Taniguichi, Masahiko

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TitleBell's results on, and representations of finitely connectedplanar domains (Applications of the theory of reproducingkernels)Author(s)Taniguichi, MasahikoCitation数理解析研究所講究録 (2004), 1352: 47-53Issue Date2004-01URL http://hdl.handle.net/2433/25128RightType Departmental Bulletin PaperTextversionpublisherKyoto University47Bell’s results on, and representations offinitely connected planar domains(Masahiko Taniguchi)Department of Mathematics, Kyoto University,1 Ahlfors maps and Bergman kernelsLet $D$ be a domain in C. Consider the subspace $A^{2}(D)$ of the Hilbertspace $L^{2}(D)$ (of all square integrable functions on $D$ with respect to theLebesque meaure on C) consisting of all elements in $L^{2}(D)$ holomorphicon $D$ . Then there is the natural projection$P:L^{2}(D)arrow A^{2}(D)$ ,which is called the Bergman projection. The coresponding kernel $K(z, w)$is called the Bergman kernel.When $D$ is the unit disc,$K$ ( $z$ , $w)= \frac{1}{\pi(1-z\overline{w})^{2}}$ .Hence the Bergman kernel function $K(z, w)$ associated to a simply con-nected domain $D$ can be written by using the Riemann map $f_{a}$ (z) (de-termined uniquely by the conditions $fa(d)=0$ and $f_{a}’(a)>0)$ and itsderivative:$K$ ( $z$ , $w)= \frac{f_{a}’(z)f_{a}’(w)}{\pi(1-f_{a}(z)\overline{f_{a}(w)})^{2}}$Let $D$ be a non-degenerate multiply connected planar domain withsmooth boundary. Fix a point $a$ in $D$ , and let $f_{a}$ be the Ahlfors map1352 2004 47-5348associated with the pair $(D, a)$ . Among all holomorphic functions $h$ whichmap $D$ into the unit disc and satisfy $h$ {$a)=0,$ the Ahlfors map $f_{a}$ is theunique function which maximizes $\mathrm{h}\mathrm{f}(\mathrm{a})$ under the condition $h’(a)>0.$Such proper holomorphic maps can recover the Bergman projections andkernels in general.Theorem 1 Let $f$ : $D_{1}arrow D_{2}$ be a proper holomorphic map $b$ etweenplanar (proper) domains. Let $P_{j}$ be the Bergman projection for $D_{j}$ . Then$P_{1}(f’(\phi \mathrm{o}f))=f’((P_{2}\phi)\mathrm{o}f)$for all $0\in L^{2}(D_{2})$ .$\phi$ $\in L^{2}(D_{2})But the translation formula for the Bergman kernels is not so simplein general. For instance, it is hard to write down the following formulaexplicitly.Proposition 2 Let $f$ : $D_{1}arrow D_{2}$ be a proper holomorphic map be tweenplanar (proper) domains. Then the Bergman kernels $K_{j}$ (z, $w$ ) associatedto $D_{j}$ transform according to$m$$f’(z)K_{2}(f(z), /0)$ $=$ $\mathrm{p}$ $K_{1}(z, F_{k}(w))\overline{F_{k}’(w)}$$k=1$for $z\in D_{1}$ and $w$ $\in D_{2}-V$ where the multiplicity of the map $f$ is $m$ andthe functions $F_{k}$ , $k$ $=1$ , ( $\mathrm{f}$ . ’ $m$, denote the local inverses to $f$ and $V$ isthe set of critical values.z$ $\in D_{1}$ fm $=1_{\}(\mathrm{r}$ ,S. Bell obtained several kinds of simpler representations of Bergmankernel functions.Theorem 3 ([1]) For a non-degenarate multiply connected planar dO-main $D$ , we can find two points $a$ , $b$ in $D$ such that$K(z, w)=f_{a}’(z)f_{b}’\{w$ )R$(z, w)$with a rational combination $R(z, w)$ of $f_{a}$ and $I_{b}$ .0Here we say that a function $R$ (z, $w$ ) is a rational combination of $f_{a}$ and$f_{b}$ if it is a rational function of$\mathrm{A}$ $(z)$ , 7 $b(z)$ , $\mathrm{A}(w)$ , $f_{b}(w)$ .Such representation as above has the following variant.Theorem 4 ([5]) For a non-degenarate multiply connected planar dO-main $D$ , we can find two points $a$ , $b$ in $D$ such that$K$ ( $z$ , $w)= \frac{f_{a}’(z)\overline{f_{a}’(w)}}{(1-f_{a}(z)\overline{f_{a}\{w)})^{2}}(\sum_{-i,k}H_{j}(z)\overline{K_{k}(w)})$where $f_{a}$ , $f_{b}$ are the Ahlfors functions, $H$ and $K$ are rational functions ofthem, and the sum is a finite sum.Actually, we can use any proper holomorphic maps.Theorem 5 ([2]) Let $D$ be a non-degenarate multiply connected planardomain, and $f$ a proper holomorphic map of $D$ onto the unit disk $U\mathrm{r}$Then $K$ (z, $w$ ) is an algebraic funct\’ion of7 (z), $f’\{z)$ , $f(w)$ , $f’(w)$ .Moreover, we have the followingTheorem 6 ([2]) Let $D$ be a non-degenerate multiply connected planardomain. The following conditions are equivalent.(1) The Bergman kernel $K(z, w)$ associated to $D$ is algebraic, $i.e$ . analgebraic function of $z$ ancl $\overline{w}$ .(2) The Ahlfors map $f_{a}(z)$ is an algebraic function of $z$ .(3) There is a proper holomorphic mapping $f$ : $Darrow U$ which is analgebraic function.(4) Every proper holomorphic mapping from $D$ onto the unit disc $U$ isan algebraic function.Also we have50Theorem 7 ([4]) Let $D$ be a non-degenerate multiply connected planardomain. There are two holomorphic functions $F_{1}$ and $F_{2}$ on $D$ such thatthe Bergman kernel on $D$ is a rational combination of $F_{1}$ and $F_{2}$ if andonly if there is a proper holomorphic map $f$ of $D$ onto $U$ such that $f$and $f’$ are algebraically dependent: $i.e$ . there is a polynomial $Q$ such that$Q(f, f’)=0.$Then, for every proper holomorphic map $f$ of $D$ to $U$ , $f$ and $f’$ arealgebraically dependent.Proposition 8 ([4]) Let $D$ be a simply connected planar (proper) dO-main. The Bergman kernel on $D$ is a rational combination of a functionof a complex variable if and only if the Riemann map $f$ of $D$ and $f’$ arealgebraically dependent.Finally, we note the following facts.Proposition 9 ([2]) If $K(z, w)$ is algebraic, and $f$ be a proper holomor-phic map to U. Then $K(z, w)$ is an algebraic function of $f(z)$ and $\overline{f(w)}$.Corollary 1 ([2]) Let $D_{1}$ and $D_{2}$ have algebraic Bergman kernels, thenevery biholomorphic map of $D_{1}$ onto $D_{2}$ is algebraic.2 Bell representationsNow the issue is to find a family of canonical domains which admit asimple proper holomorphic map to $U$ . Bell proposed such a family, andactually, they are enough.Theorem 10 ([6]) Every non-degenerate $n$-connected planar domain with$n>1$ is mapped biholomorphically onto a domain $W_{\mathrm{a},\mathrm{b}}$ defined by$\mathrm{T}_{\mathrm{a}}$,b $=\{$ $z\in \mathbb{C}$ :$n-1$$z$ $+ \sum\frac{a_{k}}{z-b_{k}}$$k=1$$<1\}$with suitable complex vectors a $=$ ( $a_{1},$ $a_{2},$ $\uparrow$ t, $a_{n-1}$ ) and $\mathrm{b}=(b_{1},$ $b_{2}$ , ( $|$ T -, $b_{n-1}$ ).51The above theorem is considered as a natural generalization of theclassical Riemann mapping theorem for simply connected planar domains.The function $7_{\mathrm{a},\mathrm{b}}$ defined by$n-1$$7_{\mathrm{a},\mathrm{b}}\{z)$ $=Z$ $+ \sum\frac{a_{k}}{z-b_{k}}$$k=1$is a proper holomorphic mapping from $W_{\mathrm{a},\mathrm{b}}$ to the unit disc which isrational. Actually, it is a very classical fact that, for such an $f=f_{\mathrm{a},\mathrm{b}}$ asabove, $f$ and $f’$ are algebraically dependent. Hence the above propositionimplies the following corollary.Corollary 2 Every non-degenerate $n$ -connected planar domain $D$ with$n>1$ is biholomorphic to a domain ttyith the algebraic Bergman kernelCorollary 3 There are two holomorphic functions $F_{1}$ and $F_{2}$ such thatthe Bergman kernel on $W_{\mathrm{a},\mathrm{b}}$ is a rational combination of $F_{1}$ and $F_{2}$ .Definition The locus $\mathrm{B}_{n}$ in $\mathbb{C}^{2n-2}$ consisting of $(\mathrm{a}, \mathrm{b})$ such that the cor-responding domain $W_{\mathrm{a},\mathrm{b}}$ is a non-degenerate $n$-connected planar domain.We call this locus $\mathrm{B}_{n}$ the coefficient body for non-degenerate n-connectedcanonical domains.It is obvious that $\mathrm{B}_{n}$ is contained in the product space$(\mathbb{C}^{*})^{n-1}\mathrm{x}F_{0,n-1}\mathbb{C}$ ,which has the same homotopy type as that of$X=(S^{1})^{n-1}\mathrm{x}$ $F_{0,n-1}\mathbb{C}$ ,where$F_{0,n-1}\mathbb{C}=$ { $(z_{1},1\cdot l|$ , $z_{n-1}\in \mathbb{C}^{n-1}|z_{j}\overline{\tau}^{-Z}k\angle$ if $j\overline{7}\leq k$ }is called a configuration space.To clearify the topological structure of the coefficent body, it is moreconvenient to use the following modified representation space.Definition We set$\mathrm{B}_{\mathrm{n}}^{*}=\{(a_{1},1\Gamma =, a_{n-1}, \mathrm{b})\in\{\mathbb{C})^{2n-2}|(a_{1}^{2}, \cdot\circ \mathrm{Q} , a.\mathit{2}_{-1}, \mathrm{b})\in \mathrm{B}_{n}\}$,and call it the modified coefficient body.X}\mathbb{C}=\{(z_{1},1\cdot$ $||$ athbb{C}^{n-1}|Z_{j}\overline{7}^{- \angle z_{k}$ erline{7}\leq k\}$SC S$\mathrm{B}_{\mathrm{n}}^{*}=\{(a_{1},1\Gamma$ $=$ , $a_{n-1}$ , b) $\in\{\mathbb{C})^ 2n-2 |(a_{1}^{2}$ , $\cdot\circ \mathrm{D}$ , $a_{n-1}^{2 .’ \mathrm{b})\in \m _{n}\}$52Theorem 11 $\mathrm{B}_{n}^{*}$ is a circular domain, and has the same homotopy typeas that of the product space $X\iota$Corollary 4 The homotopy type of $\mathrm{B}_{n}$ is the same as that of $X_{\llcorner}$Remark The fundamental group of $F_{0,n-1}\mathbb{C}$ is called the pure braidgroup, and its structure is well-known.Problem1. Determine the Ahlfors locus of $\mathrm{B}_{n}$ which consists of all $(\mathrm{a}, \mathrm{b})$ suchthat $f_{\mathrm{a},\mathrm{b}}$ gives an Ahlfors map (, or more precisely, $e^{i\theta}f_{\mathrm{a},\mathrm{b}}$ with asuitable $0\in$ il is an Ahlfors map).2. Fix a point $(\mathrm{a}, \mathrm{b})$ in $\mathrm{B}_{n}$ , and let $W=W_{\mathrm{a},\mathrm{b}}$ be the corresponding n-conenncted canonical domain. Determine the leaf $E(W)$ of $\mathrm{B}_{n}$ for $W$ ,consisting of all points which correspond to $n$-connected canonicaldomains biholomorphically equivalent to $W$ .3. Determine the collision locus $C$ of $\mathrm{B}_{n}$ which consists of all $(\mathrm{a}, \mathrm{b})$ suchthat the correcponding map $f_{\mathrm{a},\mathrm{b}}$ has a pair of critical points (countedwith multiplicities) whose image is the same. (Note that $\mathrm{B}_{n}-C$is a finite-sheeted holomorphic smooth cover of the intersection of$\mathrm{F}\mathrm{Q}_{2n-2},\mathbb{C}$ and the unit polydisc.)( $\mathrm{a}$ , b)\Upsilon \ athrm{a},\mathrm{b}$Sfinite-she$F_{0,2n-2}$CExample 1$\mathrm{B}_{2}^{*}=$ {{ $a$ , $b)\in \mathbb{C}^{2}$ : a 40, $|b+2a|<1$ , $|b-2a|<1$ } $,$which is biholomorphic to the polydisc deleted the diagonalNext, the set$\{(a_{)}b)$ $\in \mathrm{B}_{2}^{*}$ : $\frac{4a^{2}}{1-\overline{(b+2a)}(b-2a)}$ $= \frac{4r}{4+r^{2}}$ }corresponds to a leaf of $\mathrm{B}_{2}$ for every given $r>2_{f}$ and the collision locusof $\mathrm{B}_{2}$ is empty.53[1] S. Bell, Ahlfors maps, the double of a domain, and complexity inpotential theory and conformal mapping, J. d’Analyse Math., 78(1999), 329-344.[2] S. Bell, Finitely generated function fields and complexity in potentialtheory in the plane, Duke Math. J., 98 (1999), 187-207.[3] S. Bell, A Riemann surface attached to domains in the plane andcomplexity in potential theory, Houston J. Math., 26, (2000), 277-297.[4] S. Bell, Complexity in Complex analysis, Adv. Math., 172 (2002),15-52.)$[4]$ co l–[5] S. Bell, M\"obius transformations, the Caratheodory metric, and theobjects of complex analysis and potensial theory in maltiply connecteddomains, preprint.[6] M. Jeong and M. Taniguchi, Bell representation offinitely connectedplanar domains, Proc. AMS., 131 (2003), 2325-2328.[7] M. Jeong and M. Taniguchi, Algebraic kernel functions and represen-tation of planar domains, J. Korea Math. Soc, 40 (2003), 447-460.[8] M. Jeong and M. Taniguchi, in preparation.$[6]$$[7]$ m $repr en-$$d ains_{f}$ s

Bell's results on, and representations of finitely connected planar domains (Applications of the theory of reproducing kernels)

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Bell's results on, and representations of finitely connected planar domains (Applications of the theory of reproducing kernels)

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