1 Introduction
A condenser in the complex plane
${\mathbb {C}}$
is a pair
$(E,F)$
where E and F are non-empty disjoint compact subsets of
${\mathbb {C}}$
. Let
$S(E,F)$
denote the family of signed measures
$\tau =\tau _{E}-\tau _{F}$
, where
$\tau _{E}$
and
$\tau _{F}$
are Borel probability measures supported on E and F, respectively. The energy of a measure
$\tau \in S(E,F)$
is defined by
$$\begin{align*}I(\tau):=\iint\log\frac{1}{|z-w|}d\tau(z)d\tau(w).\end{align*}$$
We note that
$I(\tau )>0$
, for every
$\tau \in S(E,F)$
; see e.g., [11, p. 80]. The equilibrium energy of
$(E,F)$
is defined by
$$\begin{align*}I(E,F):=\inf\limits_{\tau\in S(E,F)}I(\tau),\end{align*}$$
and the capacity of
$(E,F)$
is given by
$$\begin{align*}{\mathrm{Cap}}(E,F):=\frac{2\pi}{I(E,F)}.\end{align*}$$
When
${\mathrm {Cap}}(E,F)>0$
, there exists a unique measure
$\sigma \in S(E,F)$
, called the equilibrium measure of
$(E,F)$
, satisfying
${\mathrm {Cap}}(E,F):=2\pi /I(\sigma )$
. For more information about condenser capacity, we refer the reader to [Reference Bagby3, Reference Landkof11].
A classical object of study in geometric function theory is the behavior of several types of capacities (such as logarithmic, analytic, and Riesz capacity) of sets and condensers in
${\mathbb {C}}$
, under geometric transformations such as conformal mappings, symmetrizations, and polarization. We refer the reader to the books [Reference Dubinin4, Reference Goluzin7] and the references therein for an account of the methods and the applications of capacities in geometric function theory. In this note, we will study the behavior of condenser capacity under holomorphic motions.
A holomorphic motion is a holomorphically parameterized family of injective maps. Here is the precise description.
Definition 1.1 A holomorphic motion of a set
$A\subset {\mathbb {C}}$
, parameterized by a domain
$D\subset {\mathbb {C}}$
containing
$0$
is a map
$f:D\times A\mapsto {\mathbb {C}}$
satisfying:
-
(i) for any fixed
$z\in A$
, the map
$\lambda \mapsto f(\lambda ,z)$
is holomorphic in D; -
(ii) for any fixed
$\lambda \in D$
, the map
$z\mapsto f(\lambda ,z):=f_{\lambda }(z)$
is an injection; -
(iii) the mapping
$f(0,\cdot )$
is the identity on A.
Although there is no continuity assumption of f on
$D\times A$
in the above definition, the continuity (as a function of two complex variables) of holomorphic motions has been proved, among other properties, by Mañé, P. Sad, and D. Sullivan [Reference Mañé, Sad and Sullivan12], who introduced the notion of holomorphic motions, in a result that is known as
$\lambda $
-lemma. The joint continuity will be used several times in our proofs. A fundamental result in this theory, proved by Słodkowski [Reference Słodkowski21], is that a holomorphic motion of any set in the Riemann sphere parameterized by the unit disc
${\mathbb {D}}:=\{z\in {\mathbb {C}}:|z|<1\}$
can be extended to a holomorphic motion of the whole Riemann sphere parameterized by
${\mathbb {D}}$
. For an account of the properties of holomorphic motions (such as quasiconformality, distortion, and non-extendability properties) and their applications, we refer the reader to [Reference Astala, Iwaniec and Martin1, Reference Astala and Martin2, Reference Jiang and Mitra10, Reference Martin13] and the references therein.
The behavior of different geometric quantities under holomorphic motions has been studied by several researchers. Some examples are the Hausdorff dimension [Reference Ransford18, Reference Ruelle20], the conformal modulus of doubly connected domains [Reference Earle and Mitra5], the analytic and the logarithmic capacity [Reference Pouliasis, Ransford and Younsi16, Reference Ransford, Younsi and Ai19], and the capacity of condensers [Reference Pouliasis15].
In this paper, we will consider the behavior of condenser capacity under holomorphic motions. Let
$(E,F)$
be a condenser with positive capacity and let f be a holomorphic motion of
$E\cup F$
parameterized by a domain D containing 0. From the continuity of the injective functions
$f_{\lambda }(\cdot )$
by the
$\lambda $
-lemma, it follows that
$f_{\lambda }(E)$
and
$f_{\lambda }(F)$
are disjoint compact subsets of
${\mathbb {C}}$
, for every
$\lambda \in D$
. We will show that
$T(\lambda )={\mathrm {Cap}}(f_{\lambda }(E),f_{\lambda }(F))$
is a continuous subharmonic function on D. We note that, applying the methods used in [Reference Pouliasis15], one can show that the function
$T(\lambda )$
is upper-semicontinuous and subharmonic in D, but the continuity proved here requires different arguments. In [Reference Ransford, Younsi and Ai19], the authors showed that the logarithmic capacity of a compact set also varies continuously under holomorphic motions. Although there are estimates between condenser capacity and logarithmic capacity, the continuity of condenser capacity is not a consequence of the corresponding result for logarithmic capacity under holomorphic motions. In contrast to condenser and logarithmic capacity, the analytic capacity of a compact set may vary discontinuously (see [Reference Ransford, Younsi and Ai19]) and, in general, is neither a subharmonic nor a superharmonic function under holomorphic motions (see [Reference Pouliasis, Ransford and Younsi16]). Also, we will show that the equilibrium measures
$\{\sigma _{\lambda }\}$
,
$\lambda \in D$
, of the condensers
$(f_{\lambda }(E),f_{\lambda }(F))$
vary continuously with respect to the weak-star convergence. The proof is based on the properties of pointwise suprema of harmonic functions established in [Reference Ransford, Younsi and Ai19] and on Bagby’s formula for condenser capacity via discrete charges [Reference Bagby3].
Earle and Mitra proved a much stronger result for a certain class of condensers called rings. A condenser
$(E,F)$
is called a ring if both E and F are connected and
${\mathbb {C}}\backslash (E\cup F)$
is a doubly connected domain. It was proved in [Reference Earle and Mitra5] that if
$(E,F)$
is a ring and f is a holomorphic motion of
$E\cup F$
parameterized by a domain D, then the equilibrium energy of
$(f_{\lambda }(E),f_{\lambda }(F))$
(which coincides with the conformal modulus of the doubly connected domain
${\mathbb {C}}\backslash (f_{\lambda }(E)\cup f_{\lambda }(E))$
) is a real analytic function on D. It is not known whether the equilibrium energy of
$(f_{\lambda }(E),f_{\lambda }(F))$
is a real analytic function of
$\lambda $
for arbitrary condensers.
A notion related to rings is uniform perfectness. Let K be a compact subset of
${\mathbb {C}}$
. We recall that a ring
$(E,F)$
is said to separate K if
$K\subset E\cup F$
,
$K\cap E\neq \varnothing ,$
and
$K\cap F\neq \varnothing $
. We will denote by
$R(K)$
the set of rings that separate K. A compact set
$K\subset {\mathbb {C}}$
is called uniformly perfect if
$$\begin{align*}P(K):=\mathrm{sup}\big\{I(E,F):(E,F)\in R(K)\big\}<+\infty.\end{align*}$$
A uniformly perfect set is thick, in the above sense, close to each of its points; in particular, it does not have isolated points. Uniform perfectness can be characterized using several other quantities such as the logarithmic capacity or the density of the hyperbolic metric. For more information, we refer the reader to [Reference Järvi and Vuorinen9, Reference Pommerenke14].
In our last result, we will give an estimate of the quantity
$P(\cdot )$
for compact sets moving under holomorphic motions, involving the Harnack distance.
In the following section we describe some tools needed for the proofs of our results. In Section 3, we prove the continuity of condenser capacity, and in Section 4, we prove the continuity of the equilibrium measures with respect to weak-star convergence. The estimate of the quantity
$P(\cdot )$
under holomorphic motions is proved in Section 5.
2 Background Material
2.1 Bagby’s Formula
Let
$(E,F)$
be a condenser and suppose that both sets E and F contain infinitely many points. That holds for all condensers
$(E,F)$
having positive capacity. For any integer
$n\geq 2$
, let
$$ \begin{align*} L_{n}(E,F)&:=\{(a_{1},\dots,a_{n},b_{1},\dots,b_{n})\in E^{n}\times F^{n}: a_{i}\neq a_{j}\text{ and }b_{i}\neq b_{j}, i\neq j\}, \\ W_{n}(E,F)&=\frac{1}{n(n-1)}\inf\Bigg\{\sum\limits_{1\leq i<j\leq n}\log \Bigg(\frac{|a_{i}-b_{j}||a_{j}-b_{i}|}{|a_{i}-a_{j}||b_{i}-b_{j}|}\Bigg)\Bigg\}, \end{align*} $$
where the infimum is taken over all
$(a_{1},\dots ,a_{n},b_{1},\dots ,b_{n})\in L_{n}(E,F)$
. Although every discrete signed measure in
$S(E,F)$
has infinite energy, the above sum can be considered as a discrete version of the energy of a discrete measure having point masses at the points
$a_{i}$
and
$b_{i}$
,
$i=1,\dots ,n$
. Bagby [Reference Bagby3] proved the following theorem relating the equilibrium energy with the discrete energies
$W_{n}(E,F)$
of a condenser.
Theorem 2.1 [Reference Bagby3]
Let
$(E,F)$
be a condenser and suppose that both sets E and F contain infinitely many points. Then the sequence
$\{W_{n}(E,F)\}$
is increasing and
$$\begin{align*}I(E,F)=\lim\limits_{n\to\infty}W_{n}(E,F).\end{align*}$$
2.2 Pointwise Infima of Harmonic Functions
The following family of functions was introduced and its main properties were studied in [Reference Ransford, Younsi and Ai19].
Definition 2.2 Let
$D\subset {\mathbb {C}}$
be a domain. A function
$u:D\mapsto (-\infty ,+\infty ]$
is said to belong to the class
$\mathcal {H}^{\downarrow }(D)$
if the following properties hold:
-
(i) u is locally bounded below on D;
-
(ii) u is the pointwise infimum of a family of harmonic functions on D.
The following proposition summarizes some properties of the members of the family
$\mathcal {H}^{\downarrow }(D)$
that will be needed in the proofs of our results.
Proposition 2.3 ([19, Propositions 2.4 and 2.5])
Let D be a domain in
${\mathbb {C}}$
. Then
-
(i) if
$u_{n}$
is an increasing sequence of functions in
$\mathcal {H}^{\downarrow }(D)$
and
$u=\lim _{n\to \infty }u_{n}$
, then
$u\in \mathcal {H}^{\downarrow }(D)$
. -
(ii) if
$u\in \mathcal {H}^{\downarrow }(D)$
and
$u\not \equiv +\infty $
, then
$u<+\infty $
in D and u is a continuous superharmonic function in D.
Remark 2.4 In [Reference Ransford, Younsi and Ai19], the dual family
$\mathcal {H}^{\uparrow }(D)$
of pointwise suprema of harmonic functions is considered, and the corresponding results stated above for
$\mathcal {H}^{\downarrow }(D)$
follow by standard modifications.
3 Continuity of Condenser Capacity
In this section, we will state and prove our first result concerning the continuity of condenser capacity under holomorphic motions.
Theorem 3.1 Let
$(E,F)$
be a condenser with positive capacity and let f be a holomorphic motion of
$E\cup F$
parameterized by a domain
$D\subset {\mathbb {C}}$
containing 0. Then
$$\begin{align*}T(\lambda)={\mathrm{Cap}}\big(f_{\lambda}(E),f_{\lambda}(F)\big)\end{align*}$$
is a continuous subharmonic function on D.
Proof We note that since
${\mathrm {Cap}}(E,F)>0$
, both E and F contain infinitely many points. We will first show that the functions
$R_{n}(\lambda ):=W_{n}(f_{\lambda }(E),f_{\lambda }(F))$
are in
$\mathcal {H}^{\downarrow }(D)$
. Let
$n\geq 2$
and note that since
$f_{\lambda }$
is injective,
$$ \begin{align*} L_{n}(f_{\lambda}(E),f_{\lambda}(F)) &= \{(f_{\lambda}(a_{1}),\dots,f_{\lambda}(a_{n}),f_{\lambda}(b_{1}),\dots,f_{\lambda}(b_{n})): \\ &\quad:(a_{1},\dots,a_{n},b_{1},\dots,b_{n})\in L_{n}(E,F)\}. \end{align*} $$
Let
$(a_{1},\dots ,a_{n},b_{1},\dots ,b_{n})\in L_{n}(E,F)$
. Since
$f_{\lambda }(\cdot )$
is injective and depends holomorphically on
$\lambda $
, the functions
$$\begin{align*}\lambda\longmapsto\frac{(\,f_{\lambda}(a_{i})-f_{\lambda}(b_{j}))(\,f_{\lambda}(a_{j})-f_{\lambda}(b_{i}))}{(\,f_{\lambda}(a_{i})-f_{\lambda}(a_{j}))(\,f_{\lambda}(b_{i})-f_{\lambda}(b_{j}))}, \qquad i,j\in\{1,\dots,n\},\, i\neq j,\end{align*}$$
are holomorphic and have no zeros on D. Therefore, the function
$$\begin{align*}\lambda\longmapsto\frac{1}{n(n-1)}\sum\limits_{1\leq i<j\leq n}\log\bigg(\frac{|\,f_{\lambda}(a_{i})-f_{\lambda}(b_{j})||\,f_{\lambda}(a_{j})-f_{\lambda}(b_{i})|}{|\,f_{\lambda}(a_{i})-f_{\lambda}(a_{j})||\,f_{\lambda}(b_{i})-f_{\lambda}(b_{j})|}\bigg)\end{align*}$$
is harmonic in D, since it is a finite sum of harmonic functions. We obtain that
$$\begin{align*}R_{n}(\lambda)=\inf\bigg\{\frac{1}{n(n-1)}\sum\limits_{1\leq i<j\leq n}\log \bigg(\frac{|\,f_{\lambda}(a_{i})-f_{\lambda}(b_{j})||\,f_{\lambda}(a_{j})-f_{\lambda}(b_{i})|}{|\,f_{\lambda}(a_{i})-f_{\lambda}(a_{j})||\,f_{\lambda}(b_{i})-f_{\lambda}(b_{j})|}\bigg)\bigg\},\end{align*}$$
where the infimum is taken over all
$(a_{1},\dots ,a_{n},b_{1},\dots ,b_{n})\in L_{n}(E,F)$
, is a pointwise infimum of harmonic functions of
$\lambda $
in D. Let
$\lambda _{0}\in D$
. Since
$f(\lambda ,z)$
is jointly continuous in
$D\times (E\cup F)$
, there exist an open neighborhood V of
$\lambda _{0}$
such that
$$ \begin{align*} &\inf\limits_{\lambda\in V}{\mathrm{dist}}(\,f_{\lambda}(E),f_{\lambda}(F))>0,\\ &\mathop{\mathrm{sup}}\limits_{\lambda\in V}{\mathrm{diam}}(\,f_{\lambda}(E))<+\infty, \\ &\mathop{\mathrm{sup}}\limits_{\lambda\in V}{\mathrm{diam}}(\,f_{\lambda}(F))<+\infty. \end{align*} $$
It follows that
$R_{n}$
is locally bounded below on D. We conclude that
$R_{n}\in \mathcal {H}^{\downarrow }(D)$
. By Theorem 2.1,
$R_{n}$
is an increasing sequence of functions in
$\mathcal {H}^{\downarrow }(D)$
and
$$\begin{align*}\lim\limits_{n\to+\infty}R_{n}(\lambda)=I(\,f_{\lambda}(E),f_{\lambda}(F)),\qquad \lambda\in D.\end{align*}$$
Since
$I(E,F)<+\infty $
, by Proposition 2.3, the function
$\lambda \mapsto I(\,f_{\lambda }(E),f_{\lambda }(F))$
belongs to
$\mathcal {H}^{\downarrow }(D)$
and is a continuous superharmonic function in D. Finally, from [Reference Ransford17, p. 43], it follows that
$T(\lambda )=2\pi /I(\,f_{\lambda }(E),f_{\lambda }(F))$
is a continuous subharmonic function on D.▪
4 Weak-star Continuity of Equilibrium Measures
In this section, we show that the equilibrium measures of condensers vary continuously with respect to weak-star convergence under holomorphic motions. We recall that a sequence of Borel probability measures
$\mu _{n}$
on a compact set
$K\subset {\mathbb {C}}$
converges weak-star to a Borel probability measure
$\mu $
on K if
$$\begin{align*}\lim\limits_{n\to\infty}\int\phi(z)d\mu_{n}(z)=\int\phi(z)d\mu(z),\end{align*}$$
for every continuous function
$\phi $
on K.
Theorem 4.1 Let
$(E,F)$
be a condenser with positive capacity and let f be a holomorphic motion of
$E\cup F$
parameterized by a domain
$D\subset {\mathbb {C}}$
containing 0. Let
$\sigma _{\lambda }$
be the equilibrium measure of the condenser
$(\,f_{\lambda }(E),f_{\lambda }(F))$
,
$\lambda \in D$
. Then
$\sigma _{\lambda }$
converges to
$\sigma _{\lambda _{0}}$
in the weak-star sense, whenever
$\lambda \to \lambda _{0}$
in D.
Proof From Theorem 3.1, it follows that
${\mathrm {Cap}}(\,f_{\lambda }(E),f_{\lambda }(F))>0$
; therefore, the condenser
$(\,f_{\lambda }(E),f_{\lambda }(F))$
has a unique equilibrium measure
$\sigma _{\lambda }=\sigma _{E}^{\lambda }-\sigma _{F}^{\lambda }$
, for every
$\lambda \in D$
.
Let
$\lambda _{n}\to \lambda _{0}$
in D. From the Riesz Representation Theorem and the sequential version of Alaoglu’s Theorem (see e.g., [Reference Folland6, pp. 169 and 223]), we obtain that there exist a subsequence
$\sigma _{E}^{\lambda _{n_{m}}}$
of
$\sigma _{E}^{\lambda _{n}}$
and a measure
$\nu _{E}^{0}$
such that
$\sigma _{E}^{\lambda _{n_{m}}}\overset {w^{*}}{\to }\nu _{E}^{0}$
. Applying Alaoglu’s Theorem and passing to a subsequence if needed, we can assume that there exists a measure
$\nu _{F}^{0}$
such that
$\sigma _{F}^{\lambda _{n_{m}}}\overset {w^{*}}{\to }\nu _{F}^{0}$
. Then
$\nu _{E}^{0}$
and
$\nu _{F}^{0}$
are Borel probability measures, and from the joint continuity of
$f(\lambda ,z),$
it follows that they are supported on
$f_{\lambda _{0}}(E)$
and
$f_{\lambda _{0}}(F)$
, respectively. So
$\nu _{0}:=\nu _{E}^{0}-\nu _{F}^{0}\in S(\,f_{\lambda _{0}}(E),f_{\lambda _{0}}(F))$
and
$$ \begin{align} I(\,f_{\lambda_{0}}(E),f_{\lambda_{0}}(F))\leq I(\nu_{0}). \end{align} $$
A simple computation (see e.g. [3, p. 318]) shows that for every condenser
$(K,L)$
and for every
$\tau =\tau _{K}-\tau _{L}\in S(K,L)$
,
$$\begin{align*}I(\tau)=\iiiint\log\bigg(\frac{|z-v||w-u|}{|z-w||u-v|}\bigg) d\tau_{K}(z)d\tau_{K}(w)d\tau_{L}(u)d\tau_{L}(v). \end{align*}$$
The function
$$\begin{align*}(z,w,u,v)\longmapsto\log\bigg(\frac{|z-v||w-u|}{|z-w||u-v|}\bigg) \end{align*}$$
is lower-semicontinuous and bounded below on
$K\times K\times L\times L$
. Since
$\sigma _{E}^{\lambda _{n_{m}}}\overset {w^{*}}{\to }\nu _{E}^{0}$
and
$\sigma _{F}^{\lambda _{n_{m}}}\overset {w^{*}}{\to }\nu _{F}^{0}$
, it follows that (see e.g., [Reference Helms8, p. 224])
$$\begin{align*}\sigma_{E}^{\lambda_{n_{m}}}\times\sigma_{E}^{\lambda_{n_{m}}} \times\sigma_{F}^{\lambda_{n_{m}}}\times\sigma_{F}^{\lambda_{n_{m}}}\overset{w^{*}}{\longrightarrow}\nu_{E}^{0}\times\nu_{E}^{0}\times\nu_{F}^{0}\times\nu_{F}^{0}.\end{align*}$$
Therefore (see e.g. [Reference Landkof11, pp. 78–79]), from the lower-semicontinuity of energy integrals with respect to weak-star convergence, we get that
$$ \begin{align} I(\nu_{0})\leq \liminf\limits_{m\to\infty}I(\sigma_{E}^{\lambda_{n_{m}}} -\sigma_{F}^{\lambda_{n_{m}}}). \end{align} $$
Also, from Theorem 3.1 it follows that
$$ \begin{align} \lim\limits_{m\to\infty}I(\sigma_{E}^{\lambda_{n_{m}}} -\sigma_{F}^{\lambda_{n_{m}}})=\lim\limits_{m\to\infty}I\big(\,f_{\lambda_{n_{m}}}(E),f_{\lambda_{n_{m}}}(F)\big)= I\big(\,f_{\lambda_{0}}(E),f_{\lambda_{0}}(F)\big). \end{align} $$
From equations (4.1), (4.2), and (4.3), we conclude that
$I(\,f_{\lambda _{0}}(E),f_{\lambda _{0}}(F))=I(\nu _{0})$
and
$\nu _{0}$
is an equilibrium measure of
$(\,f_{\lambda _{0}}(E),f_{\lambda _{0}}(F))$
. From the uniqueness of equilibrium measure, we get that
$\nu _{0}=\sigma _{\lambda _{0}}$
. Since the
$w^{*}$
-convergent subsequences
$\sigma _{E}^{\lambda _{n_{m}}}$
and
$\sigma _{F}^{\lambda _{n_{m}}}$
considered above were arbitrary, we conclude that the sequence
$\sigma _{\lambda _{n}}$
has a unique
$w^{*}$
-accumulation point, the measure
$\sigma _{\lambda _{0}}$
. Since the space of Borel probability measures on a compact set equipped with the weak-star convergence is metrizable and by Alaoglu’s Theorem is
$w^{*}$
-compact,
$\sigma _{\lambda _{n}}\overset {w^{*}}{\to }\sigma _{\lambda _{0}}$
. Given that the sequence
$\lambda _{n}\to \lambda _{0}$
was arbitrary, the conclusion follows. ▪
Remark 4.2 Similarly, it can be shown that the equilibrium measure (\,for the logarithmic capacity) of a compact plane set is
$w^{*}$
-continuous under a holomorphic motion of the compact set.
5 Uniformly Perfect Sets
In this section, we will study the change of the quantity
$P(\cdot )$
, measuring the thickness of a uniformly perfect compact set, under holomorphic motions. Our estimates use the Harnack distance, which is defined as follows.
Let
$D\subset {\mathbb {C}}$
be a domain and let
$z,w\in D$
. The Harnack distance on D between z and w is the smallest number
$\tau _{D}(z,w)$
such that
$$\begin{align*}\frac{h(w)}{\tau_{D}(z,w)}\leq h(z)\leq\tau_{D}(z,w)h(w),\end{align*}$$
for every positive harmonic function h on D. Moreover, if D is a bounded domain, then
$\log \tau _{D}(z,w)$
is a complete metric on D. See [Reference Ransford17, pp. 14–21] for more information about the Harnack distance.
Theorem 5.1 Let
$K\subset {\mathbb {C}}$
be a uniformly perfect compact set and let f be a holomorphic motion of
${\mathbb {C}}$
parameterized by a bounded domain
$D\subset {\mathbb {C}}$
containing 0. Then
$$ \begin{align} \frac{P(K)}{\tau_{D}(\lambda,0)}\leq P(\,f_{\lambda}(K))\leq\tau_{D}(\lambda,0)P(K), \end{align} $$
for every
$\lambda \in D$
.
Proof Let
$(E,F)\in R(K)$
such that
$I(E,F)<+\infty $
. Let
$\nu =\nu _{E}-\nu _{F}$
be any signed measure in
$S(E,F)$
having finite energy. We note that, due to the singularity of the logarithmic kernel, we must have
$\nu \times \nu ((E\cup F)^{2}\backslash A(E,F))=0$
, where
$$\begin{align*}A(E,F):= \{(z,w)\in (E\cup F)^{2}:z\neq w\}.\end{align*}$$
For every
$\lambda \in D$
, let
$\nu _{\lambda }:=\nu _{E}\circ f_{\lambda }^{-1}-\nu _{F}\circ f_{\lambda }^{-1}$
. Since the injective functions
$f_{\lambda }$
are continuous and therefore Borel measurable on
$E\cup F$
by the
$\lambda $
-lemma,
$\nu _{\lambda }\in S(\,f_{\lambda }(E),f_{\lambda }(F))$
,
$\lambda \in D$
, and
$$\begin{align*}I(\nu_{\lambda})=\iint_{A(E,F)}\log\frac{1}{|\,f_{\lambda}(z)-f_{\lambda}(w)|}d(\nu\times\nu)(z,w). \end{align*}$$
Since
$f_{\lambda }(z)$
is a holomorphic function of
$\lambda $
and an injective function of z, we obtain that the function
$\lambda \mapsto \log [1/(|\,f_{\lambda }(z)-f_{\lambda }(w)|)]$
is harmonic on D, whenever
$(z,w)\in A(E,F)$
. It follows (see e.g., [Reference Helms8, p. 16]) that
$u_{\nu }(\lambda )=I(\nu _{\lambda })$
is a positive harmonic function on D. Therefore,
$$ \begin{align} \frac{I(\nu)}{\tau_{D}(\lambda,0)}\leq I(\nu_{\lambda})\leq\tau_{D}(\lambda,0)I(\nu),\qquad \lambda\in D, \end{align} $$
where
$\tau _{D}(\lambda ,0)>1$
, for all
$\lambda \in D\backslash \{0\}$
, since the domain D is assumed to be bounded. Noting that
$S(\,f_{\lambda }(E),f_{\lambda }(F))=\{\nu _{\lambda }:\nu \in S(E,F)\}$
and taking the infimum in (5.2) over all
$\nu \in S(E,F)$
having finite energy, we obtain that
$$ \begin{align} \frac{I(E,F)}{\tau_{D}(\lambda,0)}\leq I(\,f_{\lambda}(E),f_{\lambda}(F))\leq\tau_{D}(\lambda,0)I(E,F),\qquad \lambda\in D. \end{align} $$
Also,
$R(\,f_{\lambda }(K))=\{(\,f_{\lambda }(E),f_{\lambda }(F)):(E,F)\in R(K)\}$
. We conclude that (5.1) follows by taking the supremum in (5.3) over all
$(E,F)\in R(K)$
.▪
In the case that the parameterizing domain of the holomorphic motion is the unit disc, a precise estimate follows from Theorem 5.1.
Corollary 5.2 Let
$K\subset {\mathbb {C}}$
be a uniformly perfect compact set and let f be a holomorphic motion of K parameterized by the unit disc
${\mathbb {D}}$
. Then
$$ \begin{align} \frac{1-|\lambda|}{1+|\lambda|} P(K)\leq P(\,f_{\lambda}(K))\leq\frac{1+|\lambda|}{1-|\lambda|} P(K), \end{align} $$
for every
$\lambda \in {\mathbb {D}}$
.
Proof From Słodkowski’s Theorem [Reference Słodkowski21], f can be extended to a holomorphic motion of the Riemann sphere parameterized by
${\mathbb {D}}$
. The conclusion follows from Theorem 5.1 and the formula (see [Reference Ransford17, p. 14])
$$\begin{align*}\tau_{{\mathbb{D}}}(z,0)=\frac{1+|z|}{1-|z|},\qquad z\in{\mathbb{D}},\end{align*}$$
of the Harnack distance for
${\mathbb {D}}$
.▪
Acknowledgment
The author would like to thank Professor Paul M. Gauthier and the anonymous referees for carefully reading the paper and for providing suggestions that improved it.
