Explicit variational formulas for third-order equations on Riemann surfaces Текст научной статьи по специальности «Математика»

УДК 515.17+517.545

Explicit Variational Formulas for Third-order Equations on Riemann Surfaces

Marina I. Tulina*

Gorno-Altaisk State University Lenkin, 1, Gorno-Altaisk, Altai Republic, 649000

Russia

Received 21.01.2013, received in revised form 01.02.2013, accepted 03.04.2013 In this paper we deduce explicit variational formulas for the solution of an ordinary differential equation of third order and its monodromy group with respect to a variation in the space of cubic holomorphic differentials on a compact Riemann surface.

Keywords: holomorphic cubic differentials, compact Riemann surface, variational formulas, differential equation of third order, monodromy group.

Introduction

In this paper we deduce explicit variational formulas for the solution of an ordinary differential equation of third order and its monodromy group with respect to a variation in the space of cubic holomorphic differentials on a compact Riemann surface of genus g > 2. Since there are no explicit solutions even for second-order equations, there appeared Hejhal's approach: to understand how the generators of the monodromy group change with a small variation of the coefficients in the spaces of holomorphic differentials. Also we will find out how these objects are related to the matrix Prym differentials and to holomorphic sections of some vector bundles on compact Riemann surfaces. Variational formulas find applications in geometric theory of functions of a complex variable and in the theory of Teichmiiller spaces in connection with the uniformization of compact Riemann surfaces.

In the paper, using vector and matrix notation, we present a method for deducing explicit variational formulas for the solution vector and the monodromy group with respect to a variation in the space of cubic holomorphic differentials. These theorems extend the results of D.Hejhal [1, 2] and V.V. Chueshev [3] to the case of third-order equations.

Preliminaries

Let F be a compact Riemann surface of genus g > 2 and D an open disk on the complex plane C. Use r to denote the Fuchsian group of the first type uniformizing F on D, so F is conformally equivalent to D/r. Consider an ordinary linear differential equation

dnv , . dn-2v , . dn-3v

on + ®(t) + qs(i) + ■ ■ ■ + in(t)v = 0 ,t G d, (1)

where qj(t) are meromorphic functions on D, j = 2, ...,n. We shall say that equation (1) is of Fuchsian type on F if it has only regular singular points and is preserved under the change of variable:

n— 1

(t,v) ^ (s,v), s = L(t), LV = w(s)L'(t) — ,L G r. (2)

*[email protected] © Siberian Federal University. All rights reserved

The solution vector is the column vector consisting of all linearly independent solutions of the equation. A holomorphic differential of order q has the form $(z)dzq and is invariant under the change of coordinates on the surface, i. e. $(Lz)L'(z)q = $(z), z e D, L e r. We denote by (F) the vector space of holomorphic q—differentials on D/r, q e N.

Lemma 1 ([1]). An ordinary linear differential equation —u(j-) + q2(-) —+ q3(-)u(-) =0 is

at3 d-

defined on D/r iff q2(L-)L'(-)2 = q2(t), q3(L-)L'(-)3 = qs(-), q2(-) e 03(r), - e D, L e r.

Lemma 2 ([1]). Let the column vector U(-) consist of n linearly independent solutions to equation (1) on F = D/r. Then the equality

U(L-) = x(L)U(-)a(-r-1, &(-) = V^W, L e r, (3)

defines uniquely the homomorphism x : r ^ GL(n, C) given by L ^ x(L), L e r.

The homomorphism x in equality (3) is called the monodromy homomorphism for the solution vector U(-). The image x(r) of the group r is the monodromy group for differential equation (1). It is well known that the monodromy homomorphism x is a non-unitary irreducible representation of r in GL(n, C). Moreover, the monodromy group x(r) for U(-) on the surface D/r of genus g ^ 2 is infinite [1].

1. Taylor expansions for the solution vector and elements of the monodromy group

Consider a perturbed equation

u(3)(z) + Qo(z)u(1)(z) + (Ro(z) — Mr(z))w(z) = 0, (4)

where Q0(z)dz2, R0(z)dz3, and r(z)dz3 are fixed holomorphic quadratic and cubic abelian differentials on D/r, respectively, and ^ e C, < e for a sufficiently small e.

Denote by U(z,^) = 4(w(z,^), v(z,^), w(z,^)) the vector of linearly independent solutions of the Cauchy problem with the condition

U(z0, = t(1, 0, 0), U'(z0, = 4(0,1, 0), U''(z0, = 4(0,0,1) (5)

for every where z0 e D.

Introduce shorter matrix notation:

/ u(z) \ / u(z) 0 0 \ U (z) = u(z) = 0 u(z) 0 , V "(z) / V 0 0 w(z) /

(«m(zA / Um(z) 0 0 \

Um(z) = 0 Um(z) 0

^m(z) / \ 0 0 Um(z) /

for every m ^ 1. By Poincare's small parameter method and the Cauchy-Kowalevski theorem we

have a Taylor expansion in ^ for the solution vector: U(z, = U(z) + ^ U1(z) +-----+ ^mUm(z) +

• • • . By substituting this series in (4), we obtain the identity with respect to ^

0 = U (3)(z) + MUi(3)(z) + M2U23) (z) + • • • + MmUm3)(z) + • • • + +Q0(z)U (1)(z) + MQ0(z)Ui(1)(z) + M2Q0(z)U2(1)(z) + • • • + MmQ0(z)Umi)(z) + • • • +R0(z)U (z) + MR0(z)U1 (z) + M2R0 (z)U2(z) + • • • + Mm«0(z)Um(z) + • • •

-pr(z)U(z) - p2r(z)Ut(z) - p3r(z)U2(z)-----pm+1r(z)Um(z) + ...

Thus, we have an infinite system of vector equations

U (3)(z) + Qo(z)U W(z) + Ro (z)U (z) = 0, U(3)(z) + Qo(z)U[1)(z) + Ro(z)Ui(z) = r(z)U (z), Ug\z) + Qo(z)U№ (z) + Ro(z)Um(z) = r(z)Um—\ (z),

where Um(z0) = U(n)(zo)= 4(0,0,0), m > 2, n = 1, 2.

Denote by V(z) the solution to the Lagrange dual equation on D/r and by E the unit matrix of order three. We have the following relation (see [1]) V(Lz) = £L(z)2V(z )x(L)-\ L G r. Solving the second equation of the system by Lagrange's method of variation of constants, we

get Ui(z)

r(x)U(x)V(x)dx U(z). Denote by B(x) = r(x)U(x)V(x), B0(z)

B(x)dx.

For m = 1 we have the equality U1(z) = B0(z)U(z).

For m > 1 we introduce the notation Um(z) = Bm-l(z)U(z), where Bm-l(z) =

/ r(x)Um-i(x)V(x)dx. For m = 2 we have U2(z) = B\(z)U(z). On the other hand, U2(z) = e z°z z z

/ r(x)U\(x)V(x)dxU(z) = B0(x)B(x)dxU(z). Consequently, Bi(z) = / B0(x)B(x)dx.

J zo J zo J zo

Thus, we have proved the statement for the cases m = 1, m = 2. By the induction assumption, for m we have the equality Um(z) = Bm-l(z)U(z), where Bm-l(z) = / Bm-2(x)B(x)dx. We

zo

shall prove this statement for m + 1. In our notation we have Um+i(z) = Bm(z)U(z). On the

other hand, Um+i(z)

r(x)Um(x)V(x)dx U(z)

Bm-i(x)r(x)U(x)V(x)dx U(z)

/ Bm-i(x)B(x)dxU(z). Consequently, the equality Bm(z) = / Bm-l(x)B(x)dx holds for

J zo J zo

m + 1. Thus, we have proved the following theorem.

Theorem 1.1. For the solution vector of linear differential equation (4) with condition (5) on a compact Riemann surface F = D/Г of genus g ^ 2 the following explicit variational

formulas hold: U(z, p) = E + pBo(z) + p?B\(z) + • • • + pmBm-\(z) + • • • U(z), where B(x) = r(x)U(x)V(x), Bo(z) = / B(x)dx, Bm(z) = / Bm-i(x)B(x)dx for every m ^ 1 and |p| < e.

zo zo

In Theorem 1.1 we obtain variations of every order, i.e., the whole variational Taylor series in parameter p.

Now, using variational formulas for the solution U(z,p), we shall find variational formulas for the monodromy group of equation (4). Write down the basic relation

X(L; p)U (z, p)ib(z)2 = U (Lz, p) = Uo (Lz) + pUx(Lz) + p2U2(Lz) + ... + pmUm(Lz) + ••• .

Express Um(Lz) in terms of U(z) and the coefficients of the equation. We have

Ui(Lz)

r(x)U(x)V(x)dx U(Lz)

çLzo pbz

/ r(x)U(x)V(x)dx + / r(x)U(x)V(x)dx U(Lz)

' zo J Lzo

z

z

z

z

L

z

= Bo(Lzo)x(L)U (z)& (z)2 + x(L)Bo(z)U (z)£l(z)2 = = Bo(Lzo)x(L)U (z)cl(z)2 + X(L)Ui(z)eL(z)2.

Indeed,

' f r(x)U(x)V(x)dx)x(L)U(z)^L(z)2 =< x = L(t) >=

'./Lz™ '

r(Lt)U (Lt)V (Lt)d(Lt)J x(L)U (z)& (z)2 =

Ï0

r(t)L' (t)-3x(L)U (î)£l(î)2£l(î)2V (t)x(L-1 )L' (t)dt)x(L)U (z)£l(z)2 =

' zo

r(t)x(L)U (t)V (t)dt U (z)Cl(z)2 = x(L)Ui(z)Cl(z)2.

Next, U2(Lz) =(/zLz r(x)U1(x)V(x)dxV(Lz)

z0

fLzo f Lz

/ r(x)U1(x)V (x)dx + / r(x)U1(x)V (x)dx)U (Lz) =

' zo J Lzo

= B1(Lz0)x(L)U(z)£L(z)2+( /" r(L-)U1(L-)V(L-)dL(-))U(Lz) =

-'zo

= B1(Lz0)x(L)U(z)£L(z)2+[ /^ r(-)L'(-)-3 (B0(Lz0)x(L)U(-)&(-)2 +

zo

+x(L)U1(-)eL(-)^£l(-)2V(-)x(L-1)L '(-)d-] U(Lz) = = B1(Lz0)x(L)U(z)&(z)2+( /Zr(-)B0(Lz0)x(L)U(-)V(-)x(L-1)d-) U(Lz)+

zo

+ ( /Zr(-)x(L)U1(-)V(-)x(L-1)d^U(Lz) = B1(Lz0)x(L)U(z)£L(z)2 +

zo

+B0(Lz0)x(L)[ / r(-)U(-)V(-)d-]a(z)2U(z)+

zo

+x(L)( /" Zr(-)U1(-)V (-)d^CL (z)2 U (z) = B1(Lz0)x(L)U (z)£L (z)2 +

zo

+B0(Lz0)x(L)B0(z)eL(z)2U (z) + x(L)B1(z)£L(z)2U (z) = = B1 (Lz0)x(L)U (z)£L(z)2 + B0(Lz0)x(L)U1 (z)£L(z)2 + x(L)U2(z)£L(z)2. Assume that the statement holds for m, i.e.,

m— 1

Um (Lz) =53 Bm—1—j (Lzo)x(L)Uj (z)£l(z)2 + x(L)Um(z)^L(z)2.

j=o

z

z

We will show that it holds for m + 1. We have

Um+l(Lz) = fzo r(x)Um(x)V(x)dx U(Lz) =

/■Lz0 r-Lz .

f r(x)Um(x)V(x)dx + r(x)Um(x)V(x)dx U(Lz) =

zo ZLzo

= Bm(Lzo)x(L)U(z)£l(z)2+\ / r(Lt)Um(Lt)V(Lt)dL(t)] U(Lz) =

zo m-1

= Bm(Lzo)x(L)U(z)iL(z)2 +J2 Bm-1-j(Lzo)x(L)Uj+i(z)^L(z)2+

j=o

+x(L)Um+i(z)^L(z)2 = J2 Bm-j(Lzo)x(L)Uj(z)Çl(z)2 + x(L)U„î+i(z)Çl(z)2.

j=o

Since

rz . rz m_

r(Lt)Um(Lt)V(Lt)d(L(t)) U (Lz)= r(t)L'(t)-3^ Bm-i-j (Lzo)+x(L)Uj (t)^L(t)2+

■'zo Jz o j=o

+x(L)Um(t)ÏL(t)2)ÎL(t)2V(t)x(L-1 )L'(t)] dtx(L)U(z)^l(z)2 =

m-1 r-z r-z n

]T / r(t)Bm-l-j(Lzo)x(L)Uj(t) + r(t)x(L)Um(t) V(t)dtU(z)&(z)2 = j=0 zo J zo

m-1 . z _

Y,Bm-!-j(Lz0)x(L) r(t)Uj(t)V(t)dt U(z)£l(z)2+ j=0 Jzo

+x(L) \ (Z r(t)Um(t)V(t)dt] U(z)Cl(z)2 =

zo

m- 1

= J2 Bm-1-j(Lz0)x(L)Uj+i(z)iL(z)2 + x(L)Um+i(z)$L(z)2.

j=0

Using the expression for Um(Lz) proved by induction, we obtain an explicit variational formula for the elements of the monodromy group. We have

x(L; p)U(z, p)iL(z)2 = U(Lz, p) = U(Lz) + pUi(Lz)+ +p2U2(Lz) + p3U3(Lz) + ■■■ + pmUm(Lz) + ■ ■■ = = Ex(L)U (z)^l(z)2 + pB0(Lz0 )x(L)U (z)&(z)2 + px(L)Ui(z)^(z)2+ +p2Bi(Lz0 )x(L)U (z)^l(z)2 + p2B0(Lz0)x(L)Ui(z)^L(z)2 + P2x(L)U2(z)£l(z)2 + +p3B2(Lz0)x(L)U (z)Cl(z)2 + p3Bi(Lz0)x(L)Ui(z)^L (z)2 + +p3B0(Lz0)x(L)U2 (z)^L(z)2 + p3x(L)U3(z)^L(z)2 + ■■■ +

m- 1

+pmY, Bm-1-j(Lz0)x(L)Uj(z)^L(z)2 + pmx(L)Um(z)iL(z)2 + ■■■ = j=0

= Ex(L) \lJ (z) + pUi(z) + p2U2(z) + p3U3(z) + ■■■ + +pmUm(z) + ■ ■ ■ I &(z)2 + pB0(Lz0)x(L) ¡U(z) + pUi(z) + p2U2(z)+

+p3U3(z) + •••+ pm-1Um-i(z) + •• • Ïl(z)2 + p2Bi(Lzo)x(L)(U (z)+

m

+MUi(z) + M2U2(z) + M3U3(z) + • • • + Mm-2Um-2(z) + • • • )£l(Z)2 + • • • + +MmBm-i(Lzo)X(L)(U (z) + (z) + • • • + (z) + • • • )£L(z)2 + • • • Thus we have proved

Theorem 1.2. For the elements of the monodromy group of equation (4) on a compact Riemann surface F of genus g ^ 2 there hold the following explicit variational formulas:

X(L;m) = E + ^Bo(Lzo) + m2Bi(Lzo) + ••• + MmBm-i(Lzo) + ••• x(L), L G Г, |m| < e.

2. Elements of the monodromy group for the variation with respect to the base of cubic differentials

Consider the perturbed differential equation

d

u(3)(z) + Qo(z)u(1)(z) + (Ro(z) Mjrj)u(z) = 0 (6)

j=i

on the surface F = D/Г, where ri,...,rd is the basis for cubic holomorphic differentials in the space Q3(F), d = 5g — 5. Put ^ = ). As before, we denote by U(z,^) =

t(u(z, v(z, w(z, the vector of three linearly independent solutions to the Cauchy problem with the conditions at the point z0

u(zq,m)= ¿(1, 0,0); U(1) (zq,m)= 4(0,1, 0); u(2)(zq,m)= ¿(0,0,1), (7)

for every

Rewrite scalar equation (6) in the vector form

d

U(3)(z,M) + Qo(z)U(1)(z,M) + (Ro(z) - Mjrj)U(z,M) = 0. (8)

j=i

By Poincare's small parameter method and the Cauchy-Kowalevski theorem we have a Taylor series for the solution of equation (8)

œ œ

U(z,M) = U(z) +53 U|fc|.(fcli...,fcd)(z)Mk1 ■ ■ ■ № = U(z) ^53 U|fchfc(z)Mk (9)

|k| = 1 |k| = 1

for sufficiently small m G Cd, ||m|| < e, where ||m|| = max |Mj|, k = (k1, ...,kd) is a vector with

integer nonnegative coordinates, |k| = k1 + ■ ■ ■ + kd, and U(z) = U|0|;(o,...,o)(z). By substituting series (9) in equation (8), we obtain the vector equality

^ ^ d ^

53 Ug)fc(z)Mk + Qo(z) 53 U|(fc1)fc(z)Mk + (Ro(z) - 53Mjrj(z)) £ U^k(z)Mk = 0.

|k|=0 |k| = 0 j=1 | k | =0

Note that for every |k| ^ 1 the following conditions hold

U|k|;k (z0) = U(k1)k (z0) = U|(k2|)k (z0) = 0. (10)

Hence, we have an infinite system of vector linear differential equations of the form:

U (3)(z) + Q0(z)U(1) (z) + R0(z)U (z) = 0;

Ug(z) + Q0(z)U11fc) (z) + R0(z)U1;fc (z) = rj (z)U (z) for k = ej (the unit is on jth place, while others equal zero), j = 1,..., d;

d

U|(k3)k(z)+ Q0(z)U| (fc1|) fc + R0(z)U|fc|;fc(z)= rj(z)U|k|-1. fc_e.(z)

j = 1, kj =0

for k = (k1, ...,kd), |k| > 2.

Solving the second equation by Lagrange's method of variation of constants, we get U1 ;k(z) =

j rj-(-)U(-)V(-)d-U(z) for k = ej-,j = 1,...,d. Let B^(z) = r^-(z)U(z)V(z), j = 1,...,d, B0;ej =

J zo

/ Bj (-)d-. Then, U1;ej (z) = B0;ej (z)U(z) for every k = e^-, j = 1, ...,d.

zo

For k = (k1,..., kd), |k| = 2, we have

d z

U2;k (z)= / B0;k-ej (-)Bj (-)d-U (z) = B1;k (z)U (z).

j=1, kj =0^zo

For k = (k1,..., kd), |k| > 3, we obtain the equality

U| k | ;k (z)= f B| k | — 2; k — e j (-)Bj (-)d-U (z) = B| k |— 1;k(z)U (z).

(j: kj =0) 7zo

Thus, we have proved the following theorem.

Theorem 2.1. For the solution vector of equation (6) with normalization (7) and with a perturbation with respect to the base of rj, j = 1,...,d = 5g — 5, one has the following explicit variational formulas:

U (z,M)= [e +53 B0;k (z)Mk +53 B1;k (z)Mk + ••• +53 Bn—1;k (z)Mk + •••] U (z),

|k| = 1 |k| = 2 |k|=n

where B0;k(z) = Bj(-)d- for k = ej, Bj(-) = rj(-)U(-)V(-), j = 1,...,d, B|k|;(k1,...,kd)(z) =

J z zo

i 53 B|k| —1;k—ej(-)Bj(-)d- for |k| > 2, IHI < e. ,/zo (j: kj =0)

To deduce variational formulas for the monodromy group, we need some relations associated with the group r. First, for k = (k1, ...,kd) = ej, |k| = 1, j = 1, ...,d, we find that

Lz Lz0 Lz

U1;k(Lz) = Bj(t)dtU(Lz) = cl(z)m Bj(t) x(L)dtU(z) + £l(z)2 / Bj(t)x(L)dtU(z)

z0 z0 Lz0

= £L(z)2Bo;fc(Lzo)x(L)U(z) + £l(z)2x(l) f Bj(t)dtU(z) =

z0

= £L(z)2Bo;fc(Lzo) x(L)U(z) + eL(z)2x(L)U1;fc(z),

since

Lz /*z /*z

/ Bj-(x) x(L)dx =< x = L(t) >= / Bj-(Lt) x(L)d(Lt) = r,(Lt)U(Lt)V(Lt) x(L)L'(t)dt =

-'Lzo z -'zo •'zo

= / rj(t)L'(t)-3L'(t) x(L)U(t)L'(t)V(t)x(L—1 ) x(L)L'(t)dt =

zo

x(L) f rj(t)U(t)V(t)dt = x(L)Bo;ej (z).

zo

For k = (ki,..., kd), |k| = 2, we find that

U2;k(Lz) = ÏL(z)2Bi;k(Lz) x(L)U(z) = ^l(z)2BUk(Lzo) x(L)U(z)+

/Lz _

]T Bo-k-j (x)Bj (x)dx x(L)U (z). (j: kj =o)

Next

Lz

/Lz ______pz

Y Bo;k-ej (x)Bj (x)dxx(L) =<x = L(t) >= Y Bo-k-e, (Lt)Bj (Lt)d(Lt)x(L) -z° (j: k, =o) (j: k, =o)

= f Bo ; k-ej (Lt)rj (Lt)U (Lt)V (Lt)L' (t)dtx(L) =

z Jz° (j: kj =o)

= i Y Bo;k-ej (Lt)rj(t)L'(t)-3L'(t) x(L)U(t)L'(t)V(t)L'(t)dt =

Jz° (j: kj =o) z

= f E Bo;k-ej (Lt)rj (t) x(L)U (t)V (t)dt = z Jz° (j: kj =o) z

I J2 Bo-k-e, (Lt)x(L)Bj(t)dt = I J2 Bo-k-e, (Lzo) x(L)Bj (t)dt+ z° (j: kj =o) Jz° (j: kj =o)

/z ____f Lt

Y Y, Bi(s) x(L)dsBj(t)dt =

° (j: kj =o)(i:(k-e, )i = o)J Lz° z

Y Bo,k-e, (Lzo)x(L)Bo;ej (z) + x (L) ( Y Bo-k-e, (t)Bj (t)dt.

, z° ,, ,

(j: kj =o) ■Jz° (j: kj =o)

Thus,

U2-k (Lz) = iL (z)2 Bi-k(Lzo) x(L)U (z) + Uz)2 Y Bo-k-e, (Lzo)x(L)B0.Cj (z)U (z)+

z (j: kj =o)

+iL(z)2x(L) Y Î Bo-k-e, (t)Bj(t)dtU(z)= iL(z)2Bi-k(Lzo) x(L)U(z)+

z°

+iL(z)2 Y Bo-k-ej (Lzo) x(L)Ui.e, (z) + iL(z)2 x(L)U2,k(z). (j: kj =o)

By induction, for every n > 1 we obtain the equality:

) Lz

Un+i-k(Lz) = Bn-k(Lz)U(Lz)= Y / rj(s)Un.k-e, (s)V(s)dsU(Lz)

z°

= iL(z)2Bn-k(Lzo)x(L)U(z)+ iL(z)2 Y r rj(x)Un-k-ej (x)V(x)dxx(L)U(z) =

(j: kj =o)JLz°

)z

= iL(z)2Bnk (Lzo) x(L)U (z) + iL(z)2 Y / rj (s)L ' (s)-1Un;k-j (Ls)V (s)dsU (z)

z°

iL(z)2Bn-k(Lzo)x(L)U (z) + iL (z)2 Y / rj (s) Bn-i-k-ej (Lzo) x(L)U (s)+

(j: kj =o)Jz°

z

+ E Bn — 2;k — ej — ej (Lz0)x(L)U^j (s) +

(ji:(k —ej )ji =0)

+ X) E Bn—3;k —ej — ej — ej2 (Lz0) x(L)U2;ej +e32 (s) +

(j1: (k —ej j =0) (j2 : (k —ej — ej )j2 =0)

+ E E •••

(j1: (k —ej ) j =0) (j2: (k —ej — ej )j2 =0)

E B0;k —ej — ej — ej2 —...— e^- (Lz0) x(L)Un—1j +ej2 +-- + e3„-1 (s

(jn-1: (k —ej —ej1 —— ej„-2 )j„-1 =0)

+x(L)U„;k—ej (s)] V(s)dsU(z) =

= CL(z)2B„;k(Lz0)x(L)U (z) + Cl (z)2[ 53 Bn—1;k—ej (Lz0)x(L)U1;ej (z)+

(j: kj =0)

+ E E Bn—2;k —ej —ej1 (Lz0)x(L)U2;ej +ej1 (z) +

(j: kj = 0) (j1:(k— ej j =0)

+ E E E Bn—3;k —ej — ej —ej2 (Lz0)x(L)U3;ej +ej +e,2 (z

(j: kj = 0) (j1: (k— ej )j1 =0) (j2: (k —ej — ej j =0)

••• + EE E •••

(j: kj = 0) (j1: (k — ej j =0) j (k—ej — ej j =0) 53 B0;k —ej —ej1 —ej2 —...— e^— (Lz0) x(L)Un;ej + ej1 +ej2 + ... + ejn — (z) +

(jn-1 : (k —ej —ej1 —...— ej„-2 )j„-1 =0)

+x(L)Un+1;k(z) . Taking into account the previous equalities we obtain the following

Cl(z)2x(L; M)U(z, M) = U(Lz, M) = U(Lz) +53 /U^(Lz)+

|k|=1

+ E ^k(Lz) + • • • ^53 /U„;k(Lz) + • • • = £l(z)2 [x(L)U(z)+

|k|=2 |k|=n

+ 53 MkB0;k(Lz0)x(L)U(z)+ 53 Mkx(L)U1;k(z) + 53 /B1;k(Lz0)x(L)U(z)+ |k|=1 |k|=1 |k|=2

+ E Mk E B0;k—ej (Lz0)x(L)U1;ej (z) + 53 /x(L)U2;k(z)+

|k|=2 (j: kj = 0) |k|=2

+ 53 B2;k(Lz0)x(L)U(z) + 53 / E B1;k—ej (Lz0) x(L)U1;ej (z)+

| k | =3 |k|=3 (j: kj =0)

+ E ^ E E B0;k —ej —ej1 (Lz0)x(L)U2;ej +ej1 (z) +

|k|=3 (j: kj =0) (j1: (k —ej )j1 =0)

+ 53 / x(L)U3;k (z) + ••• +53 / Bn—1;k(Lz0) x(L)U (z)+

|k|=3 |k|=n

+ E E Bn — 2; k — e j (Lzo) x(L)Ul;e3 (z) +

|k|=n (j: kj =0)

+ E E E Bn—3;k — ej —e31 (Lzo)x(L)U2;e3 +e31 (z) + ••• +

|k|=n (j: kj = 0)(ji:(k —ej j =0)

+ EMk E E

|k|=n (j: kj = 0) (ji: (k —ej j =0)

E B0;k —ej —eji —...— e^- (Lz°) x(L)Un—^ +eji +...+e3„_2 (z) +

(jn-2: (k —ej —eji —ejn-3 )j„_2 =0)

+ 53 Mk x(L)Un;k (z) + ••• = £l(z)2 |x(L)U (z,m)+ E <"k B0;k M x(L)U (z,m)+

|k|=l

+ MkBi;k(Lz0)x(L)U(z, m) + 53 MkB2;k(Lz0) x(L)U(z, m) + • • • +

|k|=2

|k|=3

+ 53 mkbn—i;k(lz0)x(l)u(z,m) +

Thus, we have proved the following theorem.

Theorem 2.2. For the elements of the monodromy group of equation (6) with normalization (7) on compact Riemann surface F = D/r of genus g ^ 2 the following explicit variational formulas hold

X(L; m) = [e ^53 B0;k(Lz0 )Mk ^53 Bl;k (Lz0)Mk + ^53 Bn — 1;k (Lz0)Mk + •••] x(L),

| k| =n

|k|=l

|k|=2

n

k

n

where L G r, for ||m|| < £•

Remark 2.1. Variational formulas show how the monodromy group and the solution to the equation of third order depend on the parameters (m1, •••, Md) for a variation with respect to the base of cubic holomorphic differentials on F. In particular, they give explicit variational formulas for the generators x(A1),..., x(Ag), x(B1), • • •, x(Bg) of the monodromy group.

Remark 2.2. It follows from the relation dB0(Lz) = x(L)dB0(z)x(L) —1 that the matrix differential form dB0(z) can be considered as a kind of Prym differentials, in Gunning's sense, with respect to the character that is equal to the monodromy homomorphism of equation (6) for a fixed group r with values in GL(3, C) [4].

Remark 2.3. The equality U(Lz)(L'(z)) —1 = x(L)U(z), L G r means that the solution vector U(z) for the Cauchy problem at the point z0 is a kind of vector Prym 1-differentials of third order on F = D/r with respect to the matrix character x of the group r with values in GL(3, C). More precisely, U(z) is a holomorphic section of the vector bundle x ® K— 1, where K is the canonical bundle on F = D/r [4].

References

[1] D.A.Hejhal, Monodromy groups for higher-order differentials equation, Bull. Amer. Math. Soc., 81(1975), no. 3, 590-592.

[2] D.A.Hejhal, The variational theory of linearly polymorphic functions, J. d'Analyse Math., 30(1976), 215-264.

[3] V.V.Chueshev, An explicit variational formula for the monodromy group on a compact Riemann surface, Sib. Adv. Math., 15(2005), no. 2, 1-32 .

[4] R.C.Gunning, Lectures on vector bundles over Riemann surfaces, Princeton, Princeton Univ. Press., 1967.

Точные вариационные формулы для уравнения третьего порядка на римановой поверхности

Марина И. Тулина

В статье выводятся точные вариационные формулы для решения и его группы монодромии обыкновенного дифференциального уравнения третьего порядка при вариации в пространстве голоморфных кубических дифференциалов на компактной римановой поверхности.

Ключевые слова: голоморфные кубические дифференциалы, компактная риманова поверхность, вариационные формулы, группа монодромии, дифференциальное уравнение третьего порядка.