ∩D are
regular real analytic simple arcs through 0 and tangent at 0.
(2c) Ω∩D is simply connected and Γ∩D is a regular real analytic simple arc except for a cusp
at 0. The cusp points intoΩ.
The converse is also true in the sense that if any of these four cases hold, then there exists a Schwarz
function S onΩ∪Γ at 0.
His proof is heavily dependent on the Phragmén–Lindelöf Principle and on certain auxiliary
functions. In particular, depending on which one ofΦ1(z) = zS(z) andΦ2(z) =
p
zS(z) is univalent
around 0, we get the cases (1), or (2a), (2b) and (2c), respectively.
Driven by Sakai’s results and by motivation from complex dynamics, we ask whether we can
relax the condition S(ζ) = ζ and get the same regularity for the boundary Γ, and if not, how regular
can Γ be. We are interested in three different scenarios, but we always restrict our attention to a
simply connected domainΩ since otherwise too many complications arise.
Nevanlinna Domains and Model Spaces
Initially, we investigate a scenario where we have two holomorphic functions f1 and f2 (instead
of only one) on the simply connected Ω continuous on the boundary Γ = ∂Ω∩D(ζ0,r ), where
ζ0 ∈ ∂Ω, which satisfy
f1(ζ) = ζ f2(ζ) whenever ζ ∈ Γ. (1)
We may assume this happens around any point ζ0 simply by substituting
f T1 (z) = f1(z +ζ0)−ζ0 f2(z +ζ0) and f T2 (z) = f2(z +ζ0),
which translates our problem at 0.
If it so happens that f2 is a polynomial or it somehow extends analytically over the boundary Γ,
then the ratio f = f1/ f2 turns out to be a Schwarz function onΩ thanks to the Phragmén–Lindelöf
principle. However, if this is not the case, we need stronger tools in order to proceed.
Suppose Ω is a simply connected open domain and that there exist bounded holomorphic
functions f and g on Ω satisfying that ξ = f (ξ)/g (ξ) for ξ ∈ ∂Ω in the sense of conformal
mappings, i.e.ϕ(ζ) = f (ϕ(ζ))/g (ϕ(ζ)) for almost every ζ ∈Twhereϕ is a conformal mapping from
C. R. Mathématique — 2021, 359, n 10, 1233-1238
Dimitris Vardakis and Alexander Volberg 1235
D onto Ω. In this case, Ω is called a Nevanlinna domain (cf. [5, Definition 2.1]). This definition is
independent of the choice of ϕ and does not guarantee additional regularity for f or g ; not even
continuity [5, Example 5.8], which is in our interest in this note.
Now, consider such a conformal ϕ :D→Ω ontoΩ and construct the functions F1 = f1 ◦ϕ and
F2 = f2 ◦ϕ which, because of (1), satisfy
F1(ζ)
F2(ζ)
=ϕ(ζ) for almost every ζ ∈T.
This turnsΩ into a Nevanlinna domain.
The question arises as to whether such functions f1, f2 and ϕ actually exist and if they do,
how “good” or “bad” can Γ be? It turns out Sakai’s result does not in general hold in this setup
as (we will see below) we can find functions f1, f2 holomorphic on Ω and continuous on Ω∪Γ
satisfying (1), but such that Γ is only C∞-smooth and not real analytic.
Decomposing F1 and F2 into inner and outer parts, we see that ϕ satisfies
ϕ= F
θ
= F̃
θ̃
almost everywhere on T,
where θ is an inner function and F ∈ H∞, and where h̃(z) = h(z). This means that ϕ admits a
(Nevanlinna-type) pseudo-continuation and as such it belongs to some model space Kθ. See [8]
for the proof of this and [6] for more details about pseudo-continuations.
A lot of results are already known for conformal maps in model spaces when the inner function
θ has a Blaschke part. In fact, the boundaries of such domains may be C 1 but not C 1,α for any
α ∈ (0,1) [2, Theorem 2], may be unrectifiable [11, Example 1] or nowhere analytic [10], or may
be rectifiable but admitting no such functions continuable up to the boundary [5, Example 5.8].
Furthermore, the Hausdorff dimension of Nevanlinna domains’ boundaries can be any number
from 1 up to 2 [3, Theorem 1]. Additionally, it is always possible to find conformal maps smooth
up to T, for example ϕ(z) = 1/(1− z0z) ∈ Kθ∩C∞(T) where θ(z0) = 0.
In the case of a purely singular inner function θ, it was recently found through a series of
papers that Kθ contains bounded univalent functions if, and only if, the singular measure, µθ,
associated with θ is positive on some Carleson set E ⊂ T. When the latter happens we say θ has
the Carleson property. (See [1] or [4] for a nice exposition on this.)
More explicitly, take singular θ with the Carleson property and a non-trivial g ∈ Kθ ∩C∞(T)
(such functions always exist thanks to [7, Theorem 2.1]) and let G be its analytic extension on
Ĉ\ supp(µθ). Then, as shown in [4], for appropriate |α| > 1 the function
ϕ(z) = G (z)−G (α)
z −α
lives, in fact, inside Kθ∩C∞(T) and it is additionally univalent there yet not real-analytic on T.
This ϕ ∈ Kθ sends D onto some simply connected domain Ω and we will see that Sakai’s type
real analyticity result fails for this Ω. Indeed, it is true that F = ϕθ for some F ∈ H∞. Multiplying
this equality with a suitable outer function H , we find that the functions f1 = (H F ) ◦ϕ−1
and f2 = (H θ) ◦ϕ−1 are holomorphic on Ω, continuous on Ω∪Γ and satisfy (1). However, by
construction Γ cannot be real analytic.
Holomorphic function on the complex 2-plane
Next, we want to describe the boundary Γ= ∂Ω∩D(ζ0,r ) ofΩ under a different setup; we replace
the ζ f2(ζ) from the equality f1(ζ) = ζ f2(ζ) withΦ(ζ,ζ) whereΦ a function of two complex variables
this time holomorphic in a neighbourhood of (ζ0,ζ0). In particular, we prove the following:
C. R. Mathématique — 2021, 359, n 10, 1233-1238
1236 Dimitris Vardakis and Alexander Volberg
Theorem 2. Let Ω ⊂ D(ζ0,r ) be a bounded simply connected domain such that the set Γ =
∂Ω∪ D(ζ0,r ) is a (union of) Jordan arc(s) for some (small) r > 0. Also, let Φ be a (non-trivial)
function of two variables holomorphic on a C2 neighbourhood of (ζ0,ζ0), and suppose there exists
a function R
(i) holomorphic onΩ,
(ii) continuous onΩ, and such that
(iii) R(ζ) =Φ(ζ,ζ) for all ζ ∈ Γ.
Then, there exists a closed set, E ⊂ Γ, of zero harmonic measure so that Γ\E is a countable union of
regular real-analytic simple arcs except possibly for some cusps. The cusps (if they exist) point into
Ω and may only accumulate on E.
The first thing to note is that the problem can be “translated” from any ζ0 to 0. Also, Φ(z, w)
must have a non-zero partial derivative of some order in terms of w .
The proof is essentially a “chase” of the roots of the function Φ(z, z)−R(z) and particularly
those lying on the boundary Γ. To begin with, we would want to “isolate” the zeroes of the func-
tionΦ(z, w)−R(z) using the Weierstraß Approximation Theorem around 0, but R is not holomor-
phic on Γ. So, instead we first deploy Mergelyan’s Theorem to find a polynomial sequence, pn ,
approaching R and then use Weierstraß’ theorem onΦ(z, w)−pn(z).
Taking limits, this leads to a polynomial of two variables of the form
P (z, z) = zk +ak−1(z)zk−1 +·· ·+a0(z),
where an are holomorphic functions on Ω and continuous on Ω∪Γ. Additionally, P (ζ,ζ) = 0 for
ζ ∈ Γ.
Eventually, the roots of P (z, · ) for any fixed z give us a multivalued holomorphic function.
Therefore, we need to consider several simply connected domains around Γ in order to split
the different branches. Each such domain and branch gives us a Schwarz function which, using
Sakai’s theorem, gives us the desired regularity for Γ. The set E which was excluded is simply the
the zeroes of the discriminant, D(z) of P (z, · ) that fall on Γ.
Harmonic functions and the one-phase problem
The final part of this note deals with a one-phase problem of harmonic functions that appears
naturally in a problem of dimension of harmonic measure on limits sets of a wide class of
holomorphic dynamical systems; see [13]. The theorem we show is the following.
Theorem 3. Let Ω⊂ D(ζ0,r ) be a simply-connected domain of C and let Γ be an open Jordan arc
of its boundary with ζ0 ∈ Γ. Suppose there are two positive non-proportional harmonic functions
U and V onΩ, continuous onΩ∪Γwhich satisfy
U (ζ) = V (ζ) = 0 and U (ζ)
V (ζ)
= |A(ζ)|2
for all ζ ∈ Γ, where A is a non-trivial analytic function on a neighborhood of the boundary piece Γ.
Then, for all but possibly finitely many points ζ0 ∈ Γ there exist some small neighborhood D of
ζ0 such that the following holds:
Γ∩D is a regular real-analytic simple arc through ζ0 except possibly a cusp at ζ0. (2)
The finitely many points around which (2) might fail are the points ζ ∈ Γwhere A′(ζ) = 0, i.e. where
A might not to be invertible.
At first, using Harnack’s principle we show that the ratio U /V is indeed well defined and
(positive) real-analytic on Γ. What is more, this real-analytic function gives rise to a holomorphic
function, R, onΩwhich is continuous onΩ∪Γ and satisfies R = |A|2 on Γ.
C. R. Mathématique — 2021, 359, n 10, 1233-1238
Dimitris Vardakis and Alexander Volberg 1237
We immediately conclude that Γ is a union of regular real-analytic simple arcs except possibly
a closed set E ⊂ Γ of zero harmonic measure. This comes directly from Theorem 2 for the function
Φ(z, w) = A(z)A(w).
However, it is possible to say even more. Indeed, around any point where A is invertible the
function
S(z) = 1
z
R ◦ A−1(z)
turns out to be a Schwarz function on Ω∪Γ at ζ0. Again by Sakai, we conclude that Γ is either a
regular real-analytic simple arc through ζ0 itself or two regular real-analytic simple arcs forming
a cusp at ζ0.
Moreover, it possible to differentiate between the two cases. Following through Sakai’s work
we find that the following is true:
Proposition 4. With assumptions as above, around any ζ0 ∈ Γwith A′(ζ0) 6= 0 either
(1) there exist a functionΨ1 holomorphic and univalent onΩ∩D such thatΨ1 is continuous
on (Ω∪Γ)∩D, andΨ1(ζ) = |A(ζ)− A(ζ0)|2 for ζ ∈ Γ∩D, or
(2) there exist a functionΨ2 holomorphic and univalent onΩ∩D such thatΨ22 is continuous
on (Ω∪Γ)∩D, andΨ22(ζ) = |A(ζ)− A(ζ0)|2 for ζ ∈ Γ∩D.
The functionsΨ1 andΨ2 are given by
Ψ1(z) = (A(z)− A(ζ0))
(
R(z)
A(z)
− A(ζ0)
)
and
Ψ2(z) =
√
Ψ1(z).
It turns out Γ has a cusp at ζ0 if, and only if, (2) of Proposition 4 holds.
Open “free boundary” problems in the spirit of Sakai
All problems treated above are examples of the so-called (non-variational) free boundary prob-
lems.
We would like to call the attention of the reader to the open question: what if the domain Ω
for positive harmonic functions is not simply connected? Finitely connected situation presents
no difficulties, but what if, for example, Γ is a Cantor set andΩ=D\Γ? Suppose we know that the
ratio of two positive harmonic (non-proportional) functions U ,V in Ω vanishing on the Cantor
set Γ has a well defined ratio on Γ (this happens for a wide class of Cantor sets, for instance for all
regular Cantor sets of positive Hausdorff dimension).
Suppose this ratio is non-constant and equal to |A(ζ)|2 for ζ ∈ Γ, where A is a holomorphic
function on D. What we can say about the Cantor set Γ? The “desired” answer is that this is
impossible to happen on any Cantor set.
This type of problems (we may call them one-phase free boundary problems) appear naturally
in certain problems of complex dynamics, see e.g. [13]. If we would know the aforementioned
answer (we conjecture that no Cantor set would allow such a triple (U ,V , A)), then a long
standing problem of dimension of harmonic measure on Cantor repellers would be solved.
Another similar one-phase boundary problem concerns functions in Rn for n > 2. Let Ω be
a connected bounded domain in Rn , n > 2, and Γ = ∂Ω∩D(x,r ), where x ∈ ∂Ω. Again, let U ,V
be two positive (non-proportional) harmonic functions inΩ vanishing continuously on Γ. IfΩ is
assumed to be a Lipschitz domain, then [9] claims that U /V makes sense on Γ and is additionally
a Hölder function on Γ (boundary Harnack principle).
Here is a question: Let R be a real analytic function on D(x,r ), x ∈ Γ, and let U /V = R on
Γ∩ D(x,r ). Is it true that Γ∩ D(x,r ) is real analytic, maybe with the exception of some lower
dimensional singular set?
C. R. Mathématique — 2021, 359, n 10, 1233-1238
1238 Dimitris Vardakis and Alexander Volberg
References
[1] A. D. Baranov, Y. S. Belov, A. Borichev, K. Y. Fedorovskiy, “Univalent functions in model spaces: revisited”, https:
//arxiv.org/abs/1705.05930v2, 2017.
[2] A. D. Baranov, K. Y. Fedorovskiy, “Boundary regularity of Nevanlinna domains and univalent functions in model
subspaces”, Sb. Math. 202 (2011), no. 12, p. 1723-1740.
[3] Y. S. Belov, A. Borichev, K. Y. Fedorovskiy, “Nevanlinna domains with large boundaries”, J. Funct. Anal. 277 (2019),
no. 8, p. 2617-2643.
[4] Y. S. Belov, K. Y. Fedorovskiy, “Model spaces containing univalent functions”, Russ. Math. Surv. 73 (2018), no. 1, p. 172-
174.
[5] J. J. Carmona, P. V. Paramonov, K. Y. Fedorovskiy, “On uniform approximation by polyanalytic polynomials and the
Dirichlet problem for bianalytic functions”, Sb. Math. 193 (2002), no. 10, p. 1469-1492.
[6] J. A. Cima, W. T. Ross, The Backward Shift on the Hardy Space, Mathematical Surveys and Monographs, vol. 79,
American Mathematical Society, 2000.
[7] K. Dyakonov, D. Khavinson, “Smooth functions in star-invariant subspaces”, in Recent advances in operator-related
function theory, Contemporary Mathematics, vol. 393, American Mathematical Society, 2006.
[8] K. Y. Fedorovskiy, “On some properties and examples of Nevanlinna domains”, Proc. Steklov Inst. Math. 253 (2006),
p. 186-194.
[9] D. S. Jerison, C. E. Kenig, “Boundary behavior of harmonic functions in non-tangentially accessible domains”, Adv.
Math. 46 (1982), p. 80-147.
[10] M. Y. Mazalov, “An example of a nonconstant bianalytic function vanishing everywhere on a nowhere analytic
boundary”, Math. Notes 62 (1997), no. 4, p. 524-526.
[11] ——— , “Example of a non-rectifiable Nevanlinna contour”, St. Petersbg. Math. J. 27 (2016), p. 625-630.
[12] M. Sakai, “Regularity of a boundary having a Schwarz function”, Acta Math. 166 (1991), no. 3-4, p. 263-297.
[13] A. Volberg, “On the dimension of harmonic measure of Cantor repellers”, Mich. Math. J. 40 (1993), no. 2, p. 239-258.
C. R. Mathématique — 2021, 359, n 10, 1233-1238


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