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UDC 517.54
MATHEMATICS
S. L. KRUSHKAL
ON A CERTAIN CLASS OF EXTREMAL QUASICONFORMAL MAPPINGS
(Presented by Academician M. A. Lavrent’ev, 2 VI 1967)
In this note a new class of extremal quasiconformal mappings is considered, connected with one generalization of the well-known Teichmüller theorem \((^{1-4})\).
Let \(S\) and \(S'\) be two identically oriented closed Riemann surfaces of genus \(g > 1\), and let \(a\) be a given homotopy class of orientation-preserving homeomorphisms \(f : S \to S'\). On the surface \(S\), let there be distinguished \(n \ge 1\) distinct finitely connected domains \(D_1, \ldots, D_n\) with nondegenerate Jordan boundaries, with \(\overline{D_i}\cap \overline{D_j}=\phi\) for \(i\ne j\). Put
\[
D=\bigcup_{j=1}^{n} D_j.
\]
It is assumed that the surfaces \(S\) and \(S'\) are represented in the form
\[
S=U/\Gamma,\qquad S'=U'/\Gamma',
\]
where \(U\) and \(U'\) are respectively the disks \(|z|<1\) and \(|w|<1\), and \(\Gamma\) and \(\Gamma'\) are Fuchsian groups of the first kind corresponding to the surfaces \(S\) and \(S'\). The groups \(\Gamma\) and \(\Gamma'\) are henceforth regarded as fixed.
We consider the following problems, which generalize Teichmüller’s problem.
Problem I. Let a divisor \(\Delta=q_1^{\alpha_1}q_2^{\alpha_2}\ldots q_l^{\alpha_l}\) be given on the surface \(S\), where \(q_i\in D\) and \(\alpha_i\ge 0\) are integers. Let \(P\subset U\) be a fixed fundamental polygon corresponding to the surface \(S\), and let \(z_1,\ldots,z_l\) be points of \(P\) corresponding to the points \(q_1,\ldots,q_l\). Denote by \(Q(a,\Delta,D)\), for some \(Q<\infty\), the class of quasiconformal homeomorphisms \(f:S\to S'\) with the following properties: 1) \(f\in a\); 2) the homeomorphisms \(f\) are conformal in the domains \(D_j\), \(j=1,\ldots,n\); 3) the maximal dilatation* \(K[f]\le Q\); 4) the homeomorphisms \(f\) take prescribed values at the points \(z_i\), together with their derivatives up to order \(\alpha_i\):
\[
f^{(m)}(z_i)=w_{m,i},\qquad m=0,1,\ldots,\alpha_i;\quad i=1,2,\ldots,l.
\tag{1}
\]
Let the class \(Q(a,\Delta,D)\) be nonempty. It is required to find a mapping \(f\in Q(a,\Delta,D)\) for which \(\inf K[f]\) is attained in this class.
Problem II. It is required to find a mapping with the least dilatation \(K[f]\) in the class of homeomorphisms \(f:S\to S'\) satisfying conditions 1)—3) and the condition
\[
f^{(\alpha_i)}(z_i)=w_{\alpha_i,i};\qquad i=1,2,\ldots,l.
\tag{1′}
\]
The solution of these problems is given by the following theorem.
Theorem. Let \(S\) and \(S'\) be two closed Riemann surfaces of genus \(g>1\). Then, if the class \(Q(a,\Delta,D)\) of quasiconformal homeomorphisms \(f:S\to S'\) is nonempty, each of its extremal mappings \(w=f_0(z)\) has the following properties: either \(f_0(z)\) is an analytic function, or there exists a quadratic differential \(\psi(w)\,dw^2\not\equiv 0\) on \(S'\), having, possibly, at the points \(q_j'=f_0(q_j)\) poles of order not higher than
\[
\text{*}
\]
By the maximal dilatation of a quasiconformal mapping \(w=f(z)\) with complex characteristic \(\mu(z)=w_{\bar z}/w_z\) is meant
\[
K[f]=(1+k)/(1-k),
\]
where \(k=\|\mu\|_{L_\infty(S)}\). In other words, \(K[f]\) is the essential maximum of the characteristic \(p(z)\) in the sense of M. A. Lavrent’ev \((^5)\).
\(\alpha_j+1\) \((j=1,\ldots,l)\) and holomorphic at the remaining points of the surface \(S'\), and such a positive constant \(k<1\) that the characteristic
\(\mu_0(w)=(f_0^{-1})_{\bar w}/(f_0^{-1})_w\) of the inverse mapping \(z=f_0^{-1}(w)\) has the form
\[ \mu_0(w)= \begin{cases} 0, & w\in D'=f_0(D),\\ k\overline{\psi(w)}/|\psi(w)|, & w\in S'\setminus D'. \end{cases} \tag{2} \]
The quadratic differential \(\psi\,dw^2\) is determined uniquely up to a positive factor.
The extremal functions of Problem II possess analogous properties.
The existence of extremal mappings follows from the compactness of the classes under consideration. Let \(w=f_0(z)\) be an extremal mapping in the class \(Q(\alpha,\Delta,D)\). Put \(q_j'=f_0(q_j)\), \(D_i'=f_0(D_i)\), \(D'=\bigcup_{i=1}^n D_i'\), \(\Delta'=q_1'^{\alpha_1}\cdots q_l'^{\alpha_l}\), and denote by \(A_1(U',\Gamma',\Delta',D')\) the Banach space of quadratic differentials \(\psi\,dw^2\) on the surface \(S'\), having at the points \(q_j'\) poles of orders not exceeding \(\alpha_j+1\) and holomorphic at the remaining points of \(S'\), with norm
\[ \|\psi\|_{A_1(U',\Gamma',\Delta',D')}= \iint_{S'\setminus D'} |\psi(w)|\,du\,dv<\infty \quad (w=u+iv). \tag{3} \]
By the Riemann—Roch theorem, \(\dim A_1(U',\Gamma',\Delta',D')<\infty\). By \(B(\Gamma')\) denote the Banach space of Beltrami differentials \(\mu(w)\,d\bar w/dw\) on \(S'\) with norm
\(\|\mu\|_{B(\Gamma')}=\|\mu\|_{L_\infty(S')}\), and by \(N(\Gamma',\Delta',D')\) the set of differentials \(\nu(w)\,d\bar w/dw\in B(\Gamma')\) satisfying the conditions:
\[ \nu(w)=0,\quad w\in D'; \qquad \langle \nu,\psi\rangle= \iint_{S'\setminus D'} \nu(w)\psi(w)\,du\,dv=0, \]
\[ \psi\in A_1(U',\Gamma',\Delta',D'). \tag{4} \]
The proof of the theorem is carried out by the method of the work \((^4)\) and is based on the following lemmas.
Lemma 1. Let \(E_0\) be a set of positive (Lebesgue) measure on the surface \(S'=U'/\Gamma'\), with \(E_0\cap D'=\varnothing\), and let \(\mu(w)d\bar w/dw\) be a Beltrami differential on \(S'\), equal to zero for \(w\in E_0\) and \(w\in D'\). Then there exists a differential \(\hat\mu(w)d\bar w/dw\in N(\Gamma',\Delta',D')\), coinciding with \(\mu(w)d\bar w/dw\) on \(S'\setminus E_0\), and such that
\[ \|\hat\mu\|_{L_\infty(E_0)}\le C\|\mu\|_{L_\infty(S')}, \tag{5} \]
where \(C\) is a constant depending only on \(S'\), \(\Delta'\), \(D'\), \(E_0\).
Fix on the surface \(S'=U'/\Gamma'\) a canonical dissection \(c\), unique up to a homeomorphism of \(S'\) onto itself homotopic to the identity. This dissection determines in the fundamental group \(\pi_1(S')\) of the surface \(S'\), up to an inner automorphism, a system of generators \(\Sigma'\): \(a_1,b_1,\ldots,a_g,b_g\), satisfying the relation
\[
\prod_{j=1}^{g} a_jb_ja_j^{-1}b_j^{-1}=1.
\]
Denote by \(\tau_1,\tau_2,\ldots,\tau_{3g-3}\) the moduli (coordinates), introduced by L. Ahlfors \((^2)\), of the pair \((S',\Sigma')\), i.e., the coordinates of the point \(\bar S'\) determined by this pair in the Teichmüller space \(T_g(S)\). The pair \((S',\Sigma')\) is called a marked Riemann surface. If \(S^*\) is another closed surface of genus \(g\), then every homeomorphism \(f:S'\to S^*\) determines the corresponding marked surface \((S^*,\Sigma^*)\), \(\Sigma^*=f(\Sigma')\) (see \((^2)\)). Fix fundamental polygons \(P'\) and \(P^*\), corresponding to the marked surfaces \((S',\Sigma')\) and \((S^*,\Sigma^*)\).
Lemma 2. Let the marked Riemann surfaces \((S',\Sigma')\) and \((S^*,\Sigma^*)\) have respectively the coordinates \(\{\tau_j'\}\) and \(\{\tau_j^*\}\), \(j=1,\ldots,3g-3\), satisf...
satisfying the inequalities
\[ |\tau'_j-\tau^*_j|<\delta,\quad j=1,\ldots,3g-3. \tag{6} \]
Let a system of numbers \(\Omega=\{\omega_{mi}\}\), \(m=0,1,\ldots,\alpha_i\), \(i=1,\ldots,l\), be given, satisfying the conditions: \(\omega_{0i}\in P^*\) and \(|\omega_{0i}-w_i|<\delta\), where \(w_i\) are points of \(P'\) corresponding to the points \(q'_i\in D'\); \(|\omega_{1i}-1|<\delta\), \(|\omega_{mi}|<\delta\), \(m=2,3,\ldots,\alpha_i\); \(i=1,\ldots,l\).
Then, for sufficiently small \(\delta>0\), there exists a Beltrami differential \(\nu(w)\,d\overline w/dw\) such that \(\nu(w)=0\) for \(w\in D'\), and the quasiconformal homeomorphism \(\omega=f_\delta(w)\) with characteristic \(\nu(w)\) maps the surface \((S',\Sigma')\) onto \((S',\Sigma^*)\) and has the following properties:
\[ f^{(m)}(w_i)=\omega_{mi}\quad (m=0,1,\ldots,\alpha_i;\ i=1,\ldots,l);\quad \|\nu\|_{L_\infty(S')}=O(\delta). \tag{7} \]
In particular, if the surfaces \(S'\) and \(S^*\) coincide, then we obtain a quasiconformal homeomorphism of the surface \(S'\) onto itself satisfying the conditions (7).
Lemma 3. Let the quasiconformal mapping \(w=f_0(z):S\to S'\) be extremal in the class \(Q(a,\Delta,D)\), and let \(f_{0\overline z}/f_{0z}=\nu_0(z)\). Then almost everywhere in \(S\setminus D\) the inequality
\[ |\nu_0(z)|=k_0=(K_0-1)/(K_0+1),\quad K_0=\inf_{f\in Q(a,\Delta,D)} K[f]. \tag{8} \]
holds.
Lemma 4. Let the quasiconformal mapping \(w=f_0(z):S\to S'\) be extremal in the class \(Q(a,\Delta,D)\) and different from a conformal one on the set \(S\setminus D\). Then there exists a quadratic differential \(\psi_0(w)\,dw^2\in A_1(U',\Gamma',\Delta',D')\) such that the inverse mapping \(z=f_0^{-1}(w)\) has, at the points \(w\in S'\setminus D'\), \(D'=f_0(D)\), the characteristic
\[ \mu_0(w)=k_0\overline{\psi_0(w)}/|\psi_0(w)|. \tag{2'} \]
The differential \(\psi_0\,dw^2\) is determined up to a positive constant factor.
Lemmas 1, 3, and 4 are proved analogously to the corresponding lemmas of the work [4]. The proof of Lemma 2 is based on the solution of a certain moment problem.
Along with closed surfaces, one may also consider compact Riemann surfaces bounded by a finite number of analytic curves, and require that the homeomorphisms under consideration also take, at a finite number of prescribed points \(p_1,\ldots,p_r\) lying in \(S\setminus D\), prescribed values \(p'_1,\ldots,p'_r\). However, this more general case is reduced in a known way to the one considered above. In this case the assertion of the theorem is preserved, only the quadratic differential \(\psi\,dw^2\) must take real values on the boundary of the surface and may also have simple poles at the points \(p'_1,\ldots,p'_r\).
Remark 1. Analogous theorems hold for mappings of the complex plane and of compact surfaces of genus \(g=1\) (tori).
We note the following assertion, analogous to Lemma 2 for \(l=1\).
Lemma 5. Let \(d_0,d_1,\ldots,d_n\) be prescribed complex numbers satisfying the inequalities \(|d_0|<\varepsilon\), \(|d_1-1|<\varepsilon\), \(|d_s|<\varepsilon\), \(s=2,\ldots,n\), and let \(b_1,\ldots,b_m\) be arbitrary finite points in the \(w\)-plane, distinct from \(w=0\). Then, for every \(R>\max(|b_1|,\ldots,|b_m|)\), for sufficiently small \(\varepsilon>0\), there exists a quasiconformal mapping \(\omega=f_\varepsilon(w)\) of the \(w\)-plane onto itself having the following properties: 1) the mapping \(f_\varepsilon(w)\) is conformal in the disk \(|w|<R\); 2) \(f_\varepsilon^{(s)}(0)=d_s\), \(s=0,1,\ldots,n\); 3) \(f_\varepsilon(b_j)=b_j\), \(j=1,\ldots,m\); 4) the characteristic \(\mu_\varepsilon(w)=f'_{\varepsilon\overline w}/f'_{\varepsilon w}\) for \(|w|>R\) satisfies the inequality \(|\mu_\varepsilon(w)|<C(R)\varepsilon\), where \(C(R)\) is a constant depending only on \(R\).
Indeed, for \(|w|\le R\) one may put
\[ \omega=f_\varepsilon(w)=\sum_{k=0}^{n+m} d_k w^k, \]
where the numbers \(d_{n+1},\ldots,d_{n+m}\) are uniquely determined from the conditions \(f_\varepsilon(b_j)=b'_j,\ j=1,\ldots,m\). By the conditions of the lemma we obtain that \(f'_\varepsilon(w)=1+O(\varepsilon)\) for \(|w|\le R\), and \(f_\varepsilon(w)\) is univalent in this disk for small \(\varepsilon\). Then it is not difficult to construct a quasiconformal continuation of the mapping \(f_\varepsilon(w)\) to the whole plane, in such a way that the characteristic \(\mu_\varepsilon(w)\) for \(|w|>R\) satisfies the inequality \(|\mu_\varepsilon(w)|=1+O(\varepsilon)\).
Remark 2. One may also consider quasiconformal homeomorphisms \(f:S\to S'\) with the following property. Let \(E\) be a set of positive measure on the surface \(S\), with \(m(S\setminus E)>0\). It is required that the characteristic \(\mu(z)=f_{\bar z}/f_z\) satisfy on the set \(E\) the inequality \(|\mu(z)|\le q(z)\), where \(q(z)\) is a prescribed measurable nonnegative function on \(E\) such that \(\|q\|_{L_\infty(E)}<1\). If the class of such homeomorphisms is nonempty, then, analogously to the preceding, it is established that for the characteristics \(\mu_0(z)\) of the extremal mappings \(w=f_0(z)\), minimizing the quantity \(\|\mu\|_{L_\infty(S\setminus E)}\), the equalities
\[ |\mu_0(z)|= \begin{cases} q(z), & z\in E,\\ \mathrm{const}<1, & z\in S\setminus E. \end{cases} \tag{9} \]
hold. Such mappings for the case of rectangles were considered in \((^6)\).
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
23 V 1967
CITED LITERATURE
- O. Teichmüller, Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl., 22, 1 (1940).
- L. Ahlfors, in: L. Ahlfors, L. Bers, Riemann Surfaces and Quasiconformal Mappings, IL, 1961, pp. 51, 104.
- L. Bers, ibid., p. 9.
- S. L. Krushkal, Siberian Mathematical Journal, 8, No. 2, 338 (1967).
- M. A. Lavrent’ev, Matematicheskii Sbornik, 42, 407 (1935).
- R. Kühnau, Math. Zs., 94, No. 3, 178 (1966).