Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian
We introduce the class of controlled Loewner-Kufarev equations and consider aspects of their algebraic nature. We lift the solution of such a controlled equation to the (Sato)-Segal-Wilson Grassmannian, and discuss its relation with the tau-function. We briefly highlight relations of the Grunsky mat...
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| description | We introduce the class of controlled Loewner-Kufarev equations and consider aspects of their algebraic nature. We lift the solution of such a controlled equation to the (Sato)-Segal-Wilson Grassmannian, and discuss its relation with the tau-function. We briefly highlight relations of the Grunsky matrix with integrable systems and conformal field theory. Our main result is the explicit formula that expresses the solution of the controlled equation in terms of the signature of the driving function through the action of words in generators of the Witt algebra.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 108, 25 pages
Controlled Loewner–Kufarev Equation Embedded
into the Universal Grassmannian
Takafumi AMABA † and Roland FRIEDRICH ‡
† Fukuoka University, 8-19-1 Nanakuma, Jônan-ku, Fukuoka, 814-0180, Japan
E-mail: fmamaba@fukuoka-u.ac.jp
‡ ETH Zürich, D-GESS, CH-8092 Zurich, Switzerland
E-mail: roland.friedrich@gess.ethz.ch
Received June 30, 2020, in final form October 22, 2020; Published online October 29, 2020
https://doi.org/10.3842/SIGMA.2020.108
Abstract. We introduce the class of controlled Loewner–Kufarev equations and consider
aspects of their algebraic nature. We lift the solution of such a controlled equation to
the (Sato)–Segal–Wilson Grassmannian, and discuss its relation with the tau-function. We
briefly highlight relations of the Grunsky matrix with integrable systems and conformal field
theory. Our main result is the explicit formula which expresses the solution of the controlled
equation in terms of the signature of the driving function through the action of words in
generators of the Witt algebra.
Key words: Loewner–Kufarev equation; Grassmannian; conformal field theory; Witt algebra;
free probability theory; Faber polynomial; Grunsky coefficient; signature
2020 Mathematics Subject Classification: 35Q99; 30F10; 35C10; 58J65
1 Introduction
C. Loewner [21] and P.P. Kufarev [20] initiated a theory which was then further extended by
C. Pommerenke [29], and which shows that given any continuously increasing family of simply
connected domains containing the origin in the complex plane, the inverses of the Riemann
mappings associated to the domains are described by a partial differential equation, the so-
called Loewner–(Kufarev) equation
∂
∂t
ft(z) = zf ′t(z)p(t, z),
where the ft are the inverses of the Riemann map and p(z, t) is a function with positive real part
(see Section 2.2 for details). More recently, I. Markina and A. Vasil’ev [25, 27] considered the
so-called alternate Loewner–Kufarev equation, which describes not necessarily increasing chains
of domains.
We introduce a further generalisation, namely, the class of controlled Loewner–Kufarev equa-
tions
dft(z) = zf ′t(z){dx0(t) + dξ(x, z)t}, f0(z) ≡ z ∈ D,
where D is the unit disc in the complex plane centred at zero, x0, x1, x2, . . . are given functions
which will be called the driving functions, x = (x1, x2, . . .) and ξ(x, z)t :=
∑∞
n=1 xn(t)zn. The
controlled Loewner–Kufarev equation can be transformed, after a calculation, into
dft(z) = −
∞∑
n=0
(Lnf)(z)dxn,
mailto:fmamaba@fukuoka-u.ac.jp
mailto:roland.friedrich@gess.ethz.ch
https://doi.org/10.3842/SIGMA.2020.108
2 T. Amaba and R. Friedrich
where the Ln := −zn+1∂/(∂z), n ∈ Z, are the generators of the Witt algebra, i.e., the central
charge zero Virasoro algebra, satisfying the commutation relations
[Lm, Ln] = (m− n)Lm+n. (1.1)
Therefore, we are going to consider an extension of [10], where the second author established
and studied the role of Lie vector fields, boundary variations and the Witt algebra in connection
with the Loewner–Kufarev equation.
Let us recall first some of the classical work of A.A. Kirillov and D.V. Yuriev [15] / G.B. Segal
and G. Wilson [32] / N. Kawamoto, Y. Namikawa, A. Tsuchiya and Y. Yamada [14] which will
be also fundamental in the present context, in particular in understanding the appearance of
the Virasoro algebra with nontrivial central charge.
A.A. Kirillov and D.V. Yuriev [15], constructed a highest weight representation of the Virasoro
algebra, where the representation space is given by the space of all holomorphic sections of an
analytic line bundle over the orientation-preserving diffeomorphism group Diff+S
1 of the unit
circle S1 (modulo rotations). They also gave an embedding of
(
Diff+S
1
)
/S1 into the infinite
dimensional Grassmannian. In fact, this embedding is an example of a construction of solutions
to the KdV hierarchy found by I. Krichever [19], which we address in Section 3.2. If we embed
a univalent function on the unit disc D into the infinite dimensional Grassmannian, by the
methods of Kirillov–Yuriev [15], Krichever [19], or Segal–Wilson [32], then one needs to track
the Faber polynomials and Grunsky coefficients associated to the univalent function. In general,
it is not easy to calculate them from the definition. One of our main results is, however, the
following.
Theorem 1.1 (see Propositions 2.12 and 2.14). The Faber polynomials and Grunsky coefficients
associated to solutions of the controlled Loewner–Kufarev equation satisfy linear differential equa-
tions, and the Grunsky coefficients can be explicitly calculated.
In [10], the second author proposed to lift the embedded Loewner–Kufarev equation to the
determinant line bundle over the Sato–Segal–Wilson Grassmannian Gr(H), as a natural ex-
tension of the “Virasoro uniformisation” approach by M. Kontsevich [16] / R. Friedrich and
J. Kalkkinen [11] to construct generalised stochastic / Schramm–Loewner evolutions [31] on
arbitrary Riemann surfaces, which would also yield a connection with conformal field theory in
the spirit of [14, 32]. Let us also mention the work of B. Doyon [6], who uses conformal loop
ensembles (CLE), and which is related to the content of the present article.
In [27], I. Markina and A. Vasil’ev established basic parts of this program, by considering
embedded solutions to the Loewner–Kufarev equation into the Segal–Wilson Grassmannian and
related the dynamics therein with the representation of the Virasoro algebra, as discussed by
Kirillov–Yuriev [15]. Further, they considered the tau-function associated to the embedded
solution as a lift to the determinant line bundle. As observed and briefly discussed in [11, 16],
the generator of the stochastic Loewner equation is hypo-elliptic.
I. Markina, I. Prokhorov and A. Vasil’ev [24] observed and discussed the sub-Riemannian
nature of the coefficients of univalent functions. As the second author pointed out [10], this
connects with the general theory of hypo-elliptic flows, as explained in the book by F. Baudoin [4],
and led him to propose a connection of the (stochastic) Loewner–Kufarev equation with rough
paths. Now, in the theory of rough paths (see, e.g., the introduction in [22]), one of the central
objects of consideration is the following controlled differential equation:
dYt = ϕ(Yt)dXt, (1.2)
where Xt is a continuous path in a normed space V , called the input of (1.2). On the other
hand, the path Yt is called the output of (1.2). When we deal with this equation, an important
Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian 3
object is the signature of the input Xt, with values in the (extended) tensor algebra associated
with V and which is written in the following form:
S(X)s,t :=
(
1, X1
s,t, X
2
s,t, . . . , X
n
s,t, . . .
)
, s 6 t.
If Xt has finite variation with respect to t, then each Xn
s,t is the nth iterated integral of Xt
over the interval [s, t]. With this object, a combination of the Magnus expansion and the Chen–
Strichartz expansion theorem (see, e.g., [4, Section 1.3]) tells us that the output Yt is given
as the result of the action of S(X)0,t applied to Y0. Heuristically, we may say that a ‘group
element’ S(X) in some big ‘group’ acts on some element in the (extended) tensor algebra T ((V ))
which gives the output Yt, or it might be better to say that the vector field ϕ defines how the
‘group element’ acts on the algebra. In this spirit, we would like to describe such a picture in
the context of controlled Loewner–Kufarev equations.
For this, we extract the algebraic structure of the controlled Loewner–Kufarev equation. If
we regard the driving functions x0, x1, x2, . . . just as letters in an alphabet then it turns out that
explicit expressions for the associated Grunsky coefficients are given by the algebra of formal
power series, where the space of coefficients is given by words over this alphabet. It is worth
mentioning that the action of the words over this alphabet will be actually given by the negative
part of the Witt generators. Thus the action of the signature encodes many actions of such
negative generators. This can be used to derive a formula for ft(z) as the signature ‘applied’ to
the initial data f0(z) ≡ z (see Theorem 3.8).
Now, given a diffeomorphism of the unit circle S1, the solution to the associated conformal
welding problem is a solution to the dispersionless Toda lattice hierarchy [34, 36]. Also in this
case, the corresponding tau-function is described by the (full) Grunsky coefficients and this
generates the solution via an explicit formula. This gives us the possibility to explicitly describe
the solution to the conformal welding problem associated to Malliavin’s canonic diffusion [23] by
means of a controlled Loewner–Kufarev equation; a topic to which we intend to return elsewhere.
Since the canonic diffusion is a natural object ‘on’ the diffeomorphism group of S1, as well as
the Brownian motion on a Euclidean space, it would describe a natural universal class in the
infinite-dimensional situation.
However, the story so far lets us ask how the signature associated with the driving functions
describes the corresponding tau-function rather than ft itself.
Theorem 1.2 (see Theorem 3.9). Along the solution of the controlled Loewner–Kufarev equa-
tion, the associated tau-function can be written as the determinant of a quadratic form of the
signature.
Let us now summarise the structure of the paper. In Section 2, we formulate solutions ft(z)
to controlled Loewner–Kufarev equations. We add also a brief review of the classical Loewner–
Kufarev equation, and then explain how the classical one is recovered from the controlled
Loewner–Kufarev equation. We track the variation of the Taylor-coefficients of ft and also
the Faber polynomials and Grunsky coefficients. In Section 3, we first recall briefly basics of
the Segal–Wilson Grassmannian and Krichever’s construction. After that, we describe how a
univalent function on D is embedded into the Grassmannian. We extract the algebraic structure
of the controlled Loewner–Kufarev equation in order to obtain Theorem 3.9. In Appendix A,
we give the proofs of Theorems 2.10 and 3.9, respectively, and of Proposition 2.14.
2 The controlled Loewner–Kufarev equation
General assumption: N denotes the set of all positive integers, i.e., {1, 2, 3, . . .}, (without zero).
4 T. Amaba and R. Friedrich
2.1 Definition of solutions to controlled Loewner–Kufarev equations
Given functions x1, x2, . . . : [0, T ]→ C, we will write x := (x1, x2, . . .) and
ξ(x, z)t :=
∞∑
n=1
xn(t)zn, for z ∈ C,
if it converges. If A : [0, T ]→ C is of bounded variation, we write dA or A(dt) (when emphasising
the coordinate t on [0, T ]) for the associated complex-valued Lebesgue–Stieltjes measure on [0, T ],
and the total variation measure will be denoted by |dA|.
Definition 2.1. Let T > 0. Suppose that x0 : [0, T ]→ R, as well as x1, x2, . . . : [0, T ]→ C, are
continuous and of bounded variation, and x0(0) = 0. Let ft : D→ C be conformal mappings for
0 6 t 6 T . We say {ft}06t6T is a solution to
dft(z) = zf ′t(z){dx0(t) + dξ(x, z)t}, f0(z) ≡ z ∈ D (2.1)
if
(1) f0(z) ≡ z for z ∈ D,
(2)
∑∞
n=1 n
∫
[0,T ] |dxn|(t)r
n converges for all r ∈ (0, 1),
(3) for each compact set K ⊂ D, the mapping [0, T ] 3 t 7→ f ′t |K ∈ C(K) is continuous with
respect to the uniform norm on K,
(4) it holds that
ft(z)− z =
∫ t
0
zf ′s(z)
{
dx0(s) + dξ(x, z)s
}
, (t, z) ∈ [0, T ]× D.
In the sequel, we refer to equation (2.1) as a controlled Loewner–Kufarev equation (with
driving paths x0 and x := (x1, x2, . . .)).
In joint work with T. Murayama [3], we proved that a solution to the controlled Loewner–Ku-
farev equation is unique if it exists [3, Theorem 3.1]. In the ω-controlled case, for ω(0, T ) < 1/2,
a solution exists (and hence uniquely exists), cf. [3, Theorem 3.2]. More specifically, we have
Proposition 2.2 ([3, Lemma 2.1]). Under the assumptions (1)–(3) above,
(i) the series ξ(x, z)t in z has convergence radius one for each t ∈ [0, T ],
(ii) the family {ξ(x, z)}06t6T of holomorphic functions on D is continuous in the topology of
locally uniform convergence,
(iii) the function t 7→ ξ(x, z)t is of bounded variation and satisfies
dξ(x, z)t =
∞∑
k=1
zkdxk(t),
for each z ∈ D.
Furthermore, in [3, equations (3.1) and (3.2)] we proved that f ′t(0) = ex0(t) > 0.
Definition 2.3. We say {ft}06t6T is a univalent solution to the controlled Loewner–Kufarev
equation if it is a solution to (2.1) and ft is a univalent function on D for each 0 6 t 6 T .
Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian 5
2.2 Loewner–Kufarev equation as a controlled Loewner–Kufarev equation
Definition 2.4. Suppose that Ω(t) ⊂ C is given for each 0 6 t 6 T . {Ω(t)}06t6T is called
a Loewner subordination chain if
(1) 0 ∈ Ω(s) ( Ω(t) for each 0 6 s < t 6 T ,
(2) Ω(t) is a simply connected domain (i.e., open, connected and simply connected) for each
t ∈ [0, T ],
(3) (Continuity in the sense of Carathéodory, under the conditions (1) and (2)): For each
t ∈ [0, T ] and any sequence 0 6 tn ↑ t, ∪∞n=1Ω(tn) = Ω(t).
For the following Definition 2.5, cf. specifically [29, Chapter 6, Section 6.1, pp. 156–157;
Chapter 2, Section 2.1, p. 35 and Lemma 2.1].
Definition 2.5 ([29]). Let ft : D → C be given for 0 6 t 6 T . Then {ft}06t6T is called
a Loewner chain if
(1) ft is analytic and univalent on D, for each 0 6 t 6 T ,
(2) ft(z) = etz + a2(t)z2 + · · · , for z ∈ D,
(3) fs(D) ⊂ ft(D), for each 0 6 s < t 6 T .
The above chains {Ω(t)} and {ft} are known to be in one-to-one correspondence via the
relation Ω(τ) = ft(D), where t = log f ′τ (0) is a time-reparametrisation to satisfy Definition 2.5(2)
(see [29, Chapter 6, Section 6.1]).
Theorem 2.6 ([29, Theorem 6.2]). Let ft : D → C be given for 0 6 t 6 T . Then {ft}06t6T is
a Loewner chain if and only if there exist constants r0,K0 > 0, and a function p(t, z), analytic
in z ∈ D, and measurable in t ∈ [0, T ] such that
(i) for each 0 6 t 6 T , the function ft(z) = etz + · · · is analytic in |z| < r0, the mapping
[0, T ] 3 t 7→ ft(z) is absolutely continuous for each |z| < r0, and
|ft(z)| 6 K0et, for all |z| < r0 and t ∈ [0, T ].
(ii) Re{p(t, z)} > 0, for all (t, z) ∈ [0, T ]× D, and
∂
∂t
ft(z) = zf ′t(z)p(t, z), (2.2)
for all |z| < r0 and for almost all t ∈ [0, T ].
According to the terminology in [5] we call the equation (2.2) the Loewner–Kufarev equation
(if we regard p(t, z) as given and ft(z) as unknown).
Because of equation (2.2), it holds that p(t, 0) = lim
z→0
(
∂
∂tft(z)
)
/(zf ′t(z)) = 1, and hence the
‘Herglotz representation theorem’ applies, which permits us to conclude that, for every t ∈ [0, T ],
there exists a probability measure νt on S1 = ∂D (which is naturally identified with [0, 2π] as
measurable spaces, and then the induced probability measure is still denoted by νt) such that
p(t, z) =
∫ 2π
0
eiθ + z
eiθ − z
νt(dθ) for z ∈ D.
Substituting this into (2.2), the Loewner–Kufarev equation becomes
∂ft
∂t
(z) = zf ′t(z)
∫ 2π
0
eiθ + z
eiθ − z
νt(dθ). (2.3)
6 T. Amaba and R. Friedrich
Assuming that νt(dθ) =: νt(θ)dθ, we write the Fourier series of νt(θ) as
νt(θ) =
1
2π
{
a0(t) +
∞∑
k=1
(
ak(t) cos(kθ) + bk(t) sin(kθ)
)}
.
We temporarily introduce the notation x0(t) :=
∫ t
0 a0(s)ds and
uk(t) :=
∫ t
0
ak(s)ds, vk(t) := −
∫ t
0
bk(s)ds,
for k = 1, 2, . . .. Because of the relations
1
2π
∫ 2π
0
eiθ + z
eiθ − z
cos(kθ)dθ = zk,
1
2π
∫ 2π
0
eiθ + z
eiθ − z
sin(kθ)dθ = −izk,
for k = 1, 2, . . . and |z| < 1, equation (2.3) assumes the following form:
∂ft
∂t
(z) = zf ′t(z)
{
ẋ0(t) +
∞∑
k=1
(
u̇k(t) + iv̇k(t)
)
zk
}
.
This can be rewritten as the following controlled differential equation
dft(z) = zf ′t(z){dx0(t) + dξ(x, z)t},
where xk(t) = uk(t) + ivk(t) for k > 1, and ξ(x, z)t =
∑∞
k=1 xk(t)z
k.
If we omit the condition Re{p(t, z)} > 0, that is, we allow the real part of p(t, z) to have
an arbitrary sign, then equation (2.2) is called the alternate Loewner–Kufarev equation, as
considered by I. Markina and A. Vasil’ev [25]. Intuitively, this describes evolutions of conformal
mappings whose images of D are not necessary increasing, i.e., not strict subordinations. It
appears that the general theory with respect to the existence and uniqueness of solutions is not
yet fully developed. However, our controlled Loewner–Kufarev equation (2.1) deals with this
alternate case because we have not assumed that p(t, z) := d
dt(x0(t) + ξ(x, z)t) has a positive
real part.
Remark 2.7. Readers focusing on radial Loewner equations might feel puzzled by the heuristic
assumption that the Radon–Nikodym density νt(dθ)
dθ = νt(θ) exists, because the radial Loewner
equation describes the case νt(dθ) = δeiw(t)(dθ) where w(t) is a continuous path in R, so that
there does not exist a Radon–Nikodym density. However, several explicit examples of Loewner–
Kufarev equations within this setting, are presented with simulations in Sola [33].
2.3 Taylor coefficients along the controlled Loewner–Kufarev equation
Suppose that x0 : [0,+∞) → R, x1, x2, . . . : [0,+∞) → C are continuous and of bounded vari-
ation. Let {ft}06t6T be a solution to the controlled Loewner–Kufarev equation (2.1). We
parametrise ft as
ft(z) = C(t)
(
z + c1(t)z2 + c2(t)z3 + c3(t)z4 + · · ·
)
, (2.4)
with the additional convention that c0(t) ≡ 1.
The dynamics of the coefficients (c1, c2, . . .) has been previously studied by Vasil’ev and his
co-authors [12, 24, 25, 26]. The (stochastic/Schramm)-Loewner (equation/evolution) (SLE)
case is discussed by Friedrich [10]. A complementary, conformal field theoretic perspective of
the Bieberbach–de Branges theorem is given by Duplantier et al. [7]. Within our framework, we
get the following similarly:
Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian 7
Proposition 2.8. Let {ft}06t6T be a solution to the controlled Loewner–Kufarev equation (2.1)
with the parametrisation (2.4). Then we have
dC(t) = C(t)dx0(t),
and
dc1(t) = dx1(t) + c1(t)dx0(t),
dc2(t) = dx2(t) + 2c1(t)dx1(t) + 2c2(t)dx0(t),
...
dcn(t) = dxn(t) +
n−1∑
k=1
(k + 1)ck(t)dxn−k(t) + ncn(t)dx0(t), for n > 2,
(2.5)
with the initial conditions C(0) = 1 and c1(0) = c2(0) = · · · = 0. In particular, C = {C(t)}06t6T
takes its values in R.
As f ′t(0) = C(t) = ex0(t)−x0(0) 6= 0, we get
Corollary 2.9. Let {ft}06t6T be a solution to the controlled Loewner–Kufarev equation (2.1).
Then ft is univalent in a neighbourhood of 0, for each 0 6 t 6 T .
Theorem 2.10. Let {ft}06t6T be a solution to the controlled Loewner–Kufarev equation (2.1).
Then for each n ∈ N, the coefficient cn in (2.4) is given by
cn(t) =
n∑
p=1
∑
i1,...,ip∈N:
i1+···+ip=n
w̃(n)i1,...,ipenx0(t)
×
∫
06s1<s2<···<sp6t
e−i1x0(s1)dxi1(s1)e−i2x0(s2)dxi2(s2) · · · e−ipx0(sp)dxip(sp),
where
w̃(n)i1,...,ip :=
{
(n− i1) + 1
}{
(n− (i1 + i2)) + 1
}
· · ·
{
(n− (i1 + i2 + · · ·+ ip−1)) + 1
}
,
and n = i1 + · · ·+ ip.
The proof can be found in Appendix A.1.
2.4 Variation of Grunsky coefficients induced
by a Loewner–Kufarev equation
There are several different ways to introduce the Faber polynomials. Here we give a derivation
by utilising Teo [35], and, an alternative one, in Section 3.3, which serves our purpose better.
For a (formal) power series f(z) = a1z + a2z
2 + a3z
3 + · · · , a1 6= 0, the (generalised) Faber
polynomials Qn(w), n ∈ N, associated to f , are defined as
log
w − f(z)
w
= log
f(z)
a1z
−
∞∑
n=1
Qn(w)
n
zn. (2.6)
By differentiating equation (2.6), and reordering it, we obtain, via the Residue theorem, the
Faber polynomials (cf. also expression (3.1)), as
Qn(w) = Res
z=0
[
wz−n
w − f(z)
f ′(z)
f(z)
]
dz =
ζ=f(z)
Res
ζ=0
[(
f−1(ζ)
)−n
1− ζw−1
1
ζ
]
dζ.
8 T. Amaba and R. Friedrich
The coefficients (b−m,−n)∞m,n=1 in the series expansion
log
f(z)− f(ζ)
z − ζ
= −
∞∑
m=0
∞∑
n=0
b−m,−nz
mζn, (2.7)
at (z, ζ) = (0, 0), are called the (generalised) Grunsky coefficients of f . Equivalently, these are
defined via the Laurent series at z = 0,
Qn(f(z)) = z−n + n
∞∑
m=1
b−n,−mz
m.
Proposition 2.11. Let {ft}06t6T be a solution to the controlled Loewner–Kufarev equation (2.1).
Then there exists an open neighbourhood U of the origin, such that
(i) U ⊂ D,
(ii) ft|U is univalent for each t ∈ [0, T ],
(iii) V :=
⋂
06t6T ft(U) is an open neighbourhood of the origin,
(iv) for each ζ ∈ V , [0, T ] 3 t 7→ f−1
t (ζ) is continuous and of bounded variation,
(v) for each ζ ∈ V , with f−1(t, ζ) := f−1
t (ζ) and df−1
t (ζ) := f−1(dt, ζ), we have
df−1
t (ζ) = −f−1
t (ζ)
{
dx0(t) +
∞∑
k=1
(
f−1
t (ζ)
)k
dxk(t)
}
,
as Lebesgue–Stieltjes measures on [0, T ].
Let {ft}06t6T be a solution to the controlled Loewner–Kufarev equation (2.1). Because of
Corollary 2.9, associated to each ft(z) are the corresponding Faber polynomials and Grunsky
coefficients, which will be denoted by Qn(t, w), and b−n,−m(t), respectively.
Proposition 2.12.
(i) (Variation of Faber polynomials): We have for each n ∈ N,
dQn(t, w) = ndxn(t) + n
n∑
k=1
Qk(t, w)dxn−k(t).
(ii) (Variation of Grunsky coefficients): For each n,m ∈ N,
db−n,−m(t) = −dxn+m(t) +
∑
k,l∈Z>0;
k+l=m−1
(k + 1)b−n,−(k+1)(t)dxl(t)
+
∑
k,l∈Z>0;
k+l=n−1
(k + 1)b−m,−(k+1)dxl(t), (2.8)
with the initial condition b−n,−m(0) = 0, for all n,m ∈ N.
Proof. (i) Let n ∈ N. Let U and V be as in Proposition 2.11. Then f−1
t (ζ), ζ ∈ V , satisfies
the equation
df−1
t (ζ) = −f−1
t (ζ)
{
dx0(t) +
∞∑
k=1
(
f−1
t (ζ)
)k
dxk(t)
}
.
Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian 9
Let X0 ⊂ V be an open disc centred at 0. By using Cauchy’s integral formula, we have for
w ∈ X0,
dQn(t, w) =
1
2πi
∫
∂X0
dζ
ζ
d
(
f−1
t (ζ)
)−n
1− ζw−1
=
1
2πi
∫
∂X0
(−n)
(
f−1
t (ζ)
)−n−1
1− ζw−1
(
−f−1
t (ζ)
) ∞∑
k=0
(
f−1
t (ζ)
)k
dxk(t)
dζ
ζ
=
n∑
k=0
n
2πi
(∫
∂X0
(
f−1
t (ζ)
)−n+k
1− ζw−1
dζ
ζ
)
dxk(t)
=
ndxn(t)
2πi
∫
∂X0
1
1− ζw−1
dζ
ζ
+ n
n−1∑
k=0
Qn−k(t, w)dxk(t).
By noting that the orientation of ∂X0 is anti-clockwise, we get
1
2πi
∫
∂X0
1
1− ζw−1
dζ
ζ
= 1,
and hence the result.
(ii) By putting p(dt, z) := dx0(t)+dξ(x, z)t, and since ft(z) satisfies the controlled Loewner–
Kufarev equation, we have
dQn(t, ft(z)) = Qn(dt, ft(z)) +Q′n(t, ft(z))dft(z)
= Qn(dt, ft(z)) +Q′n(t, ft(z))
{
zf ′t(z)p(dt, z)
}
= Qn(dt, ft(z)) + z
[
∂zQn(t, ft(z))
]
p(dt, z),
so that
dQn(t, ft(z)) = Qn(dt, ft(z)) + z
[
∂zQn(t, ft(z))
]
p(dt, z). (2.9)
By recalling that Qn(t, ft(z)) = z−n + n
∑∞
m=1 b−n,−m(t)zm, we have, by substitution, the
following sequence of identities
(LHS of (2.9))>1 = (LHS of (2.9)) = n
∞∑
m=1
zmdb−n,−m(t). (2.10)
Here, (· · · )>1 is the operator which forgets those terms in (· · · ), whose degree is less than one.
On the other hand, by Proposition 2.12(i), we have
dQn(t, ft(z)) = ndxn(t) + n
n∑
k=1
Qk(t, ft(z))dxn−k(t)
= ndxn(t) + n
n∑
k=1
(
z−k + k
∞∑
m=1
b−k,−m(t)zm
)
dxn−k(t)
= ndxn(t) + n
n∑
k=1
z−kdxn−k(t) + n
∞∑
m=1
(
n∑
k=1
kb−k,−m(t)dxn−k(t)
)
zm,
so that
(
dQn(t, ft(z))
)
>1
= n
∞∑
m=1
(
n∑
k=1
kb−k,−mdxn−k(t)
)
zm. (2.11)
10 T. Amaba and R. Friedrich
We further have
z
[
∂zQn(t, ft(z))
]
p(dt, z) = z
(
−nz−n−1 + n
∞∑
k=1
kb−n,−kz
k−1
)(
dx0(t) +
∞∑
l=1
dxl(t)z
l
)
= n
(
−dx0(t)z−n −
∞∑
m=1−n
dxm+n(t)zm
+
∞∑
m=1
mb−n,−m(t)dx0(t)zm +
∞∑
m=2
∑
k,l>1;
k+l=m
kb−n,−k(t)dxl(t)z
m
)
,
from which we conclude
(
z
[
∂zQn(t, ft(z))
]
p(dt, z)
)
>1
= n
∞∑
m=1
−dxn+m(t) +
∑
k>1, l>0;
k+l=m
kb−n,−k(t)dxl(t)
zm. (2.12)
Combining (2.11) and (2.12), we obtain
(RHS of (2.9))>1 = n
∞∑
m=1
(
−dxn+m(t) +
∑
k,l∈Z>0;
k+l=m−1
(k + 1)b−n,−(k+1)(t)dxl(t)
+
∑
k,l∈Z>0;
k+l=n−1
(k + 1)b−m,−(k+1)dxl(t)
)
zm,
and then by comparing with (2.10), we get the result. Furthermore, the initial condition is
derived from f0(z) ≡ z. �
In order to derive an explicit formula for the Grunsky coefficients b−n,−m(t), cf. equation (2.7),
we shall introduce some notation. In [2], we study analytic aspects of these coefficients.
Definition 2.13. Let p, q ∈ N.
(1) A bijection σ : {1, 2, . . . , p+ q} → {1, 2, . . . , p+ q} is called a (p, q)-shuffle if it holds that
σ(1) < σ(2) < · · · < σ(p) and σ(p+ 1) < σ(p+ 2) < · · · < σ(p+ q).
(2) Suppose that x1, x2, . . . , xp+q : [0, T ]→ C are continuous and of bounded variation. Then
for each 0 6 t 6 T , we set(
(x1 · · ·xp)� (xp+1 · · ·xp+q)
)
(t)
:=:
∫
06sq6···6s16tp6···6t16t
(
dx1(t1) · · · dxp(tp)
)
�
(
dxp+1(s1) · · · dxp+q(sq)
)
:=
∑
σ−1: (p, q)-shuffle
∫ t
0
dxσ(1)(t1)
∫ t1
0
dxσ(2)(t2) · · ·
∫ tp−1
0
dxσ(p)(tp)
×
∫ tp
0
dxσ(p+1)(s1)
∫ s1
0
dxσ(p+2)(s2) · · ·
∫ sq−1
0
dxσ(p+q)(sq).
The general formula for the Grunsky-coefficients along the controlled Loewner–Kufarev equa-
tion (2.1) is stated as next, and which is crucial for the embedding into the Grassmannian, cf.
Section 3. The proof is given in Appendix A.2.
Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian 11
Proposition 2.14. For n,m ∈ N and t > 0,
b−m,−n(t) = −e(n+m)x0(t)
∫ t
0
e−(n+m)x0(s)dxm+n(s)
−
n+m−2∑
k=2
∑
16i<m;
16j<n:
i+j=k
m−i∑
p=1
n−j∑
q=1
∑
i1,...,ip∈N :
i1+···+ip=m−i
∑
j1,...,jq∈N :
j1+···+jq=n−j
w(i, j)i1,...,ip;j1,...,jq
× e(m+n)x0(t)
∫
06uq6···6u16sq6···6s16t
(
e−i1x0(s1)dxi1(s1) · · · e−ipx0(sp)dxip(sp)
)
�
(
e−j1x0(u1)dxj1(u1) · · · e−jqx0(uq)dxjq(uq)
) ∫ uq
0
e−kx0(s)dxk(s)
−
n+m−1∑
k=m+1
n+m−k∑
q=1
∑
j1,...,jq∈N :
j1+···+jq=n+m−k
w(k −m)∅;j1,...,jq
× e(m+n)x0(t)
∫
06sq6···6s16t
(
e−j1x0(s1)dxj1(s1) · · · e−jqx0(sq)dxjq(sq)
)
×
∫ sq
0
e−kx0(s)dxk(s)−
n+m−1∑
k=n+1
m+n−k∑
p=1
∑
i1,...,ip∈N :
i1+···+ip=m+n−k
w(k − n)i1,...,ip;∅
× e(m+n)x0(t)
∫
06up6···6u16t
(
e−i1x0(u1)dxi1(u1) · · · e−ipx0(up)dxip(up)
)
×
∫ up
0
e−kx0(u)dxk(u), (2.13)
where, for m = i1 + · · ·+ ip + r, and n = j1 + · · ·+ jq + s, we have put
w(r)i1,...,ip;∅ = (m− i1)(m− (i1 + i2)) · · · (m− (i1 + i2 + · · ·+ ip)),
w(s)∅;j1,...,jq = (n− j1)(n− (j1 + j2)) · · · (n− (j1 + j2 + · · ·+ jq)),
and w(r, s)i1,...,ip;j1,...,jq := w(r)i1,...,ip;∅w(s)∅;j1,...,jq .
3 The controlled Loewner–Kufarev equation embedded
into the Segal–Wilson Grassmannian
3.1 Segal–Wilson Grassmannian
Let H := L2
(
S1,C
)
be the Hilbert space which consists of all square-integrable complex func-
tions on the unit circle S1. It decomposes orthogonally into H = H+ ⊕H−, where H+ and H−
are the closure of span
{
zk : k > 0
}
and span
{
zk : k < 0
}
, respectively.
Definition 3.1 (G. Segal and G. Wilson [32, Section 2]). The Segal–Wilson Grassmannian
Gr := Gr(H) is the set of all closed subspaces W of H satisfying the following:
(1) The orthogonal projection pr+ : W → H+ is Fredholm,
(2) The orthogonal projection pr− : W → H− is compact.
12 T. Amaba and R. Friedrich
The Fredholm index of the orthogonal projection pr+ : W → H+ is called the virtual dimension
of W . For d ∈ Z, we set
Gr
(∞
2 + d,∞
)
:= {W ∈ Gr: the virtual dimension of W is d},
and Gr
(∞
2 ,∞
)
:= Gr
(∞
2 + 0,∞
)
.
If we take W = H+, then the corresponding projections are given by pr+ = idH+ and pr− = 0,
which are Fredholm and compact operators, respectively. Therefore we have H+ ∈ Gr
(∞
2 ,∞
)
.
Definition 3.2 ([32, Section 5]). Let Γ+ denote the set of all continuous functions g : S1 → C∗,
such that g(z) = e
∑∞
k=1 tkz
k
, z ∈ S1 for some t = (t1, t2, t3, . . .).
The set Γ+ acts on H by pointwise multiplication. In particular, Γ+ forms a group. This
action induces the action of Γ+ on Gr: Γ+ ×Gr 3 (g,W ) 7→ gW ∈ Gr (see [32, Lemma 2.2 and
Proposition 2.3]), where gW = {gf : f ∈W}. For any g = e
∑∞
k=1 tkz
k ∈ Γ+, the action of g on H
is of the form
g =
(
a b
0 d
)
along H = H+ ⊕H−,
where a : H+ → H+ is invertible and b : H− → H+ is of trace class (see [32, Proposition 2.3]).
Let U be the set of all W ∈ Gr
(∞
2 ,∞
)
such that the orthogonal projection W → H+ is an
isomorphism. Then, associated to each W ∈ U is the tau-function τW (t) of W , a function of
infinitely many “times” t = (t1, t2, . . .). It is known that the following holds:
Proposition 3.3 ([32, Proposition 3.3]). Let W ∈ U . For g = e
∑∞
n=1 tnzn ∈ Γ+, we have
τW (t) = det
(
1 + a−1bA
)
,
where t = (t1, t2, t3, . . .),
g−1 =
(
a b
0 d
)
along H = H+ ⊕H−,
and A : H+ → H− is the linear operator such that graph(A) = W .
3.2 Krichever’s construction
In connection with algebraic geometry and infinite-dimensional integrable systems, a fundamen-
tal observation / construction of Krichever [17, 18, 19] states the following. A solution of the
KdV equation is associated with each non-singular algebraic curve, equipped with some addi-
tional algebro-geometric data. Segal and Wilson [32] developed and formalised, after a remark
by Mumford [28], this construction further.
The specific algebro-geometric datum is given by a quintuple (X,L, x∞, z, ϕ), consisting of
the following parts. X is a complete, irreducible and complex algebraic curve with a rank-
one, torsion-free coherent sheaf L. Additionally, a non-singular point x∞ ∈ X, and a closed
neighbourhood X∞, are chosen, such that there exists a local parameter 1/z : X∞ → D ⊂ Ĉ,
with x∞ 7→ 0, and a trivialisation ϕ : L|X∞ → D×C, of L|X∞ . Each section of L|X∞ is identified
with a complex function on D under ϕ. For X0 := X \Xo
∞, with Xo
∞ the interior of X∞, the
closed sets X0 and X∞ cover X, and X0 ∩X∞ is identified with S1 under z.
Given this algebro-geometric datum, one can associate a closed subspace W ⊂ H, consisting
of all analytic functions S1 → C which, under the above identification, extend to a holomorphic
section of L on an open neighbourhood of X0. More explicitly, one can write
W =
{
the second component
of ϕ ◦ s ◦ (1/z)−1|S1
:
s is a holomorphic section
on a neighbourhood of X0
}H
,
Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian 13
where (1/z)−1 : D → X∞ is the inverse function of 1/z. It is known that W ∈ Gr (see [32,
Proposition 6.1]), and if X is a compact Riemann surface (then L is automatically a complex
line bundle, hence a maximal torsion-free sheaf), this correspondence (X,L, x∞, z, ϕ) 7→W ∈ Gr
is one-to-one (see [32, Proposition 6.2]).
3.3 The appearance of Faber polynomials and Grunsky coefficients
Let f : D → C be a univalent function such that f(0) = 0, and f(D) is bounded by a Jordan
curve. We set β : Ĉ → Ĉ by β(w) := 1/w. For a subset A ⊂ Ĉ, we shall write A−1 := β(A),
and let D̂∞ := Ĉ \D. We obtain an algebro-geometric datum (X,L, x∞, z, ϕ) by setting X = Ĉ,
L = Ĉ × C, x∞ := ∞, X∞ := f
(
D
)−1
, z := β ◦ f−1 ◦ β−1 : X∞ → D̂∞, and ϕ = (1/z) × idC.
Correspondingly, we have X0 = Ĉ\
(
f(D)−1
)
. Further, by the Caratheodory extension theorem,
z extends continuously to X∞, and therefore we can embed f , by assigning a Hilbert space
W = Wf to it, into the Grassmannian. In this case, we have Ĉ \
(
f(D)−1
)
, and hence
Wf =
{
F ◦ (1/z)−1|S1 :
F is a holomorphic function
on a neighbourhood of Ĉ \
(
f(D)−1
)}H .
In order to start this paper’s main calculation, let us specify this more explicitly. For a closed
subset V in Ĉ, we denote by O(V ) the space of all holomorphic functions defined on an open
neighbourhood of V . For a univalent function g : D̂∞ → Ĉ, with g(∞) = ∞, and for each
h ∈ O
(
D
)
, we call
(F[h])(z) :=
1
2πi
∫
∂(C\g(D∞))
h
(
g−1(ξ)
)
ξ − z
dξ, z ∈ C \ g(D∞)
the Faber transform of h (with respect to g). If the boundary ∂(C \ g(D∞)) is analytic, it is
known that h ∈ O
(
D
)
iff Fh ∈ O(C\g(D∞)) (see [13, Theorem 1]) and F : O
(
D
)
→ O(C\g(D∞))
is bijective. In our case, we put
g := (1/z)−1 = β ◦ f ◦ β−1 : D̂∞ → f(D)−1,
and then we can describe O(X0) by O
(
D̂∞
)
through the transformation
F ◦
(
β−1
)∗
=
(
β−1
)∗ ◦Adβ∗(F) : O
(
D̂∞
)
→ O(X0),
where Adβ∗(F) := β∗ ◦ F ◦
(
β−1
)∗
: O
(
D̂∞
)
→ O
(
Ĉ \ f(D)
)
. A direct calculation shows that for
each h(η) =
∑∞
k=0 akη
−k ∈ O
(
D̂∞
)
, we have
(Adβ∗(F)[h])(w) =
1
2πi
∫
∂f(D)
h
(
f−1(ζ)
)
1− ζw−1
dζ
ζ
, w ∈ Ĉ \ f(D).
As a result, (Adβ∗(F)[h])(w) is a power series in 1/w. Actually, in view of the Cauchy integral
formula
1
2πi
∫
S1
ζn
1− ζη−1
dζ
ζ
=
{
ηn if n 6 0,
0 if n > 1,
η ∈ D∞,
we have
(Adβ∗(F)[h])(w) =
n∑
k=0
ak
2πi
∫
∂X0
(
f−1(ζ)
)−k
1− ζw−1
dζ
ζ
=
n∑
k=0
ak
[(
f−1(w)
)−k]
60
,
14 T. Amaba and R. Friedrich
where
[(
f−1(w)
)−k]
60
denotes the constant-part plus the principal-part of the Laurent series
for
(
f−1(w)
)−k
=
(
1/f−1(w)
)k
; hence every element in O
(
Ĉ \ f(D)
)
can be written as a series
in 1/w. The quantity
Qk(w) :=
1
2πi
∫
∂X0
(
f−1(ζ)
)−k
1− ζw−1
dζ
ζ
=
[(
f−1(w)
)−k]
60
, (3.1)
for k ∈ N, is called the k-th Faber polynomial associated to the domain C\f(D) (or simply to f),
and it is a polynomial of degree k in 1/w, cf. also Section 2.4.
We conclude that
[(
β−1
)∗ ◦Adβ∗(h)
]
◦ (1/z)−1 = [Adβ∗(h)] ◦ f ◦ β−1, and hence
Wf = span
(
{1} ∪ {Qn ◦ f ◦ (1/z)|S1}n>1
)H
,
where z is the identity map on D̂∞; note, if f(z) ≡ z then Wf = H+.
Remark 3.4.
(a) The Faber polynomials appeared first (with a different formalism, but equivalent to our
presentation) in the context of approximations of functions in one complex variable by
analytic functions (see [8] and [9]). Since then, they also play an important role in the
theory of univalent functions (see [30]). We introduced the Faber polynomials in a slightly
non-standard way in order to have them in a form which is suitable for embedding univalent
functions into the Grassmannian by using Faber polynomials.
(b) In the context of Abelian function theory, the exterior derivatives
ω(n)
∞ := dQn(f(1/z)),
n = 1, 2, . . . are known as Abelian differentials of the second kind on the Riemann sphere.
In general, Krichever’s embedding of the algebro-geometric datum (X,O, Q, z, ϕ), where
(X,α1, . . . , αg, β1, . . . , βg)
is a homologically marked compact Riemann surface with genus g, O is the structure sheaf
of X, Q ∈ X, z and ϕ are local uniformisers, and a local trivialisation of O, is described
by using multivalued meromorphic functions ϕ(0)(z) ≡ 1,
ϕ(n)(z) :=
∫ z
ω
(n)
Q =: zn −
∞∑
m=1
qnm
z−m
m
,
(modulo periods) where ω
(n)
Q ’s are (normalised) abelian differentials of the second kind [14,
Section 2.27 and p. 304]. These multivalued meromorphic functions can be regarded as
a generalisation of the Faber polynomials (see [37, p. 131]).
(c) Given again a homologically marked compact Riemann surface
(
X, (αi, βi)
g
i=1
)
with genus g,
Krichver’s embedding of yet another datum
(
X,Ω1/2, Q, z,
√
dz
)
or(
X,Ω1/2 ⊗ Lc, Q, z,
√
dz ⊗ sc
)
is described in [14, equation (2.34)]. Here, Ω1/2 is the so-called theta characteristic of
the compact Riemann surface X, Lc is a complex line bundle of degree 0 parametrised by
c ∈ Cg (modulo the lattice associated to (αi, βi)
g
i=1), and sc is a local trivialisation of Lc. In
particular, the embedding of the latter and the associated Fermionic state (the image under
the Plücker embedding) are described by means of the Szegő kernel of Ω1/2⊗Lc (see [1, 14],
in which, the scattering operator in [14, Section 5.12] is a special case of a Bogoliubov
transformation discussed in [1, equations (2.15)–(2.20)]), and then the corresponding tau-
function τ(t) is described as a theta function multiplied by exp
(∑∞
n,m=1 qnmtntm
)
(see
[14, Theorem 5.6]).
Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian 15
3.4 Action of words in Witt algebra generators
Let X = {x1, x2, x3, . . .} be an alphabet, consisting of a countable set of non-commuting letters.
The free monoid X∗ on X is the set of all words in the letters X, including the empty word ∅.
We denote by
C〈X〉 :=
⊕
w∈X∗
Cw = C⊕
∞⊕
n=1
C〈X〉n
the free associative and unital C-algebra on X. The unit of this algebra is the empty word which
we will denote by 1 := ∅. The set C〈X〉n stands for
⊕
|w|=nCw where the summation is taken
over all words w of length n.
Definition 3.5. We define
ξ(x, z) :=
∞∑
n=1
xkz
k ∈ C〈X〉[[z]],
and a distinguished element S(ξ(x, z)) ∈ C〈X〉[[z]] by
S(ξ(x, z)) := 1 +
∞∑
n=1
zn
n∑
p=1
∑
i1,...,ip∈N :
i1+···+ip=n
xi1 · · ·xip .
Definition 3.6. Let x0 : [0,+∞) → R and x1, x2, . . . : [0,+∞) → C be continuous and of
bounded variation. For 0 6 s 6 t, we define [
∫
1]s,t := 1 and[∫
(xip · · ·xi2xi1)
]
s,t
:=
∫
s6u1<u2<···<up6t
e−i1x0(u1)dxi1(u1)e−i2x0(u2)dxi2(u2) · · · e−ipx0(up)dxip(up).
The action of
∫
naturally extends to C〈X〉[[z]], and then we call
S(ξ(x, z))s,t :=
[∫
S(ξ(x, z))
]
s,t
,
the signature of ξ(x, z).
We define a bilinear map T : C〈X〉
((
z−1
))
× C〈X〉 → C〈X〉
((
z−1
))
, by extending the pairing
T (f, 1) := f , and T (f, xip · · ·xi1) := (L−i1 · · ·L−ipf)xip · · ·xi1 , bilinearly, for f ∈ C〈X〉
((
z−1
))
,
p > 1, and i1, . . . , ip ∈ N. Further, Lk := −zk+1∂/(∂z), for k 6 −1, forms the negative part of
the Witt algebra, cf. (1.1), and ∂/(∂z) is a formal derivation on C〈X〉
((
z−1
))
.
For f ∈ C〈X〉
((
z−1
))
and x ∈ C〈X〉, in the sequel, T (f, x), will be denoted by f.zx. The
following is clear by definition:
Proposition 3.7. T defines an action of the C-algebra C〈X〉 on C〈X〉
((
z−1
))
from the right.
The right action T can be extended to the right action
C〈X〉
((
w−1
))
× C〈X〉[[z]]→ C〈X〉
((
w−1
))
[[z]], (3.2)
under which the image of (f, znxip · · ·xi1) is mapped to zn(f.wxip · · ·xi1) =: f.w(znxip · · ·xi1).
Note that now the notation f.wS(x) makes sense.
16 T. Amaba and R. Friedrich
Theorem 3.8. Let {ft}t>0 be a solution to the Loewner–Kufarev equation. Then
ft(z) =
[∫
Res
w=0
(
ex0(t)z
1− zw
(
w−1.wS
(
ξ
(
x, ex0(t)
))))]
0,t
.
Proof. By setting
w̃(n)i1,...,ip :=
{
(n− i1) + 1
}{
(n− (i1 + i2)) + 1
}
· · ·
{(
n− (i1 + i2 + · · ·+ ip−1)
)
+ 1
}
,
where n = i1 + · · ·+ ip, we have
w−1.w1 = w−1,
w−1.wxip · · ·xi1 = w̃(n)i1,...,ipxip · · ·xi1w−(i1+···+ip+1).
Therefore Res
w=0
(∑∞
m=0 z
mwm
(
w−1.w1
))
= 1 (i.e., the empty word ∅), and
Res
w=0
( ∞∑
m=0
zmwm
(
w−1.wxip · · ·xi1
))
= z(i1+···+ip)w̃(n)i1,...,ipxip · · ·xi1 .
Hence we get
Res
w=0
(
ex0(t)z
1− zw
(
w−1.wS
(
ξ
(
x, ex0(t)
))))
= ex0(t)z +
∞∑
n=1
e(n+1)x0(t)zn+1
n∑
p=1
∑
i1,...,ip∈N :
i1+···+ip=n
w̃(n)i1,...,ipxip · · ·xi1 .
Now, in view of Theorem 2.10, we obtain the result. �
By tensoring the right action (3.2) this gives rise to(
C〈X〉
((
w−1
))
⊗ C〈X〉
((
u−1
)))
×
(
C〈X〉[[z]]⊗ C〈X〉[[z]]
)
→ C〈X〉
((
w−1
))
⊗ C〈X〉
((
u−1
))
,
under which the image of (f ⊗ g, x⊗ y) will be denoted by (f.wx)⊗ (g.uy) in the sequel.
We recall (see [32, Proposition 3.3 and pp. 50–51]) that the tau-function corresponding to
W ∈ Gr, is given by
τW (t) = det(w+) = det
(
1 + a−1bA
)
,
up to a multiplicative constant, where w+ : eξ(t,z)W → H+, is the orthogonal projection, and
eξ(t,z) : H → H, is the multiplication operator by eξ(t,z), with matrix representation
e−ξ(t,z) =
(
a b
0 d
)
along H = H+ ⊕H−,
and A : H+ → H− is such that graph(A) = W . Given a bounded univalent function f : D→ C,
with f(0) = 0, we denote by Af : H+ → H− the linear map such that graph(Af ) = Wf .
Theorem 3.9. Let {ft}06t6T be a univalent solution to the Loewner–Kufarev equation such
that ft(D) is bounded for every t ∈ [0, T ]. Then for each h ∈ H+ and |z| > 1, we have
(Afth)(z) =
[∫
Res
w=0,
u=0
(
h′(u)
w − z
∞∑
r,s=1
e(r+s)x0(t)xr+s
(
w−r.wS
(
ξ
(
x, ex0(t)
)))
�
(
u−s.uS
(
ξ
(
x, ex0(t)
))))]
0,t
.
Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian 17
The proof can be found in Appendix A.3. From this, we obtain
Corollary 3.10. For each n,m ∈ N, the coefficient b−n,−m(t), is equal to[∫
Res
z=0,
u=0
{
Res
w=0
zm−1un−1
w − z
∞∑
r,s=1
e(r+s)x0(t)xr+s
(
w−r.wS
(
ξ
(
x, ex0(t)
)))
�
(
u−s.uS
(
ξ(x, ex0(t)
)))}]
0,t
.
A Appendix
A.1 Proof of Theorem 2.10
By applying variation of constants to (2.5), we obtain the following recurence relation
cn(t) = enx0(t)
∫ t
0
e−nx0(s)dxn(s) +
n−1∑
k=1
(k + 1)enx0(t)
∫ t
0
e−nx0(s)ck(s)dxn−k(s),
for n > 2. Multiplying by e−nx0(t), this transforms to
e−nx0(t)cn(t) =
∫ t
0
e−nx0(t)dxn(s) +
n−1∑
k=1
(k + 1)
∫ t
0
e−(n−k)x0(t)dxn−k(s)
(
e−kx0(s)ck(s)
)
.
By assuming that x1, x2, . . . are non-commutative indeterminates, and the cn’s are polynomials
in the xi’s, we shall consider the following equation:
cn = xn + 2xn−1c1 + 3xn−2c2 + · · ·+ (n− 1)x2cn−2 + nx1cn−1, (A.1)
for n > 1 (roughly speaking, the polynomial cn means e−nx0(t)cn(t) and ‘applying the indeter-
minate xk from the left’ means ‘applying
∫ t
0 e−kx0(s)dxk(s)× to functions of s’) and then we
shall make some observations about the equation (A.1) and introduce some notations: If we
apply (A.1) to cn, we get
(a) The terms (n− k + 1)xkcn−k for each k = 1, 2, . . . , n. We shall denote these situation by
cn
w̃n,kxk→ cn−k,
respectively (note that the multiplication by the x∗’s must sit just left to the next c∗’s),
where w̃n,k := ((n− k) + 1).
(b) The term x0, to which we can not apply (A.1) anymore. This means, consider the situation
that we apply (A.1) iteratively to c∗’s which appeared at a previous stage. Suppose we
have the term cn at some stage. Then chasing the term multiplied by x∗ which arose from
the first term on the right-hand side in (A.1), lets us to get out of the loop of iterations;
we shall symbolise this situation by
cn
xn
⇒ end.
Let p ∈ N be such that 1 6 p 6 n. We fix i1, . . . , ip ∈ N, so that i1 + · · · + ip = n. This data
permits one to get out of the loop of iterations of (A.1) as the following diagram shows:
cñ
wn,i1xi1// cn−i1̃
wn−i1,i2xi2// cn−i1−i2
w̃n−i1−i2,i3xi3 // · · ·
w̃n−(i1+···+ip−2),ip−1
xip−1// cn−(i1+i2+···+ip−1)
= cip
xip //// end.
18 T. Amaba and R. Friedrich
Hence we have a single path from cn to the ‘end’ in the above diagram. This path produces at
the ‘end’ the term
w̃(n)i1,...,ipxipxip−1 · · ·xi2xi1 ,
where, by using the relation w̃n−k,l = w̃n,k+l, the coefficient w̃(n)i1,...,ip is given by
w̃(n)i1,...,ip = w̃n,i1w̃n−i1,i2w̃n−i1−i2,i3 · · · w̃n−(i1+i2+···+ip−2),ip−1
= w̃n,i1w̃n,i1+i2w̃n,i1+i2+i3 · · · w̃n,i1+i2+···+ip−2+ip−1
=
{
(n− i1) + 1
}{
(n− (i1 + i2)) + 1
}
· · ·
{
(n− (i1 + i2 + · · ·+ ip−1)) + 1
}
.
Collecting all possibilities, we have
cn =
n∑
p=1
∑
i1,...,ip∈N :
i1+···+ip=n
w̃(n)i1,...,ipxipxip−1 · · ·xi2xi1 ,
which yields the result by reinterpreting it in the language of paths xk(t)’s, as claimed.
A.2 Proof of Proposition 2.14
By applying variation of constants to (2.8), we have
b−m,−n(t) = −e(n+m)x0(t)
∫ t
0
e−(n+m)x0(t)dxn+m(s)
+ e(n+m)x0(t)
∫ t
0
{
(n− 1)b−m,−(n−1)(s)dx1(s) + · · ·+ b−m,−1(s)dxn−1(s)
}
+ e(n+m)x0(t)
∫ t
0
{
(m− 1)b−(m−1),−n(s)dx1(s) + · · ·+ b−1,−n(s)dxm−1(s)
}
.
By assuming that x1, x2, . . . are non-commutative indeterminates, and the b−m,−n’s polynomials
in the xi’s, we shall consider the following equation:
b−m,−n = −xn+m +
{
(n− 1)b−m,−(n−1)x1 + · · ·+ 2b−m,−2xn−2 + b−m,−1xn−1
}
+
{
(m− 1)b−(m−1),−nx1 + · · ·+ 2b−2,−nxm−2 + b−1,−nxm−1
}
, (A.2)
(roughly speaking, the polynomial b−m,−n means e−(m+n)x0(t)b−m,−n(t) and ‘applying the in-
determinate xk from the right’ means ‘applying
∫ t
0 e−kx0(s)dxk(s)× to functions of s’). If we
apply (A.2) to b−m,−n, we get:
(a) The terms (n−k)b−m,−(n−k)xk and (m−k)b−(m−k),−nxk for each k. We shall denote these
cases by
b−m,−n
(n− k)xk×→ b−m,−(n−k) and
b−m,−n
(m− k)xk× ↓
b−(m−k),−n
,
respectively (Note that the multiplication by the x∗’s must sit just right to the next b∗,∗’s).
(b) The term −xn+m, to which we can not apply (A.2) anymore. This means, consider the
situation that we apply (A.2) iteratively to the b∗,∗’s which appeared at a previous stage.
Suppose that we have the term b−m,−n at some stage. Then chasing the term, multiplied
Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian 19
by −x∗, which arose from the first term on the right-hand side in (A.2), permits us to get
out of the loop of iterations. We shall denote this situation by
b−m,−n
−xn+m×
⇒ end or
b−m,−n
−xn+m× �
end
.
Note that the multiplication by the x∗’s must be from the left. Hence in particular, to get
the term of the form xk(· · · ) in the polynomial expression of b−m,−n in the xi’s, we have
to escape the loop by passing to the cases
b−i,−j
−xk×
⇒ end or
b−i,−j
−xk× �
end
,
where i, j ∈ N with i+ j = k.
(c) If we have b−1,−1, applying (A.2) does not produce b∗,∗’s. Namely we must have
b−1,−1
−x2×
⇒ end or
b−1,−1
−x2× �
end
.
Again, the multiplication by x2 must be from the left. In particular, b−m,−n does not
contain the term x1(· · · ) and hence b−m,−n is a linear combination of xk(· · · )’s for k > 2,
though the factor (· · · ) may involve x1.
Let k ∈ N be such that 2 6 k 6 n + m. We shall find the term of the form xk(· · · ) in
the polynomial expression of b−m,−n in the xi’s. For this, we shall fix i ∈ {1, . . . ,m} and
j ∈ {1, . . . , n} such that i + j = k. Suppose that p, q ∈ N and i1, . . . , ip, j1, . . . , jq ∈ N satisfy
i1 + · · ·+ip = m−i and j1 + · · ·+jq = n−j. We then put ar := m−(i1 + · · ·+ir) for r = 1, . . . , p
and cs := n − (j1 + · · · + js) for s = 1, . . . , q. Note that ap = i and cq = j. According to this
notation, we distinguish the following three cases:
(1) If there exist such p, q, (i1, . . . , ip) and (j1, . . . , jq), then we can consider the following
diagram:
b−m,−n
a1xi1×
��
c1xj1×// b−m,−c1
a1xi1×
��
c2xj2×// · · ·
cqxjq×// b−m,−cq = b−m,−j
a1xi1×
��
b−a1,−n
a2xi2× ��
c1xj1×// b−a1,−c1
a2xi2× ��
c2xj2×// · · ·
cqxjq×// b−a1,−cq = b−a1,−j
a2xi2× ��
...
apxip×
��
...
apxip×
��
...
apxip×
��
b−i,−n
c1xj1× // b−i,−c1
c2xj2× // · · ·
cqxjq× // b−i,−cq = b−i,−j
(−1)xk×
''''
end
During the loop of iterations of (A.2), we have
(
p+q
p
)
=
(
p+q
q
)
-paths from b−m,−n to the ‘end’ in
the above diagram, each of which produces terms
−wi1,...,ip;j1,...,jqxk(· · · )’s,
20 T. Amaba and R. Friedrich
where
wi1,...,ip;j1,...,jq = a1a2 · · · apb1b2 · · · bq
= (m− i1)(m− (i1 + i2)) · · · (m− (i1 + i2 + · · ·+ ip))
× (n− j1)(n− (j1 + j2)) · · · (n− (j1 + j2 + · · ·+ jq)),
(note that wi1,...,ip;j1,...,jq depends only on i1, . . . , ip and j1, . . . , jq but not on the choice of paths
in the diagram) and (· · · ) is a monomial consisting of xip , xip−1 , . . . , xi1 and xjq , xjq−1 , . . . , xj1 ,
which is interlacing according to a riffle shuffle permutation (note that we should distinguish,
for example xi1xj1 and xj1xi1 even if i1 = j1). Hence, in total all paths produce
−wi1,...,ip;j1,...,jqxk
(
(xipxip−1 · · ·xi1)� (xjqxjq−1 · · ·xj1)
)
.
(2) If there exist such a p and (i1, . . . , ip) but not for q and (j1, . . . , jq) (then we have j = n),
then the diagram which we can have is the following:
b−m,−n
a1xi1×
��
b−a1,−n
a2xi2× ��
...
apxip×
��
b−i,−n = b−i,−j
��
(−1)xk×
��
end
Hence we have a single path from b−m,−n to the ‘end’ in the above diagram. This path produces
the term
−wi1,...,ipxk(xip · · ·xi2xi1),
where wi1,...,ip;j1,...,jq = a1a2 · · · ap = (m− i1)(m− (i1 + i2)) · · · (m− (i1 + i2 + · · ·+ ip)).
(3) If there exist such a q and (j1, . . . , jq) but not for p and (i1, . . . , ip) (then we have i = m),
then the diagram which we can have is the following:
b−m,−n
c1xj1×// b−m,−c1
c2xj2×// · · ·
cqxjq×// b−m,−cq = b−i,−j
(−1)xk×//// end.
Hence we have a single path from b−m,−n to the ‘end’ in the above diagram. This path produces
the term
−wj1,...,jqxk(xjq · · ·xj2xj1),
where
wj1,...,jq = c1c2 · · · cq = (n− j1)(n− (j1 + j2)) · · · (n− (j1 + j2 + · · ·+ jq)).
Now by reinterpreting it in the language of paths xk(t)’s, we obtain the result.
Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian 21
A.3 Proof of Theorem 3.9
Since {un}n>1 forms a basis of H+, it is enough to show that
n
∫ Res
w=0,
u=0
un−1
w − z
∞∑
r,s=1
e(r+s)x0(t)xr+s
(
w−r.wS
(
ξ
(
x, ex0(t)
)))
�
(
u−s.uS
(
ξ
(
x, ex0(t)
)))
t
= n
∞∑
m=1
b−n,−m(t)z−m, (A.3)
where b−n,−m(t) are the Grunsky coefficients associated with ft.
According to the decomposition
S(ξ(x, ex0(t))) = 1 +
∞∑
m′=1
em
′x0(t)
m′∑
p=1
∑
i1,...,ip∈N:
i1+···+ip=m′
xi1xi2 · · ·xip ,
we have(
w−r.wS
(
ξ
(
x, ex0(t)
)))
�
(
u−s.uS
(
ξ
(
x, ex0(t)
)))
=
(
w−r.w1
)
�
(
u−s.u1
)
+ Fr,s(w, u) +Gr,s(w, u) +Hr,s(w, u),
where
Fr,s(w, u) :=
[
w−r.w
(
S
(
ξ
(
x, ex0(t)
))
− 1
)]
�
[
u−s.u
(
S
(
ξ
(
x, ex0(t)
))
− 1
)]
=
∞∑
m′=1
∞∑
n′=1
m′∑
p=1
n′∑
q=1
∑
i1,...,ip∈N :
i1+···+ip=m′
∑
j1,...,jq∈N :
j1+···+jq=n′
e(m′+n′)x0(t)
×
(
w−r.wxi1xi2 · · ·xip
)
�
(
u−s.uxj1xj2 · · ·xjq
)
,
Gr,s(w, u) :=
(
w−r.w1
)
�
[
u−s.u
(
S
(
ξ
(
x, ex0(t)
))
− 1
)]
=
∞∑
n′=1
en
′x0(t)
n′∑
q=1
∑
j1,...,jq∈N :
j1+···+jq=n′
(
w−r.w1
)
�
(
u−s.uxj1xj2 · · ·xjq
)
,
Hr,s(w, u) :=
[
w−r.w
(
S
(
ξ
(
x, ex0(t)
))
− 1
)]
�
(
u−s.u1
)
=
∞∑
m′=1
em
′x0(t)
m′∑
p=1
∑
i1,...,ip∈N :
i1+···+ip=m′
(
w−r.wxi1xi2 · · ·xip
)
�
(
u−s.u1
)
.
Since w−r.w1 = w−r, we get
(
w−r.w1
)
�
(
u−s.u1
)
= w−ru−s. Then, by using
1
w − z
= −
∞∑
m=1
z−mwm−1 for |z| > |w|,
we have
Res
w=0;
u=0
un−1
w − z
∞∑
r,s=1
e(r+s)x0(t)xr+s
((
w−r.w1
)
�
(
u−s.u1
))
22 T. Amaba and R. Friedrich
= −Res
w=0;
u=0
un−1
∞∑
m=1
z−mwm−1
∞∑
r,s=1
e(r+s)x0(t)xr+sw
−ru−s
= −
∞∑
m=1
z−me(m+n)x0(t)xm+n.
Let
w(r)i1,...,ip;∅ := r(ip + r)(ip + ip−1 + r) · · · (ip + ip−1 + · · ·+ i2 + r)
=
(
m− (i1 + · · ·+ ip)
)
· · ·
(
m− (i1 + i2)
)
(m− i1),
where m = i1 + · · ·+ ip + r,
w(s)∅;j1,...,jq := s(jq + s)(jq + jq−1 + s) · · · (jq + jq−1 + · · ·+ j2 + s)
=
(
n− (j1 + · · ·+ jq)
)
· · ·
(
n− (j1 + j2)
)
(n− j1),
where n = j1 + · · ·+ jq + s, and
w(r, s)i1,...,ip;j1,...,jq := w(r)i1,...,ip;∅w(s)∅;j1,...,jq .
For Fr+s(w, u), we first observe that
w−r.wxip · · ·xi2xi1 = xip · · ·xi2xi1L−i1L−i2 · · ·L−ipw−r
= w(r)i1,...,ip;∅xip · · ·xi2xi1w−(i1+i2+···+ip+r),
and similarly
u−s.uxjq · · ·xj2xj1 = w(s)∅;j1,...,jqxjq · · ·xj2xj1u−(j1+j2+···+jp+s).
Therefore we have
Res
w=0;
u=0
(
un−1
w − z
xr+s
(
(w−r.wxi1xi2 · · ·xip)� (u−s.uxi1xi2 · · ·xip)
))
= −1{16n−s=j1+···+jq}
∞∑
m=1
z−m1{16m−r=i1+···+ip}
× w(r, s)i1,...,ip;j1,...,jqxr+s[(xip · · ·xi2xi1)� (xjq · · ·xj2xj1)],
so that
Res
w=0;
u=0
(
un−1
w − z
xr+sFr,s(w, u)
)
= −
∞∑
m=1
z−m
∞∑
m′=1
∞∑
n′=1
e(m′+n′)x0(t)
m′∑
p=1
n′∑
q=1
∑
i1,...,ip∈N:
i1+···+ip=m′
1{16m−r=i1+···+ip} · · ·
· · ·
∑
j1,...,jq∈N :
j1+···+jq=n′
1{16n−s=j1+···+jq}w(r, s)i1,...,ip;j1,...,jqxr+s[(xip · · ·xi2xi1)� (xjq · · ·xj2xj1)]
= −1{16n−s}
∞∑
m=1
z−me((m−r)+(n−s))x0(t)1{16m−r}
m−r∑
p=1
n−s∑
q=1
∑
i1,...,ip∈N:
i1+···+ip=m−r
· · ·
Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian 23
· · ·
∑
j1,...,jq∈N:
j1+···+jq=n−s
w(r, s)i1,...,ip;j1,...,jqxr+s[(xip · · ·xi2xi1)� (xjq · · ·xj2xj1)].
Hence we have reached
Res
w=0;
u=0
un−1
w − z
∞∑
r,s=1
e(r+s)x0(t)xr+sFr,s(w, u)
= −
∞∑
m=1
z−me(m+n)x0(t)
m+n−2∑
k=2
∑
16r<m;
16s<n :
r+s=k
m−r∑
p=1
n−s∑
q=1
∑
i1,...,ip∈N :
i1+···+ip=m−r
· · ·
· · ·
∑
j1,...,jq∈N:
j1+···+jq=n−s
w(r, s)i1,...,ip;j1,...,jqxk[(xip · · ·xi2xi1)� (xjq · · ·xj2xj1)].
Similarly, we find that
Res
w=0;
u=0
un−1
w − z
∞∑
r,s=1
e(r+s)x0(t)xr+sGr,s(w, u)
= −
∞∑
m=1
z−m
∞∑
r,s=1
e(r+s)x0(t)
∞∑
n′=1
en
′x0(t)
n′∑
q=1
· · ·
· · ·
∑
j1,...,jq∈N :
j1+···+jq=n′
1{16n−s=j1+···+jq}1{m=r}w(s)∅;j1,...,jqxr+s(xjq · · ·xj1)
= −
∞∑
m=1
z−me(m+n)x0(t)
m+n−1∑
k=m+1
n+m−k∑
q=1
∑
j1,...,jq∈N:
j1+···+jq=n+m−k
w(k −m)∅;j1,...,jqxk(xjq · · ·xj1),
and
Res
w=0;
u=0
un−1
w − z
∞∑
r,s=1
e(r+s)x0(t)xr+sHr,s(w, u)
= −
∞∑
m=1
z−me(m+n)x0(t)
m+n−1∑
k=n+1
n+m−k∑
p=1
∑
i1,...,ip∈N:
i1+···+ip=m+n−k
w(k − n)i1,...,ip;∅xk(xip · · ·xi1).
Now, in view of Theorem 2.13, we obtain (A.3), and hence the result.
Acknowledgements
T.A. was supported by JSPS KAKENHI Grant Number 15K17562. R.F. was previously sup-
ported by the ERC advanced grant “Noncommutative distributions in free probability”. Both
authors thank Theo Sturm for the hospitality he offered to T.A. at the University of Bonn. T.A.
thanks Roland Speicher for the hospitality offered him in Saarbrücken. R.F. thanks Roland
Speicher for discussions, and Fukuoka University and the MPI in Bonn for their hospitality. We
both thank Takuya Murayama for the discussions, comments and collaboration. We thank the
anonymous referees for their comments which helped us to improve the paper.
24 T. Amaba and R. Friedrich
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1 Introduction
2 The controlled Loewner–Kufarev equation
2.1 Definition of solutions to controlled Loewner–Kufarev equations
2.2 Loewner–Kufarev equation as a controlled Loewner–Kufarev equation
2.3 Taylor coefficients along the controlled Loewner–Kufarev equation
2.4 Variation of Grunsky coefficients induced by a Loewner–Kufarev equation
3 The controlled Loewner–Kufarev equation embedded into the Segal–Wilson Grassmannian
3.1 Segal–Wilson Grassmannian
3.2 Krichever's construction
3.3 The appearance of Faber polynomials and Grunsky coefficients
3.4 Action of words in Witt algebra generators
A Appendix
A.1 Proof of Theorem 2.10
A.2 Proof of Proposition 2.14
A.3 Proof of Theorem 3.9
References
|
| id | nasplib_isofts_kiev_ua-123456789-211012 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T10:41:53Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Amaba, Takafumi Friedrich, Roland 2025-12-22T09:28:15Z 2020 Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian. Takafumi Amaba and Roland Friedrich. SIGMA 16 (2020), 108, 25 pages 1815-0659 2020 Mathematics Subject Classification: 35Q99; 30F10; 35C10; 58J65 arXiv:1809.00534 https://nasplib.isofts.kiev.ua/handle/123456789/211012 https://doi.org/10.3842/SIGMA.2020.108 We introduce the class of controlled Loewner-Kufarev equations and consider aspects of their algebraic nature. We lift the solution of such a controlled equation to the (Sato)-Segal-Wilson Grassmannian, and discuss its relation with the tau-function. We briefly highlight relations of the Grunsky matrix with integrable systems and conformal field theory. Our main result is the explicit formula that expresses the solution of the controlled equation in terms of the signature of the driving function through the action of words in generators of the Witt algebra. T.A. was supported by JSPS KAKENHI Grant Number 15K17562. R.F. was previously supported by the ERC advanced grant "Noncommutative distributions in free probability". Both authors thank Theo Sturm for the hospitality he offered to T.A. at the University of Bonn. T.A. thanks Roland Speicher for the hospitality offered himin Saarbrucken. R.F. thanks Roland Speicher for discussions, and Fukuoka University and the MPI in Bonn for their hospitality. We both thank Takuya Murayama for the discussions, comments, and collaboration. We thank the anonymous referees for their comments, which helped us to improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian Article published earlier |
| spellingShingle | Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian Amaba, Takafumi Friedrich, Roland |
| title | Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian |
| title_full | Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian |
| title_fullStr | Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian |
| title_full_unstemmed | Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian |
| title_short | Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian |
| title_sort | controlled loewner-kufarev equation embedded into the universal grassmannian |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211012 |
| work_keys_str_mv | AT amabatakafumi controlledloewnerkufarevequationembeddedintotheuniversalgrassmannian AT friedrichroland controlledloewnerkufarevequationembeddedintotheuniversalgrassmannian |