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Explicit reconstruction of Riemann surface with given boundary in complex projective space
Alexey Agaltsov, Guennadi Khenkine
To cite this version:
Alexey Agaltsov, Guennadi Khenkine. Explicit reconstruction of Riemann surface with given bound- ary in complex projective space. Journal of Geometric Analysis, 2015, 25 (4), pp.2450-2473.
�10.1007/s12220-014-9522-1�. �hal-00912925v3�
Explicit reconstruction of Riemann surface with given boundary in complex projective space 1
A. D. Agaltsov
2, G. M. Henkin
3In this paper we propose a numerically realizable method for reconstruction of a complex curve with known boundary and without compact compo- nents in complex projective space.
Keywords: Riemann surface, reconstruction algorithm, Burgers equation, Cauchy-type formulas
1 Introduction
Let C P
2be the complex projective plane with homogeneous coordinates (w
0: w
1: w
2).
Let X ⊂ C P
2be a complex curve with rectifiable boundary γ = ∂X. Without loss of generality, we suppose that the following conditions of general position hold:
(0 : 1 : 0) 6∈ X, w
0|
γ6= 0.
Put C
2= {w ∈ C P
2: w
06= 0} with coordinates z
1=
ww10
, z
2=
ww20
. For almost all ξ = (ξ
0, ξ
1) ∈ C
2the points of intersection of X with complex line C
1ξ= {z ∈ C
2: ξ
0+ ξ
1z
1+ z
2= 0} form a finite set of points
z
1(j)(ξ), z
2(j)(ξ)
= h
j(ξ), −ξ
0− ξ
1h
j(ξ)
, ξ = (ξ
0, ξ
1), j = 1, . . . , N
+(ξ).
By Darboux lemma [2, 4] functions {h
j} satisfy the following equations
∂h
j(ξ)
∂ξ
1= h
j(ξ) ∂h
j(ξ)
∂ξ
0, ξ = (ξ
0, ξ
1), j = 1, . . . , N
+(ξ), (1) which are often called the shock-wave equations, the inviscid Burgers equations or the Riemann-Burgers equations. In this interpretation ξ
1is the time variable and ξ
0is the space variable.
1
This is an improved version of the article A. D. Agaltsov, G. M. Henkin, Explicit reconstruction of Riemann surface with given boundary in complex projective space, Journal of Geometric Analysis 55 (10), 2015, 2450-2473
2
CMAP, Ecole Polytechnique, CNRS, Universit´ e Paris-Saclay, 91128 Palaiseau, France; email:
alexey.agaltsov@polytechnique.edu
3
Universit´ e Pierre et Marie Curie, Case 247, 4 place du Jussieu, 75252, Paris, France, email:
henkin@math.jussieu.fr
The following Cauchy-type formula from [4] plays the essential role in the recon- struction of X from γ:
G
m(ξ) ==
def1 2πi
Z
γ
z
1md(ξ
0+ ξ
1z
1+ z
2) ξ
0+ ξ
1z
1+ z
2=
N+(ξ)
X
j=1
h
mj(ξ) + P
m(ξ), m ≥ 1, (2)
where N
+(ξ) = |X ∩ C
1ξ| and P
m(ξ
0, ξ
1) is a polynomial of degree at most m with respect to ξ
0at fixed ξ
1. In addition, P
0(ξ
0, ξ
1) = −N
−, where N
−= |X ∩ C P
∞1|, C P
∞1= {w ∈ C P
2: w
0= 0}, and
P
1(ξ
0, ξ
1) = −
N−
X
k=1
a
kξ
0+ b
k1 − a
kξ
1, if h
k(ξ
0, 0) ∼ a
kξ
0+ b
k+ O(ξ
0−1) at ∞. (3) In particular, it follows from (2) that:
G
0(ξ) = 1 2πi
Z
γ
d(ξ
0+ ξ
1z
1+ z
2) ξ
0+ ξ
1z
1+ z
2= N
+(ξ) − N
−. (4)
Let π
2: C
2→ C be the projection onto the second coordinate: π
2(z
1, z
2) = −z
2. We have that C \ π
2γ = ∪
Ll=0Ω
l, where Ω
l≥0are connected and Ω
0is unbounded. From the definition of N
±it follows that
N
+(ξ
0, 0) = N
−, ξ
0∈ Ω
0. (5) Assume that complex curve X does not contain compact components without boundary, or equivalently, satisfies the following condition of minimality:
for any complex curve X e ⊂ C P
2such that ∂ X e = ∂X = γ
and for almost all ξ ∈ C
2we have | X e ∩ C
1ξ| ≥ |X ∩ C
1ξ|. (∗) Condition of minimality (∗) is a condition of general position and is fulfilled for X if, for example, every irreducible component of X is a transcendental complex curve. Note that from theorems of Chow [1] and Harvey-Shiffman [7] it follows that an arbitrary complex curve X e ⊂ C P
2satisfying ∂ X e = ∂X admits the unique representation X e = X ∪ V , where X is a curve satisfying (∗), and V is a compact algebraic curve, possibly, with multiple components.
The main result of [3] gives a solution to the important problem of J. King [9], when
a real curve γ ⊂ C P
2is the boundary of a complex curve X ⊂ C P
2. Let γ ⊂ C
2.
Then γ = ∂X for some open connected complex curve X in C P
2if and only if in a
neighborhood W
ξ∗of some point ξ
∗∈ C
2one can find mutually distinct holomorphic functions h
1, . . . , h
N(ξ∗)satisfying shock-wave equations (1) and also the equation
∂
2∂ξ
20G
1(ξ
0, ξ
1) −
p
X
j=1
h
j(ξ
0, ξ
1)
= 0, ξ = (ξ
0, ξ
1) ∈ W
ξ∗.
In this work in development of [3, 4] we propose a numerically realizable algorithm for reconstruction of a complex curve X ⊂ C P
2from the known boundary and sat- isfying the condition of minimality. This algorithm permits, in particular, to make applicable the result of [8] about principal possibility to reconstruct the topology and the conformal structure of a two-dimensional bordered surface X in R
3with constant scalar conductivity from measurements on ∂X of electric current densities, induced by three potentials in general position.
Our algorithm depends on the number of points at infinity N
−of the complex curve X. It was tested on many examples and admits a simple and complete justification for N
−= 0, 1, 2. Despite a cumbersome description for N
−≥ 3, we show that, in principle, there are no obstacles for the justification and numerical realization for any N
−≥ 0. Moreover, in Theorem 3.2 we propose a method for finding the parameter N
−in terms of γ . This makes the algorithm much more applicable.
2 Cauchy-type formulas and Riemann-Burgers equa- tions
We begin by giving a new proof of the Cauchy type formulas (2) from [4], which allows to obtain explicit expressions for functions P
m.
Theorem 2.1. Let X ⊂ C P
2\ (0 : 1 : 0) be a complex curve with rectifiable boundary γ ⊂ C
2and satisfying (∗). Suppose that for almost all ξ ∈ C
2all the points of intersection of X with C
1ξhave multiplicity at most one. Then the following formulas hold for almost all ξ = (ξ
0, ξ
1) ∈ C
2:
G
m(ξ) = X
N+(ξ)j=1
h
mj(ξ) + P
m(ξ), m ≥ 1, (6)
where P
m(ξ) is holomorphic in a neighborhood of almost all (ξ
0, 0), ξ
0∈ C \ π
2(γ), and
is polynomial in ξ
0of degree at most m for any fixed ξ
1. Furthermore, the following
explicit formulas hold:
P
m(ξ
0, ξ
1) =
N−
X
s=1 m−1
X
k=0
X
i1+···+im=k di1w1
dwi01
(q
s) · · ·
dimw1dwim0
(q
s) (m − k − 1)!
d
m−kdw
m−k0ln(ξ
0w
0+ ξ
1w
1+ w
2)|
qs−
µ0
X
s=1
X
i1+···+im=m
d
i1w
1dw
i01(q
s) · · · d
imw
1dw
0im(q
s), where X ∩ C P
∞1= {q
1, . . . , q
N−}. In particular, if N
−= 0 then P
m= 0, m ≥ 1.
Proof. Put e g = ξ
0w
0+ ξ
1w
1+ w
2and g =
weg0
= ξ
0+ ξ
1z
1+ z
2. Consider differential forms
ω
m==
defz
1mdg
g = w
1mw
0mw
0e g d
e g w
0= w
m1w
m0d e g
e g − w
m1w
m+10dw
0, m = 0, 1, . . . Then G
m(ξ) =
2πi1R
γ
ω
m. Let us compute this integral explicitly. Denote by p
j, j = 1, . . . , N
+(ξ) the points of intersection of X with C P
ξ1, and by q
s, s = 1, . . . , N
−the points of intersection of X with infinity C P
∞1. Denote by B
jεthe intersection of X with the ball of radius ε in C P
2centered at p
jand by D
sεthe intersection of X with the ball of radius ε centered at q
s. The restriction of form ω
mon X is meromorphic with poles at points p
jand q
s. Thus the following equality is valid:
G
m(ξ) = 1 2πi
Z
γ
ω
m=
N+(ξ)
X
j=1
1 2πi
Z
bBjε
ω
m+
N−
X
s=1
1 2πi
Z
bDεs
ω
m.
If N
−= 0, then the second group of terms is absent. The integral R
bBjε
ω
mcan be calculated as a residue at the first order pole:
Z
bBεj
ω
m= Z
bBεj
z
m1d e g e g −
Z
bBjε
w
1mw
m+10dw
0= Z
bBεj
z
m1d e g
e g = 2πi h
mj(ξ).
Let N
−> 0. Computation of integral R
bDεs
ω
1will be done in two steps. Let us calculate first R
bDεs wm1
wm+10
dw
0. Consider the expansion of w
1(w
0) into power series in w
0in the neighborhood of point q
s:
w
1(w
0) = w
1(q
s) + dw
1dw
0(q
s)w
0+ d
2w
1dw
20(q
s)w
02+ · · · .
Note further that
w
1m(w
0) =
∞
X
k=0
X
i1+···+im=k
d
i1w
1dw
i01(q
s) · · · d
imw
1dw
0im(q
s)w
0k. The coefficient near w
0mcan be presented in the form
Z
bDεs
w
1mw
0m+1dw
0= 2πi X
i1+···+im=m
d
i1w
1dw
0i1(q
s) · · · d
imw
1dw
i0m(q
s).
Now we can calculate the integral R
bDsε wm1 wm0
deg
eg
. Using relation d e g =
dwdeg0
dw
0and expansion of w
1(w
0) into power series in w
0we obtain:
Z
bDsε
w
1mw
0md e g e g =
Z
bDsε
1 w
m0w
1(q
s) + dw
1dw
0(q
s)w
0+ d
2w
1dw
02(q
s)w
20+ · · ·
md e g dw
01 e g dw
0=
∞
X
k=0
Z
bDsε
X
i1+···+im=k
d
i1w
1dw
i01(q
s) · · · d
imw
1dw
i0m(q
s)w
0k−md e g dw
01 e g dw
0=
m−1
X
k=0
Z
bDsε
X
i1+···+im=k
d
i1w
1dw
i01(q
s) · · · d
imw
1dw
0im(q
s)w
0k−md e g dw
01 e g dw
0=
m−1
X
k=0
2πi (m − k − 1)!
X
i1+···+im=k
d
i1w
1dw
0i1(q
s) · · · d
imw
1dw
i0m(q
s) lim
w0→0
d
m−k−1dw
m−k−10d g e dw
01 e g
.
From here, taking into account the relation
dwdge0
1
eg
=
dlndweg0
, we obtain, finally Z
bDsε
w
m1w
m0d e g g e = 2πi
m−1
X
k=0
X
i1+···+im=k di1w1
dwi01
(q
s) · · ·
dimw1dw0im
(q
s) (m − k − 1)!
d
m−kdw
0m−kln(ξ
0w
0+ ξ
1w
1+ w
2)|
qs. It is a polynomial of degree at most m with respect to ξ
0.
Remark 2.1. We excluded the case (0 : 1 : 0) ∈ X because the point (0 : 1 : 0) can lead to a non-polynomial contribution in ξ
0in functions P
m.
Consider, for example, the projective curve X: e w
02= w
1w
2and the meromorphic 1-form Ω
ξ0on it:
Ω
ξ0=
zz1dz22+ξ0
, ξ
0∈ C \ 0, where z
1=
ww10
, z
2=
ww20
. The form Ω
ξ0is holomorphic on X except the points
q
1= (0 : 0 : 1), q
2= (0 : 1 : 0) and p
ξ0= (−ξ
0: 1 : ξ
02).
One can see that the residues of Ω
ξ0at q
1, q
2are equal to 0 and 1/ξ
0, respectively.
It follows that the residue at p
ξ0is equal to −1/ξ
0.
Now, consider the part X of X e obtained by cutting off some small neighborhood of the point q
1. Let γ denote the boundary of X. Choose any ξ = (ξ
0, 0) such that X ∩ C
1ξ6= ∅ . Then X ∩ C
1ξ= {(−1/ξ
0, −ξ
0)} and, as far as the residue of Ω
ξ0at q
1is zero, we have that
0 = G
1(ξ
0, 0) = 1 2πi
Z
γ
Ω
ξ0= − 1
ξ
0+ P
1(ξ
0, 0), so that P
1(ξ
0, 0) = 1/ξ
0.
We will also use the following result of [8], which gives an effective characterization of functions h
j≥1of Theorem 2.1.
Theorem 2.2 (Remark 4 to Theorem 3a of [8]). Let X ⊂ C P
2\ (0 : 1 : 0) be a complex curve with rectifiable boundary γ ⊂ C
2and satisfying (∗). Fix any ξ
0∗∈ Ω
0and let W
ξ∗be a neighborhood of ξ
∗= (ξ
0∗, 0) in C
2. Let G
k≥1, h
j≥1be defined as in (2) and (6). Suppose that there exist functions e h
1, . . . , e h
p, holomorphic in W
ξ∗and satisfying
∂2
∂ξ20
(G
1− e h
1− · · · − e h
p) = 0 in W
ξ∗, (7)
∂ehk
∂ξ1
= e h
k∂e∂ξhk0
in W
ξ∗, k = 1, . . . , p. (8) Then p ≥ N
−. Furthermore, if p = N
−:= |X ∩ C P
∞1|, then h
1, . . . , h
N−coincide with e h
1, . . . , e h
N−in W
ξ∗(up to order).
3 Reconstruction algorithm
We now pass to the reconstruction algorithm for a complex curve X ⊆ C P
2with given boundary ∂X and satisfying the condition of minimality (∗). Let us consider the cases N
−= 0, 1, 2.
The reconstruction algorithm is based on formulas (6). The next theorem permits to find the functions P
mof (6). We will use the notation
æ
kl= 1 2πi
Z
γ
z
1kz
2ldz
2, k, l ≥ 0. (9)
Theorem 3.1. Let X ⊂ C P
2, (0 : 1 : 0) 6∈ X, be a complex curve with rectifiable
boundary γ ⊂ C
2and satisfying (∗). Let h
j≥1and P
m≥1be the functions defined in
Theorem 2.1. Then the following statements are valid:
1. If N
−= 0, then P
m= 0 for all m ≥ 1. Besides, G
1= 0 in a neighborhood of any ξ
∗= (ξ
0∗, 0) with ξ
0∗∈ Ω
0.
2. If N
−= 1, then P
1(ξ
0, 0) = c
11+ c
12ξ
0, where constants c
11and c
12satisfy the following identity in ξ = (ξ
0, 0) ∈ Ω
0× 0:
c
11∂G1∂ξ0
(ξ) + c
12ξ
0∂G1∂ξ0
(ξ) + G
1(ξ)
= G
1(ξ)
∂G∂ξ10
(ξ
0, 0) −
∂G∂ξ11
(ξ). (10)
3. If N
−= 2, then P
1(ξ
0, 0) = c
11+ c
12ξ
0, P
2(ξ
0, 0) = c
21+ c
22ξ
0+ c
23ξ
02, where constants c
11, c
12, c
21, c
22, c
23satisfy the following identity in ξ = (ξ
0, 0) ∈ Ω
0×0:
æ
10(c
212+ c
23) =
∂G∂ξ21
− 2
∂G∂ξ11
(G
1− c
11− c
12ξ
0) + G
1(c
22+ 2c
23ξ
0) + ∂G
1∂ξ
0· (G
1− c
11− c
12ξ
0)
2− G
2+ c
21+ c
22ξ
0+ c
23ξ
02+ G
21− 2c
11G
1− 2c
12G
1ξ
0− G
2· (−c
12),
(11)
where all the functions are evaluated at point ξ = (ξ
0, 0).
Proof. 1. By Theorem 2.1, if N
−= 0 then P
m= 0 for all m ≥ 1. It also follows from formula (6) that G
1= 0 in a neighborhood of any ξ
∗= (ξ
0∗, 0) with ξ
∗0∈ Ω
0.
2. It follows from Theorem 2.1 that P
1(ξ
0, ξ
1) = C
11(ξ
1) + C
12(ξ
1)ξ
0in a neigh- borhood of almost all ξ
∗= (ξ
0∗, 0) ∈ Ω
l× 0, where C
11, C
12are holomorphic in a neighborhood of zero. We need to find constants c
11= C
11(0) and c
12= C
12(0).
We differentiate the equation (6) with respect to ξ
0and restrict this equation and its differentiated version to Ω
l× 0:
h
1(ξ
0, 0) = G
1(ξ
0, 0) − c
11− c
12ξ
0,
∂h
1∂ξ
1(ξ
0, 0) = ∂G
1∂ξ
1(ξ
0, 0) − C ˙
11(0) − C ˙
12(0)ξ
0,
∂h
1∂ξ
0(ξ
0, 0) = ∂G
1∂ξ
0(ξ
0, 0) − c
12,
(12)
where ξ
0∈ Ω
0. By (1) there is the equality
∂h∂ξ11
(ξ
0, 0) = h
1(ξ
0, 0)
∂h∂ξ10
(ξ
0, 0). We substitute h
1(ξ
0, 0),
∂h∂ξ11
(ξ
0, 0),
∂h∂ξ10
(ξ
0, 0) in this equation by their expressions (12), and we obtain the equation
∂G
1∂ξ
1(ξ
0, 0) − C ˙
11(0) − C ˙
12(0)ξ
0=
=
G
1(ξ
0, 0) − c
11− c
12ξ
0∂G
1∂ξ
0(ξ
0, 0) − c
12.
(13)
This equation is valid, in particular, for ξ
0∈ Ω
0. We divide it by ξ
0and tend ξ
0→ ∞.
As a result, we obtain the equality C ˙
12(0) = −c
212. Taking into account this equality, we can rewrite equation (13) in the form
∂G
1∂ξ
1(ξ
0, 0) − C ˙
11(0) =
=
G
1(ξ
0, 0) − c
11− c
12ξ
0∂G
1∂ξ
0(ξ
0, 0) −
G
1(ξ
0, 0) − c
11c
12.
(14)
Taking into account that ξ
0∂G∂ξ10
(ξ
0, 0) → 0 as ξ
0→ ∞ and passing ξ
0→ ∞ in (14), we obtain the equality C ˙
11(0) = −c
11c
12. Substituting ths explicit expression for C ˙
11(0) into (14), we get (10).
3. By (1) functions h
1(ξ) and h
2(ξ) satisfy the Riemann-Burgers equation in a neighborhood of any ξ
∗= (ξ
0∗, 0) ∈ Ω
0× 0, so that the following equalities are valid:
∂ (h
1h
2)
∂ξ
1= h
1∂h
2∂ξ
1+ ∂h
1∂ξ
1h
2= h
1h
2∂(h
1+ h
2)
∂ξ
0, (15)
∂(h
21+ h
22)
∂ξ
0= 2h
1∂h
1∂ξ
0+ 2h
2∂h
2∂ξ
0= 2 ∂(h
1+ h
2)
∂ξ
1. (16)
Note that h
1h
2=
12h
1+ h
22−
12h
21+ h
22. Therefore the system (15)–(16) is equivalent to the system
∂ (h
1+ h
2)
2∂ξ
1− ∂(h
21+ h
22)
∂ξ
1=
h
1+ h
2 2− h
21+ h
22∂(h
1+ h
2)
∂ξ
0, (17)
∂(h
21+ h
22)
∂ξ
0= 2 ∂(h
1+ h
2)
∂ξ
1. (18)
We substitute the expressions of (6) for h
21+ h
22and h
1+ h
2into (17), (18), using the notations P
1(ξ
0, ξ
1) = C
11(ξ
1) + C
12(ξ
1)ξ
0, P
2(ξ
0, ξ
1) = C
21(ξ
1) + C
22(ξ
1)ξ
0+ C
23(ξ
1)ξ
02. Then, equation (18) restricted to Ω
0× 0 takes the form
∂G
2∂ξ
0(ξ
0, 0) − c
22− 2c
23ξ
0= 2 ∂G
1∂ξ
1(ξ
0, 0) − C ˙
11(0) − C ˙
12(0)ξ
0. (19) We divide this equation by ξ
0and tend ξ
0→ ∞. It leads to the equality C ˙
12(0) = c
23. Taking this equality into account and passing ξ
0→ ∞ in (19), we obtain the equality C ˙
11(0) =
12c
22.
Next, we substitute the expressions of (6) for h
21+ h
22and h
1+ h
2into (17) and restrict the obtained formula to Ω
0× 0. It leads to the equality
2 G
1−c
11−c
12ξ
0∂G
1∂ξ
1− C ˙
11(0)− C ˙
12(0)ξ
0− ∂G
2∂ξ
1+ ˙ C
21(0)+ ˙ C
22(0)ξ
0+ ˙ C
23(0)ξ
02=
=
G
1− c
11− c
12ξ
02− G
2+ c
21+ c
22ξ
0+ c
23ξ
02∂G
1∂ξ
0− c
12. (20)
We divide this equation by ξ
02and tend ξ
0→ ∞. This leads to the equality 2c
12C ˙
12(0) + ˙ C
23(0) = − c
212+ c
23c
12.
Taking this equality into account, dividing (20) by ξ
0and passing ξ
0→ ∞, we obtain the equality
2c
11C ˙
12(0) + 2c
12C ˙
11(0) + ˙ C
22(0) = − 2c
11c
12+ c
22c
12. Using the obtained equalities, one can rewrite (20) in the form
2 G
1− c
11− c
12ξ
0∂G
1∂ξ
1− 2G
1C ˙
11(0) + ˙ C
12(0)ξ
0+ 2c
11C ˙
11(0) − ∂G
2∂ξ
1+ ˙ C
21(0) =
=
G
1− c
11− c
12ξ
02− G
2+ c
21+ c
22ξ
0+ c
23ξ
02∂G
1∂ξ
0+ +
G
1− c
112− 2G
1c
12ξ
0− G
2+ c
21(−c
12). (21) We pass ξ
0→ ∞ and note that the following relations are valid
ξ
lim
0→∞ξ
0∂G
1∂ξ
1= lim
ξ0→∞
ξ
01 2πi
Z
γ
z
1dz
1ξ
0+ z
2= 1 2πi
Z
γ
z
1dz
1= 0,
ξ0
lim
→∞ξ
0G
1= lim
ξ0→∞
ξ
01 2πi
Z
γ
z
1dz
2ξ
0+ z
2= 1
2πi Z
γ
z
1dz
2= æ
10,
ξ0
lim
→∞ξ
02∂G
1∂ξ
0= − lim
ξ0→∞
ξ
021 2πi
Z
γ
z
1dz
2(ξ
0+ z
2)
2= − 1 2πi
Z
γ
z
1dz
2= −æ
10.
As a result, we obtain
−2æ
10C ˙
12(0) + 2c
11C ˙
11(0) + ˙ C
21(0) = − c
212+ c
23æ
10− c
211− 2c
12æ
10+ c
21c
12. Due to the obtained relations, we can express constants C ˙
ij(0) as functions of c
ij:
C ˙
11(0) = 1 2 c
22, C ˙
12(0) = c
23,
C ˙
23(0) = −c
312− 3c
12c
23,
C ˙
22(0) = −2 c
11c
212+ c
12c
22+ c
11c
23,
C ˙
21(0) = æ
10(c
212+ c
23) − c
12(c
211+ c
21) − c
11c
22.
(22)
Substituting these constants to (21), we obtain the third statement of Theorem 3.1.
Complement 3.1. The statement of Theorem 3.1 admits a development for the case N
−≥ 3. In this case
P
k(ξ
0, ξ
1) = C
k1(ξ
1) + C
k2(ξ
1)ξ
0+ · · · + C
k,k+1(ξ
1)ξ
0k, k = 1, . . . , N
−. Denote C ˙
ij(0) =
∂C∂ξij1
(0) and c
ij= C
ij(0) for i = 1, . . . , N
−and j = 1, . . . , i + 1.
Let us indicate the following general procedure for finding of constants c
ij. Due to the Riemann–Burgers equations (1) the following identities in ξ
0∈ Ω
0hold for k = 1, . . . , N
−− 1:
− ∂G
k∂ξ
1(ξ
0, 0) + ˙ C
k1(0) + ˙ C
k2(0)ξ
0+ · · · + ˙ C
k,k+1(0)ξ
0k= k
k + 1
− ∂G
k+1∂ξ
0(ξ
0, 0) + c
k+1,2+ 2c
k+1,3ξ
0+ · · · + (k + 1)c
k+1,k+2ξ
0k. Taking into account that
∂G∂ξk1
(ξ
0, 0) → 0 and
∂G∂ξk+10
(ξ
0, 0) → 0 as ξ
0→ +∞ we obtain the equalities
C ˙
k,m(0) = km
k + 1 c
k+1,m+1, k = 1, . . . , µ
0− 1, m = 1, . . . , k + 1.
Due to the Riemann–Burgers equations (1) the following identity in ξ
0∈ Ω
0holds:
∂e
µ0∂ξ
1(ξ
0, 0) = e
µ0(ξ
0, 0) ∂p
1∂ξ
0(ξ
0, 0), (23)
where functions e
kare given by the following formulas:
ke
k(ξ
0, ξ
1) =
k−1
X
i=1
(−1)
i+1e
k−i(ξ
0, ξ
1)p
i(ξ
0, ξ
1) + (−1)
k+1p
k(ξ
0, ξ
1), p
k(ξ
0, ξ
1) = G
k(ξ
0, ξ
1) − C
k1(ξ
1) − C
k2(ξ
1)ξ
0− · · · − C
k,k+1(ξ
1)ξ
0k,
(24)
where k = 1, . . . , µ
0.
Equality (23) allows to represent constants { C ˙
µ0,j(0)} as functions of constants {c
ij}. Finally, substituting the obtained expressions for constants { C ˙
ij(0)} via con- stants {c
ij} into equation (23) we obtain the identity in ξ
0∈ Ω
0for computation of constants {c
ij}.
For example, in the case N
−= 3 the identity (23) in ξ
0∈ Ω
0for finding of constants c
ijtakes the form
C ˙
31(0) + ˙ C
32(0)ξ
0+ ˙ C
33(0)ξ
20+ ˙ C
34(0)ξ
03= ∂G
3∂ξ
1+ + 3
4 (p
21− p
2) ∂p
2∂ξ
0− p
1∂p
3∂ξ
0− 1
2 p
31− 3p
1p
2+ 2p
3∂p
1∂ξ
0,
where all functions are evaluated at point (ξ
0, 0), the functions p
kare defined in formula (24) and the constants C ˙
31(0), C ˙
32(0), C ˙
33(0), C ˙
34(0) are given by formulas
C ˙
31(0) = 1
2 æ
103c
11c
212+ 3c
12c
22+ 3c
11c
23+ 2c
33− 1
2 æ
11c
312+ 3c
12c
23+ 2c
34− 3 2
1 2πi
Z
γ
z
1z
2dz
1c
212+ c
23− 1
2 c
311c
12− 3
2 c
11c
12c
21− 3
4 c
211c
22− 3
4 c
21c
22− c
12c
31− c
11c
32, C ˙
32(0) = æ
10c
312+ 3c
12c
23+ 2c
34− 3
2 c
211c
212− 3
2 c
212c
21− 3c
11c
12c
22− 3
4 c
222− 3
2 c
211c
23− 3
2 c
21c
23− 2c
12c
32− 2c
11c
33, C ˙
33(0) = − 3
2 c
11c
312− 9
4 c
212c
22− 9
2 c
11c
12c
23− 9
4 c
22c
23− 3c
12c
33− 3c
11c
34C ˙
34(0) = − 1
2 c
412− 3c
212c
23− 3
2 c
223− 4c
12c
34, where æ
10and æ
11are defined in formula (9).
The next theorem permits to find N
−= |X ∩ C P
∞1| from γ = ∂X.
Theorem 3.2. Let X ⊂ C P
2\ (0 : 1 : 0) be a complex curve with rectifiable boundary γ ⊂ C
2and satisfying (∗). Let G
m≥1be the functions defined in (2) and let N
−=
|X ∩ C P
∞1|. Fix any ξ
0∗∈ Ω
0and let W
ξ∗be a neighborhood of ξ
∗= (ξ
0∗, 0) in C
2. Then the following statements are valid:
1. If G
1= 0 in W
ξ∗, then either N
−= 0, or γ
0bounds a complex curve in C
2, where γ
0denotes γ with the opposite orientation.
2. If there exist complex constants c
11, c
12such that
∂
∂ξ1
G
1− P
1= (G
1− P
1)
∂ξ∂0
(G
1− P
1) in W
ξ∗, (25)
where P
1(ξ
0, ξ
1) =
c1+c11ξ0+c1211ξ1
, then N
−≤ 1. Furthermore, c
11, c
12are the same constants as in Theorem 3.1.
3. If there exist complex constants a
1, a
2, b
1, b
2, c
1, c
2such that c
1+ c
2= æ
10,
∂
∂ξ1
((G
1− P
1)
2− G
2− P
2)
= ((G
1− P
1)
2− G
2− P
2)
∂ξ∂0
(G
1− P
1) in W
ξ∗, P
1(ξ
0, ξ
1) = −
a1−a1ξ0+b11ξ1
−
a1−a2ξ0+b22ξ0
, P
2(ξ
0, ξ
1) = −
2
X
j=1
ajξ0+bj1−ajξ1
2+
1−a2ajcjjξ1
,
(26)
then N
−≤ 2. Furthermore, these constants are related to the constants of The- orem 3.1 by the equations:
a
1+ a
2= −c
12, b
1+ b
2= −c
11, a
21+ a
22= −c
23, a
1b
1+ a
2b
2= −
12c
22, b
21+ b
22+ 2a
1c
1+ 2a
2c
2= −c
21, c
1+ c
2= æ
10.
(27)
Proof. 1. The equality G
1= 0 in W
ξ∗implies that G
1= 0 for all ξ ∈ Ω
0× C . In turn, it implies, according to [4], the moment condition
Z
γ
z
1k1z
2k2dz
2= 0, k
1, k
2≥ 0.
Then, according to [13] and [6], for an appropriate choice of orientation, γ is the boundary of a complex curve in C
2.
2. Set h = G
1− P
1in a neighborhood W
ξ∗of ξ
∗. Then h satisfies
∂2
∂ξ02
(G
1− h) = 0 in W
ξ∗,
∂h
∂ξ1
= h
∂ξ∂h0
in W
ξ∗.
It follows from Theorem 2.2 that N
−≤ 1. Note also that equation (10) is the restric- tion of (25) to W
ξ∗∩ (Ω
0× 0).
3. Consider the following quadratic equation in variable t:
t
2− (G
1− P
1)t +
12((G
1− P
1)
2− G
2− P
2) = 0. (28) Suppose that the discriminant is non-zero at ξ
∗. Then, without loss of generality, it is non-zero in W
ξ∗(we can always choose a smaller neighborhood). We denote two different roots of this equation as e h
1= e h
1(ξ), e h
2= e h
2(ξ). Clearly, e h
1and e h
2are holomorphic in W
ξ∗. Furthermore, by the Vi?te formulas we have
e h
1+ e h
2= e e
1:= G
1− P
1in W
ξ∗,
e h
1e h
2= e e
2:=
12((G
1− P
1)
2− G
2− P
2) in W
ξ∗. Note that by definition
∂P∂ξ11
=
12∂P∂ξ20
. Note also that by Lemma 3.3.1 of [4] we have
∂G1
∂ξ1
=
12∂G∂ξ20
. It leads to the equation
∂ e e
1∂ξ
1= 1 2
∂
∂ξ
0( e e
21− 2 e e
2) in W
ξ∗. (29) Furthermore, equation (26) can be rewritten in the form
∂ e e
2∂ξ
1= e e
2∂ e e
1∂ξ
0in W
ξ∗. (30)
Now denote by b h
1, b h
2the shock-wave extensions of e h
1(·, 0) and e h
1(·, 0) to W
ξ∗which exist and are unique by the Cauchy-Kowalevski theorem. Set b e
1= b h
1+ b h
2, b e
2= b h
1b h
2. Due to the shock-wave equations for b h
1and b h
2, the functions b e
1and b e
2satisfy
∂ b e
1∂ξ
1= 1 2
∂
∂ξ
0( b e
21− 2 b e
2) in W
ξ∗,
∂ b e
2∂ξ
1= e b
1∂ b e
2∂ξ
0in W
ξ∗.
Thus, e e
1, e e
2and b e
1, b e
2are holomorphic solutions to the same system with the same restrictions at ξ
1= 0. By the Cauchy-Kowalevski theorem, e e
1= b e
1and e e
2= b e
2. It follows from the Vi?te formulas that e h
1, e h
2coincide with b h
1, b h
2(up to order). Hence, e h
1, e h
2satisfy the shock-wave equations.
Applying Theorem 2.2, we obtain that N
−≤ 2.
It remains to consider the case when the determinant of equation (28) vanishes in W
ξ∗. Otherwise, it vanishes on (at most) a dimension one analytic set and in any neighborhood of ξ
∗there are balls where it does not vanish.
The zero discriminant condition reads
(G
1− P
1)
2= 2(G
2− P
1) in W
ξ∗.
We set e h =
12(G
1− P
1). Then by definition of P
1and from the discriminant condition we get
∂
∂ξ02
(G
1− 2 e h) = 0 in W
ξ∗,
∂eh
∂ξ1
= e h
∂ξ∂eh0
in W
ξ∗.
By Theorem 2.2 it implies N
−≤ 2. Note also that if N
−= 2, then all intersections of X with C
1ξ, ξ ∈ W
ξ∗, are double.
Finally, note that equation (11) is the restriction of (26) to the set W
ξ∗∩ (Ω
0× 0).
Complement 3.2. The statement of Theorem 3.2 can be generalized to the case N
−≥ 3 in the spirit of cases N
−≤ 2. Such a generalization will be developed in a separate paper together with a statement of Theorem 3.1 for N
−≥ 3, indicated in Complement 3.1.
We pass to the description of the algorithm of reconstruction of a complex curve X ⊂ C P
2satisfying the minimality condition (∗) from the known boundary γ = ∂X ⊂ C
2.
Let {ξ
0k}
Nk=1, ξ
0k∈ C be an arbitrary grid in C , ξ
i06= ξ
0j, i 6= j, and ξ
k0∈ / π
2γ,
k = 1, . . . , N . The complex curve X intersects complex line {z
2= −ξ
0k} at points
(h
s(ξ
k0, 0), −ξ
0k), 1 ≤ s ≤ N
+(ξ
0k, 0). We are going to present the formulas for finding
these points.
The algorithm takes as input the points {ξ
k0}
Nk=1and the curve γ (for example, rep- resented as a finite number of points belonging to γ ). On the output of the algorithm we obtain the set of points (h
s(ξ
0k, 0), −ξ
0k), 1 ≤ k ≤ N ; 1 ≤ s ≤ N
+(ξ
0k, 0).
1. No points at infinity
1. Computation of N
+. According to formula (4), for every domain Ω
l≥0, the number µ
l= N
+(ξ
0, 0), ξ
0∈ Ω
l, is equal to the winding number of the curve π
2γ with respect to a point ξ
0∈ Ω
l:
µ
l≡ N
+(ξ
0, 0) = 1 2πi
Z
γ
dz
2z
2+ ξ
0≡ 1 2πi
Z
π2γ
dz
z − ξ
0, ξ
0∈ Ω
l.
2. Computation of power sums. If N
−= 0 then, according to Theorem 3.1, for every point ξ
k0∈ Ω
l≥0we have that P
m(ξ
0k, 0) = 0. Using formula (6) we obtain the following formulas for the power sums of the functions to be determined:
s
m(ξ
0k) ≡ h
m1(ξ
0k, 0) + · · · + h
mµl
(ξ
k0, 0) = 1 2πi
Z
γ
z
1mdz
2z
2+ ξ
0k, 1 ≤ m ≤ µ
l.
3. Computation of symmetric functions. For every point ξ
0k∈ Ω
≥1, the Newton identities
kσ
k(ξ
0k) =
k
X
i=1
(−1)
i−1σ
k−i(ξ
0k)s
i(ξ
0k), 1 ≤ k ≤ N
+(ξ
k0, 0).
allow to reconstruct the elementary symmetric functions:
σ
1(ξ
k0) = h
1(ξ
0k, 0) + · · · + h
µl(ξ
0k, 0),
· · · = · · ·
σ
µl(ξ
k0) = h
1(ξ
0k, 0) × · · · × h
µl(ξ
0k, 0).
4. Desymmetrisation. For every point ξ
0k∈ Ω
l, using Vi?te formulas, one can find the complex numbers h
1(ξ
0k, 0), . . . , h
µl(ξ
0k, 0) (up to order). The points h
s(ξ
0k, 0), −ξ
0k), 1 ≤ s ≤ N
+(ξ
k0, 0), 1 ≤ k ≤ N , are the required points of the complex curve X.
2. One or two points at infinity
These cases can be reduced to the case N
−= 0 in the following way. Since π
2γ ⊂ C is
a compact real curve, there exists such R > 0, that the set B
Rc(0) = {z ∈ C | |z| > R}
in contained in Ω
0. Without loss of generality, one can suppose that |ξ
0k| < R for all k = 1, . . . , N.
Consider an auxiliary complex curve X
R= {(z
1, z
2) ∈ X | |z
2| 6 R}. Its boundary γ
Rconsists of two disjoint parts (possibly, multiconnected): the first part is γ and the second part γ
R0is obtained by lifting the circle S
R= {z ∈ C | |z| = R} to X via the projection π
2: X → C .
The complex curve X
Rdoes not intersect infinity. Moreover, points of the form a, −ξ
0k, k = 1, . . . , N belong to X if and only if they belong to X
R.
Therefore, in order to reconstruct X it is sufficient to reconstruct γ
R0and then to reconstruct X
R, using the algorithm for the case when N
−= 0. The algorithm can be formulated as follows:
1. New boundary. Choose a sufficiently large R > 0, so that B
cR⊂ Ω
0and all ξ
0kbelong to B
R. In the case of N
−= 1, by virtue of formulas (6), we have that
h
1(ξ
0, 0) = 1 2πi
Z
γ
z
1dz
2z
2+ ξ
0− P
1(ξ
0, 0), |ξ
0| = R,
where P
1can be found using Theorem 3.1. This formula allows to recover γ
R0and, as a corollary, γ
R= γ t γ
R0= ∂X
R.
In the case of N
−= 2 we have two equalities:
h
1(ξ
0, 0) + h
2(ξ
0, 0) = 1 2πi
Z
γ
z
1dz
2z
2+ ξ
0− P
1(ξ
0, 0), |ξ
0| = R, h
21(ξ
0, 0) + h
22(ξ
0, 0) = 1
2πi Z
γ
