#  Rigidity theorems by capacities and kernels

Robert Xin Dong, John N. Treuer and Yuan Zhang

###### Abstract

For any open hyperbolic Riemann surface $X$, the Bergman kernel $K$, the
logarithmic capacity $c_{\beta}$, and the analytic capacity $c_{B}$ satisfy
the inequality chain $\pi K\geq c^{2}_{\beta}\geq c^{2}_{B}$; moreover,
equality holds at a single point between any two of the three quantities if
and only if $X$ is biholomorphic to a disk possibly less a relatively closed
polar set. We extend the inequality chain by showing that $c_{B}^{2}\geq\pi
v^{-1}(X)$ on planar domains, where $v(\cdot)$ is the Euclidean volume, and
characterize the extremal cases when equality holds at one point. Similar
rigidity theorems concerning the Szegö kernel, the higher-order Bergman
kernels, and the sublevel sets of the Green’s function are also developed.
Additionally, we explore rigidity phenomena related to the multi-dimensional
Suita conjecture.

††footnotetext:  Keywords. analytic capacity, Bergman kernel, Green’s
function, logarithmic capacity, open Riemann surface, Suita conjecture, Szegö
kernel Mathematics Subject Classification. Primary 32A25; Secondary 30C40,
31A15, 30C85, 32D15

##  1 Introduction

An open Riemann surface $X$ is said to be (potential-theoretically) hyperbolic
if it admits a non-constant negative subharmonic function, and parabolic if it
does not. Consider the on-diagonal Bergman kernel $K(z)|dz|^{2}$, the
logarithmic capacity $c_{\beta}(z)|dz|$, and the analytic capacity
$c_{B}(z)|dz|$ on $X$, where $z$ is some local coordinate. These three
quantities are invariant under changes of local coordinates. In 1972, Suita
determined a simple relationship between $K$ and $c_{B}$.

######  Theorem 1.1 (Suita’s Theorem in [33]).

Suppose $X$ is an open hyperbolic Riemann surface. Then for $z\in X$,

$\pi K(z)\geq c_{B}^{2}(z)$

and equality holds at some $z_{0}\in X$ if and only if $X$ is
biholomorphically equivalent to the unit disk less a (possibly empty) closed
set of inner capacity zero.

A closed set of inner capacity zero is a relatively closed polar set. In the
same paper, Suita conjectured that $c_{\beta}$ would satisfy a similar
inequality with rigidity as follows.

Suita Conjecture [33]: Suppose $X$ is an open hyperbolic Riemann surface. Then
for $z\in X$,

$\pi K(z)\geq c_{\beta}^{2}(z)$

and equality holds at some $z_{0}\in X$ if and only if $X$ is
biholomorphically equivalent to the unit disk less a (possibly empty) closed
set of inner capacity zero.

Towards the resolution of the Suita Conjecture, Ohsawa first demonstrated in
[27] that the Suita Conjecture was connected to his Ohsawa-Takegoshi $L^{2}$
extension theorem, and he proved that $750\pi K\geq c_{\beta}^{2}$. By further
considering the sharp $L^{2}$ extension problem, Błocki [6] established the
optimal inequality $\pi K\geq c_{\beta}^{2}$ for bounded domains in
$\mathbb{C}$. Later, Guan and Zhou [21] proved both the inequality and
equality parts of the conjecture for open Riemann surfaces. See also [5] for a
variational approach by Berndtsson and Lempert. Suita’s theorem and the
resolution of the conjecture may be called rigidity theorems as equality at
one point between $\pi K$ and either $c_{\beta}$ or $c_{B}$ determines the
surface up to biholomorphism.

It is straightforward to show $c_{\beta}^{2}\geq c_{B}^{2}$, so by the works
of Błocki, and Guan and Zhou, for any open hyperbolic Riemann surface it holds
that

$\pi K\geq c_{\beta}^{2}\geq c_{B}^{2}.$ (1.1)

The characterization of the equality part for the inequality $c_{\beta}\geq
c_{B}$ can be deduced from a result of Minda [26] on the behavior of
$c_{\beta}(z)|dz|$ under holomorphic mappings. His result used the sharp form
of the Lindelöf principle. In Appendix A, we provide a more direct proof of
the equality characterization without adopting this principle.

We first restrict our attention to domains in $\mathbb{C}$. Throughout this
paper, denote a disk by
$\mathbb{D}(z_{0},r):=\\{z\in\mathbb{C}:|z-z_{0}|<r\\}$ and let
$\mathbb{D}:=\mathbb{D}(0,1).$ For the Euclidean volume $v(\cdot)$, we use the
convention that $v^{-1}(\Omega)=0$ when $v(\Omega)=\infty$. Following Ahlfors
[3], a compact set $E\subset\mathbb{C}_{\infty}=\mathbb{C}\cup\\{\infty\\}$ is
said to be a null set of class $\mathcal{N}_{B}$ if all bounded holomorphic
functions on $\mathbb{C}_{\infty}\setminus E$ are constants. Compact polar
sets are always of class $\mathcal{N}_{B}$.

######  Theorem 1.2 (Rigidity theorem of $c_{B}$ and $c_{\beta}$).

Let $\Omega\subset\mathbb{C}$ be a domain. Then

$c_{B}^{2}(z)\geq{\pi\over v(\Omega)},\quad z\in\Omega.$ (1.2)

Moreover, equality holds at some $z_{0}\in\Omega$ if and only if either of the
following holds true:

  1. $1.$

$v(\Omega)<\infty$ and
$\Omega={\mathbb{D}}(z_{0},\sqrt{\pi^{-1}v(\Omega)})\setminus P$, where $P$
satisfies

$P\cap\overline{\mathbb{D}(z_{0},s)}\in\mathcal{N}_{B},\quad\hbox{for all
}s<\sqrt{\pi^{-1}v(\Omega)};$

if additionally

$c_{\beta}^{2}(z_{0})={\pi\over v(\Omega)},$

then $P$ above is a relatively closed polar set;

  2. $2.$

$v(\Omega)=\infty$ and $\Omega=\mathbb{C}\setminus P$ where
$P\in\mathcal{N}_{B}$.

It is worth pointing out that, as demonstrated in Remark 2.6, the equality
condition in (1.2) does not necessarily imply that $P$ is polar. See also
Theorem 2.4 and Lemma 2.5 for rigidity theorems with weaker assumptions than
(1.2).

Theorem 1.2 extends the chain of inequalities (1.1) to

$\pi K\geq c_{\beta}^{2}\geq c_{B}^{2}\geq\frac{\pi}{v(\Omega)},$ (1.3)

and essentially gives the equality conditions between $\pi v^{-1}(\Omega)$ and
the other quantities in (1.3) for a hyperbolic domain
$\Omega\subset\mathbb{C}$. As a corollary (see Corollary 3.1), we can reprove
the main theorem of [15], a rigidity theorem which established the equality
conditions of $K(z)\geq v^{-1}(\Omega)$. Moreover, let
$K^{(j)},j\in\mathbb{N}$, denote the higher-order Bergman kernels of a domain
$\Omega\subset\mathbb{C}$. Combining Theorem 1.2 with a result of Błocki and
Zwonek in [8], we get the inequalities

$K^{(j)}(z)\geq\frac{j!(j+1)!\pi^{j}}{v^{j+1}(\Omega)},\quad z\in\Omega,\quad
j\in\mathbb{N},$

and the following characterization of their equality conditions.

######  Corollary 1.3 (Rigidity theorem of higher-order Bergman kernels).

Let $\Omega$ be a domain in $\mathbb{C}$. Then for each $j\in\mathbb{N}$,
there exists a $z_{0}\in\Omega$ such that

$K^{(j)}(z_{0})=\frac{j!(j+1)!\pi^{j}}{v^{j+1}(\Omega)}$

if and only if either of the following holds true:

  1. $1.$

$v(\Omega)<\infty$ and
$\Omega=\mathbb{D}(z_{0},\sqrt{\pi^{-1}v(\Omega)})\setminus P$, where $P$ is a
relatively closed polar set;

  2. $2.$

$v(\Omega)=\infty$ and $\Omega=\mathbb{C}\setminus P$, where $P$ is a closed
polar set.

A classical question ([32, p. 20]) raised by Stein is what are the relations
between the Bergman kernel $K$ and the Szegö kernel $S$? On any bounded domain
$\Omega$ in $\mathbb{C}$ with Lipschitz boundary (see [4, 19] for the
$C^{\infty}$ smooth case and Proposition 3.6 for the Lipschitz case), the
Szegö kernel $S$ and the analytic capacity $c_{B}$ satisfy the identity
$c_{B}(z)=2\pi S(z)$. Using this and (1.3), we deduce the following inequality
chain, which gives a relation between the on-diagonal Bergman and Szegö
kernels:

$\sqrt{\pi K(z)}\geq c_{\beta}(z)\geq 2\pi
S(z)\geq\sqrt{\frac{\pi}{v(\Omega)}}\geq\frac{2\pi}{\sigma(\partial\Omega)},\quad
z\in\Omega.$ (1.4)

Here $\sigma(\partial\Omega)$ denotes the arc-length of $\partial\Omega$, and
the last inequality in (1.4) is the isoperimetric inequality. In particular,
the inequalities say that $K$ always dominates $4\pi S^{2}$ on Lipschitz
domains. See also [11] and the references therein for results on the
comparison of these two kernels. We characterize the equality conditions in
(1.4) as below.

######  Theorem 1.4 (Rigidity theorem of the Szegö kernel).

Suppose $\Omega\subset\mathbb{C}$ is a bounded domain with Lipschitz boundary.
Then,

  1. $1.$

there exists some $z_{0}\in\Omega$ such that

$2\pi S(z_{0})=\quad c_{\beta}(z_{0})\quad\text{or}\quad\sqrt{\pi K(z_{0})}$

if and only if $\Omega$ is simply connected;

  2. $2.$

there exists some $z_{0}\in\Omega$ such that

$2\pi
S(z_{0})=\quad\sqrt{\frac{\pi}{v(\Omega)}}\quad\text{or}\quad\frac{2\pi}{\sigma(\partial\Omega)}\quad\text{or}\quad\frac{1}{\delta(z_{0})}$

if and only if $\Omega=\mathbb{D}(z_{0},\sqrt{\pi^{-1}v(\Omega)})$.

Next, by a monotonicity result of Błocki and Zwonek in [7] concerning the
Euclidean volume of the sublevel sets of the Green’s function, for
$z\in\Omega,-\infty<t_{1}<t_{2}<0$,

$c_{\beta}^{2}(z)\geq\frac{\pi
e^{2{t_{1}}}}{v{(\\{G(\cdot,z)<t_{1}\\})}}\geq\frac{\pi
e^{2{t_{2}}}}{v{(\\{G(\cdot,z)<t_{2}\\})}}\geq\,\frac{\pi}{v(\Omega)}.$ (1.5)

See Remark 4.2. Using the isoperimetric inequality and the classical PDE
theory on the unique continuation property for harmonic functions, we obtain
the following rigidity theorem which examines the extremal cases and
characterizes domains on which equalities in (1.5) hold.

######  Theorem 1.5 (Rigidity theorem of sublevel sets of Green’s function).

Let $\Omega$ be a bounded domain in $\mathbb{C}$. Then $\Omega$ is a disk
centered at $z_{0}$ possibly less a relatively closed polar subset if and only
if either of the following holds true:

  1. $1.$

$c_{\beta}^{2}(z_{0})=\frac{\pi
e^{2{t_{0}}}}{v{(\\{G(\cdot,z_{0})<t_{0}\\})}}$

for some $z_{0}\in\Omega$, and $t_{0}\in(-\infty,0)$;

  2. $2.$

$\frac{\pi e^{2{t_{1}}}}{v{(\\{G(\cdot,z_{0})<t_{1}\\})}}=\frac{\pi
e^{2{t_{2}}}}{v{(\\{G(\cdot,z_{0})<t_{2}\\})}}$

for some $z_{0}\in\Omega$, and $t_{1}\neq t_{2}$ in $(-\infty,0)$;

  3. $3.$

$\frac{\pi
e^{2{t_{0}}}}{v{(\\{G(\cdot,z_{0})<t_{0}\\})}}=\,\frac{\pi}{v(\Omega)}$

for some $z_{0}\in\Omega$, and $t_{0}\in(-\infty,0)$.

At last, we study rigidity properties for bounded domains in
$\mathbb{C}^{n},n\geq 1$. Błocki and Zwonek [7] proved the following multi-
dimensional Suita Conjecture concerning the Bergman kernel and the Azukawa
indicatrix: for any pseudoconvex domain $\Omega\subset\mathbb{C}^{n}$,

$K(z)\geq{{v^{-1}(I^{A}(z))}},\quad z\in\Omega.$

Here $I^{A}$ denotes the Azukawa indicatrix (see (5.2) for its definition). As
the proof relied on an approximation of the domain by hyperconvex sub-domains,
the pseudoconvexity was needed in [7]. For our final result, by considering
the connection between the two involved quantities in the multi-dimensional
Suita Conjecture and the Euclidean distance function $\delta(z)$, we derive
the following rigidity theorem without requiring the domains in
$\mathbb{C}^{n}$ to be pseudoconvex when $n>1$.

######  Theorem 1.6.

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$, $n\geq 1$. Then for all
$z\in\Omega$,

$v(I^{A}(z))\geq\frac{\pi^{n}}{n!}\delta^{2n}(z),$ (1.6)

and

$K(z)\leq\frac{n!}{\pi^{n}\delta^{2n}(z)}.$ (1.7)

Moreover, either equality holds at some $z_{0}\in\Omega$ if and only if
$\Omega$ is a ball centered at $z_{0}$.

As a consequence of (1.7), we may add the distance function $\delta$ to the
inequality chains (1.3), (1.4), and (1.5) on bounded domains in $\mathbb{C}$.
See Remark 5.2 for more details.

The organization of the paper is as follows. In Section 2 we prove a rigidity
theorem of $c_{B}$ and $c_{\beta}$, and as applications, in Section 3 we prove
rigidity theorems of the Bergman kernels, and new results on the Szegö kernel
on Lipschitz domains. In Section 4, we investigate rigidity phenomena for the
logarithmic capacity and the sublevel sets of the Green’s function. In Section
5, for bounded domains in $\mathbb{C}^{n}$ we prove Theorem 1.6. In the
appendices, we prove a rigidity phenomenon between $c_{B}$ and $c_{\beta}$
based on a rigidity lemma of the Green’s function, and a stability result of
the Szegö kernel on Lipschitz domains.

##  2 Rigidity Theorem of $c_{B}$ and $c_{\beta}$

Let $SH^{-}(X)$ denote the set of negative subharmonic functions on an open
Riemann surface $X$. Given $z_{0}\in X$, let $w$ be a fixed local coordinate
in a neighbourhood of $z_{0}$ such that $w(z_{0})=0$. The (negative) Green’s
function is

$G(z,z_{0})=\sup\\{u(z):u\in SH^{-}(X),\limsup_{z\to
z_{0}}u(z)-\log|w(z)|<\infty\\}.$ (2.1)

An open Riemann surface admits a Green’s function if and only if there is a
non-constant, negative subharmonic function defined on it. The Green’s
function is strictly negative on the surface and harmonic except on the
diagonal.

We say that a Borel set $P$ is polar if there is a subharmonic function
$u\not\equiv-\infty$ defined in a neighbourhood of $P$ so that
$P\subset\\{z:u(z)=-\infty\\}$. For a domain $\Omega\subset\mathbb{C}$, a
point $\zeta_{0}\in\partial\Omega$ is said to be irregular if there is a
$\zeta\in\Omega$ such that $\lim_{z\to\zeta_{0}}G(z,\zeta)\neq 0$ and regular
otherwise. By Kellogg’s Theorem, cf. [29, Theorem 4.2.5], the set of irregular
boundary points of a domain is a polar set. The Green’s function on a Riemann
surface induces the logarithmic capacity, which is used prominently (see [29]
for its applications) in potential theory. Ahlfors introduced the analytic
capacity for domains [2, 3] in order to study Painlevé’s question: which
compact sets $E$ in the complex plane are removable for the bounded
holomorphic functions?

######  Definition 2.1.

Let $X$ be an open hyperbolic Riemann surface and $z_{0}\in X$. The
logarithmic capacity $c_{\beta}$ is defined as

$c_{\beta}(z_{0})=\lim_{z\to z_{0}}\exp(G(z,z_{0})-\log|w(z)|).$

The analytic capacity of a Riemann surface $X$ is defined as

$c_{B}(z_{0})=\sup\left\\{\left|{\partial f\over\partial
w}(z_{0})\right|:f\in\hbox{Hol}(X,\mathbb{D}),f(z_{0})=0\right\\}.$ (2.2)

When we wish to emphasize the surface $X$, we will use the notations $G_{X}$
and $c_{m;X}(\cdot)$, $m=\beta\text{ or }B$. If $h:X_{1}\to X_{2}$ is a
biholomorphism, then for $z\in X_{1}$,

$c_{m;X_{1}}(z)=|h^{\prime}(z)|c_{m;X_{2}}(h(z)),\quad m=\beta\text{ or }B.$

Thus, the logarithmic and analytic capacity are independent of the choice of
the local coordinates and define conformally-invariant metrics
$c_{\beta}(z)|dz|$ and $c_{B}(z)|dz|$. For each $z_{0}\in X$, there exists an
extremal function $f_{0}$ for the analytic capacity, unique up to rotation;
that is, $f_{0}$ belongs to the family described in (2.2) and
$\left|{df_{0}\over dw}(z_{0})\right|=c_{B}(z_{0})$.

We restrict our attention to domains in $\mathbb{C}$. With this restriction we
will be able to examine the relationship between the domain functions
$c_{\beta},c_{B}$ and the Euclidean volume of the domain. In the literature,
the analytic capacity is often defined in terms of compact sets.

######  Definition 2.2.

[19] The analytic capacity $\gamma(E)$ of a compact subset
$E\subset\mathbb{C}$ is

$\gamma(E)=\sup\\{|g^{\prime}(\infty)|:g\in\hbox{Hol}(\mathbb{C}_{\infty}\setminus
E,\mathbb{D}),g(\infty)=0\\}$

where

$g^{\prime}(\infty)=\lim_{z\to\infty}z(g(z)-g(\infty)).$

######  Notation.

In this section, $\Omega-\\{z_{0}\\}$ will denote the translation of the
domain $\Omega$ by $z_{0}$ and

$j_{z_{0}}(z):={1\over z-z_{0}}.$

For ease of notation, when $z_{0}=0$, let $j=j_{z_{0}}$.

The two definitions of the analytic capacities presented in this paper are
related as follows.

######  Lemma 2.3.

For any domain $\Omega\subset\mathbb{C}$ and $z_{0}\in\Omega$,
$c_{B}(z_{0})=\gamma(\mathbb{C}_{\infty}\setminus j_{z_{0}}(\Omega)).$

###### Proof.

Since $j(\Omega-\\{z_{0}\\})=j_{z_{0}}(\Omega)$ and
$c_{B;\Omega}(z_{0})=c_{B;\Omega-\\{z_{0}\\}}(0),$ it suffices to prove the
lemma when $z_{0}=0\in\Omega$. Let $E=\mathbb{C}_{\infty}\setminus j(\Omega)$.
Notice that there is a one-to-one correspondence between the two function sets
$\mathcal{A}:=\\{g\in\hbox{Hol}(\mathbb{C}_{\infty}\setminus
E,\mathbb{D}):g(\infty)=0\\}$ and
$\mathcal{B}:=\\{h\in\hbox{Hol}(\Omega,\mathbb{D}):h(0)=0\\}$, as $g\circ
j\in\mathcal{B}$ whenever $g\in\mathcal{A}$, and vice versa. Moreover,
$(g\circ j)^{\prime}(0)=g^{\prime}(\infty)$. The lemma then follows directly
from the definitions of the two analytic capacities.

∎

The analytic capacity is difficult to compute in general for most domains. It
does however satisfy a lower bound referred to as the Ahlfors-Beurling
Inequality (see [3], [19, Theorem 4.6, Chapter III]). Namely, for any compact
set $E\subset\mathbb{C}$,

$\gamma^{2}(E)\geq{v(E)\over\pi}.$ (2.3)

For any $z\in\Omega$, letting $E$ in (2.3) be $\mathbb{C}_{\infty}\setminus
j_{z}(\Omega)$, and combining with Lemma 2.3, we obtain

$c_{B}^{2}(z)\geq{v(\mathbb{C}_{\infty}\setminus j_{z}(\Omega))\over\pi}.$
(2.4)

The following serves as a rigidity theorem concerning (2.4).

######  Theorem 2.4.

Let $\Omega$ be a domain in $\mathbb{C}$ with $v(\Omega)<\infty$. There exists
a $z_{0}\in\Omega$ such that

$c_{B}^{2}(z_{0})={v(\mathbb{C}_{\infty}\setminus j_{z_{0}}(\Omega))\over\pi}$
(2.5)

if and only if $\Omega={\mathbb{D}}(z_{1},r)\setminus P$ for some
$z_{1}\in\mathbb{C},r>0$, where $P$ satisfies

$P\cap\overline{\mathbb{D}(z_{1},s)}\in\mathcal{N}_{B},\quad\hbox{for all
}s<r.$ (2.6)

If additionally

$c_{\beta}^{2}(z_{0})={v(\mathbb{C}_{\infty}\setminus
j_{z_{0}}(\Omega))\over\pi},$ (2.7)

then $P$ above is a relatively closed polar set.

###### Proof.

If $\Omega=\mathbb{D}(z_{1},r)\setminus Q$, where

$Q\cap\overline{\mathbb{D}(z_{1},s)}\in\mathcal{N}_{B},\quad\hbox{for all
}s<r,$

then by the classical Schwarz lemma,

$c_{B}(z_{0})=\frac{r}{r^{2}-|z_{0}-z_{1}|^{2}}.$ (2.8)

On the other hand, a direct computation gives

$v(\mathbb{C}_{\infty}\setminus j_{z_{0}}(\mathbb{D}(z_{1},r)))=\frac{\pi
r^{2}}{(r^{2}-|z_{0}-z_{1}|^{2})^{2}}.$

Equation (2.5) is proved.

For the other direction, let $E=\mathbb{C}_{\infty}\setminus
j_{z_{0}}(\Omega)$. Then $E$ is compact, and since $v(\Omega)<\infty$,
$v(E)>0$. Define $f\in\mathcal{O}(\Omega)\cap C(\overline{\Omega})$ by

$f(z)={1\over\sqrt{\pi v(E)}}\int_{E}{1\over w-{1\over z-z_{0}}}dv(w).$

Then $f(z_{0})=0$ and $f^{\prime}(z_{0})=-\sqrt{\pi^{-1}v(E)}$. The Ahlfors-
Beurling inequality states that $|f(z)|\leq 1$ for all
$z\in\mathbb{C}_{\infty}$. By the maximum modulus theorem, $|f(z)|<1$ on
$\Omega$. Thus, $f$ is an extremal function for $c_{B}$, which implies
$\sup_{z\in\Omega}|f(z)|=1$. By continuity there exists
$z_{2}\in\partial\Omega$ such that $|f(z_{2})|=1$. We observe, after a careful
inspection of the proof of the Ahlfors-Beurling inequality, as given in [29,
Lemma 5.3.6], that $E$ must be a union of a closed disk with a closed measure-
zero set. For completeness we resupply the proof since this observation is not
stated in the literature, cf. [3, 19, 29]. Indeed, after a rotation and
translation of $E$ we may assume that

$1=|f(z_{2})|={1\over\sqrt{\pi v(E)}}\int_{E}{1\over w}dv(w).$ (2.9)

Let $D:=\\{w\in\mathbb{C}:Rew^{-1}>(2a)^{-1}\\}$ be a disk such that
$v(D)=v(E)$. Then $v(E\setminus D)=v(D\setminus E)$ and so

$\int_{E\setminus D}Re\frac{1}{w}dv(w)\leq\int_{E\setminus
D}\frac{1}{2a}dv(w)=\int_{D\setminus E}\frac{1}{2a}dv(w)\leq\int_{D\setminus
E}Re\frac{1}{w}dv(w).$ (2.10)

This implies

$\begin{split}\int_{E}\frac{1}{w}dv(w)=\int_{E}Re\frac{1}{w}dv(w)=&\int_{E\cap
D}Re\frac{1}{w}dv(w)+\int_{E\setminus D}Re\frac{1}{w}dv(w)\\\ \leq&\int_{D\cap
E}Re\frac{1}{w}dv(w)+\int_{D\setminus E}Re\frac{1}{w}dv(w)\\\
=&\int_{D}Re\frac{1}{w}dv(w)\\\
=&\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{0}^{2a\cos\theta}\cos\theta
drd\theta=\pi a=\sqrt{\pi v(E)}.\end{split}$

Comparing with (2.9), both inequalities in (2.10) become equalities and thus
$v(E\setminus D)=v(D\setminus E)=0$. Since $E$ is compact,
$\overline{D}\subset E$. Consequently, $\Omega=\mathbb{D}(z_{1},r)\setminus P$
for some $z_{1}\in\mathbb{C}$ and $r=\sqrt{v(\Omega)\pi^{-1}}$, where $P$ is a
relatively closed set of measure 0.

To further show that $P$ satisfies (2.6), consider

$h(z)=\frac{r(z-z_{0})}{r^{2}-(\overline{z}_{0}-\overline{z}_{1})(z-z_{1})},\
\ z\in\Omega.$

It is not hard to verify that $h\in\hbox{Hol}(\Omega,\mathbb{D})$,
$h(z_{0})=0$ and

$h^{\prime}(z_{0})=\frac{r}{r^{2}-|z_{0}-z_{1}|^{2}}.$

Thus $h$ is an extremal function for $c_{B}$ at $z_{0}$, by (2.8). By [22,
Theorem 28], the image of the extremal function satisfies
$h(\Omega)=\mathbb{D}\setminus Q$ where

$Q=h(P),\quad Q\cap\overline{\mathbb{D}(0,r)}\in\mathcal{N}_{B},\quad\hbox{for
all }0\leq r<1.$

Thus (2.6) is proved.

If additionally (2.7) holds, then by Theorem A.1, $h$ is a biholomorphism from
$\Omega$ to $\mathbb{D}(0,1)\setminus Q$, where $Q$ is a relatively closed
polar set. This implies that $P$ is also a relatively closed polar set.
Conversely if $\Omega=\mathbb{D}(z_{1},r)\setminus P$ for a relatively closed
polar set $P$, then a direct computation shows
$c_{\beta}^{2}(z_{0})=\pi^{-1}{v(\mathbb{C}_{\infty}\setminus
j_{z_{0}}(\Omega))}$. The proof is complete.

∎

The proof of Theorem 2.4 indicates the center of $\Omega$ may not necessarily
be $z_{0}$, at which the equality (2.5) holds. Using the lemma below, we will
show that if the stronger equality $c_{B}^{2}(z_{0})=\pi v^{-1}(\Omega)$ holds
for $v(\Omega)<\infty$, then in the conclusion of the preceding theorem the
center of $\Omega$ must be $z_{0}$.

######  Lemma 2.5.

Let $\Omega\subset\mathbb{C}$ be a domain with $v(\Omega)<\infty$. Then for
all $z\in\Omega$,

${\pi\over v(\Omega)}\leq{v(\mathbb{C}_{\infty}\setminus
j_{z}(\Omega))\over\pi},$ (2.11)

and equality holds at some $z_{0}\in\Omega$ if and only if $\Omega$ is a disk
centered at $z_{0}$ less a relatively closed set of measure zero.

###### Proof.

If $B\subset\mathbb{C}$ is a set with $v(B)=0$, then $v(j_{z_{0}}(B))=0$. With
this fact it is straightforward to verify that equality holds for a disk less
a relatively closed set of measure 0. Since $v(\Omega)=v(\Omega-\\{z_{0}\\})$
and $j_{z_{0}}(\Omega)=j(\Omega-\\{z_{0}\\})$, without loss of generality we
may suppose $z_{0}=0$. Notice that for any $r>0$, $v(r\Omega)=r^{2}v(\Omega)$
and

$\displaystyle v(\mathbb{C}_{\infty}\setminus
j(r\Omega))=v(\mathbb{C}_{\infty}\setminus
r^{-1}j(\Omega))=v(r^{-1}(\mathbb{C}_{\infty}\setminus
j(\Omega)))=r^{-2}v(\mathbb{C}_{\infty}\setminus j(\Omega)).$

Here for any set $B\subset\mathbb{C}$, $rB:=\\{rz:z\in B\\}$. We may further
assume that $v(\Omega)=\pi$. So the inequality (2.11) is equivalent to

$v(j(\mathbb{C}_{\infty}\setminus\Omega))=v(\mathbb{C}_{\infty}\setminus
j(\Omega))\geq\pi.$

Set

$S_{1}={\mathbb{D}}\setminus\Omega,\quad S_{2}=\Omega\setminus{\mathbb{D}}.$

So $v(S_{1})=v(S_{2})$. Since
$\mathbb{C}_{\infty}\setminus\Omega=S_{1}\sqcup((\mathbb{C}_{\infty}\setminus{\mathbb{D}})\setminus
S_{2})$ and
$j(\mathbb{C}_{\infty}\setminus{\mathbb{D}})=\overline{\mathbb{D}}$,

$j(\mathbb{C}_{\infty}\setminus\Omega)=j(S_{1})\sqcup(\overline{\mathbb{D}}\setminus
j(S_{2})),$

where $\sqcup$ denotes the disjoint union. Noticing
$j(S_{2})\subset\overline{\mathbb{D}}$, we further have

$v(j(\mathbb{C}_{\infty}\setminus\Omega))-\pi=v(j(S_{1}))+v(\overline{\mathbb{D}})-v(j(S_{2}))-\pi=v(j(S_{1}))-v(j(S_{2})).$

Applying change of variables formula, one gets

$\begin{split}v(j(\mathbb{C}_{\infty}\setminus\Omega))-\pi=\int_{S_{1}}\frac{1}{|z|^{4}}dv(z)-\int_{S_{2}}\frac{1}{|z|^{4}}dv(z)\geq\int_{S_{1}}1dv(z)-\int_{S_{2}}1dv(z)=0.\end{split}$
(2.12)

Here we have used the fact that $|z|<1$ on $S_{1}$ and $|z|\geq 1$ on $S_{2}$.
This completes the proof of the inequality part.

If equality holds in (2.11), then the inequality in (2.12) becomes equality
and

$\int_{S_{1}}\frac{1}{|z|^{4}}dv(z)=\int_{S_{1}}1dv(z),\ \
\int_{S_{2}}\frac{1}{|z|^{4}}dv(z)=\int_{S_{2}}1dv(z).$

Since $|z|<1$ on $S_{1}$, the first equation implies $v(S_{1})=0$. Thus,
$v(S_{2})=0$. By definitions of $S_{1}$ and $S_{2}$, we know that $\Omega$ is
the unit disk centered at $0$ less a relatively closed set of measure zero.

∎

###### Proof of Theorem 1.2 (Rigidity theorem of $c_{B}$ and $c_{\beta}$).

If $v(\Omega)=\infty$, then the inequality is trivial. By [3, p. 107],
$c_{B}(z_{0})=0$ if and only if $c_{B}\equiv 0$, and thus if and only if
$\Omega=\mathbb{C}_{\infty}\setminus P$ where $P\in\mathcal{N}_{B}$ by
definition.

We now assume $v(\Omega)<\infty$. Equation (1.2) follows from (2.4) and
(2.11). If $\Omega=\mathbb{D}(z_{0},\sqrt{\pi^{-1}v(\Omega)})\setminus Q$,
where

$Q\cap\overline{\mathbb{D}(z_{0},s)}\in\mathcal{N}_{B},\quad\hbox{for all
}s<\sqrt{\pi^{-1}v(\Omega)},$

then by the classical Schwarz lemma, $c_{B}^{2}(z_{0})=\pi v(\Omega)^{-1}$.

Conversely, if equality holds at $z_{0}\in\Omega$, then

$c_{B}^{2}(z_{0})={v(\mathbb{C}_{\infty}\setminus
j_{z_{0}}(\Omega))\over\pi}={\pi\over v(\Omega)}.$

By the equality part of Lemma 2.5,
$\Omega=\mathbb{D}(z_{0},\sqrt{\pi^{-1}v(\Omega)})\setminus P$, where $P$ is a
relatively closed set of measure 0. By the equality part of Theorem 2.4, we
further conclude that $P\cap\overline{\mathbb{D}(z_{0},s)}\in\mathcal{N}_{B}$
for all $s<\sqrt{\pi^{-1}v(\Omega)}.$ The rest of the theorem follows from the
second part of Theorem 2.4.

∎

######  Remark 2.6.

If $c_{\beta}^{2}(z_{0})>c_{B}^{2}(z_{0})=\pi v^{-1}(\Omega)$, then $P$ in the
preceding theorem may not be polar. Indeed, let $Q$ be the compact four-corner
Cantor set defined in [20]. As shown therein, $Q\in\mathcal{N}_{B},$ but is
not polar. Let $\Omega=\mathbb{D}(z_{0},r)\setminus Q$ where $z_{0}$ and $r$
are chosen such that $z_{0}\not\in Q\subset\mathbb{D}(z_{0},r)$. Since
$Q\in\mathcal{N}_{B}$, all bounded holomorphic functions on $\Omega$ extend
across $Q$. Thus,

$c_{B;\Omega}(z_{0})=c_{B;\mathbb{D}(z_{0},r)}(z_{0})=\sqrt{\pi\over
v(\mathbb{D}(z_{0},r))}=\sqrt{\pi\over v(\Omega)},$

where the last equality used the fact that sets of class $\mathcal{N}_{B}$
have two-dimensional Lebesgue measure 0.

##  3 Applications of the Rigidity Theorem of $c_{B}$ and $c_{\beta}$

Let $\Omega$ be a domain in $\mathbb{C}^{n},n\geq 1$. The Bergman space of a
domain $\Omega$ is the Hilbert space

$A^{2}(\Omega)=L^{2}(\Omega)\cap\mathcal{O}(\Omega)$

with $L^{2}(\Omega)$-norm denoted by $\|\cdot\|_{\Omega}$. The Bergman kernel
is defined by

$K(z)=\sup\\{|f(z)|^{2}:f\in A^{2}(\Omega),\|f\|_{\Omega}\leq 1\\},\ \
z\in\Omega.$

By considering constant functions in the defining set of the kernel, we get

$K(z)\geq\frac{1}{v(\Omega)},\ z\in\Omega,$

which is sharp when $\Omega=\mathbb{D}(z_{0},r)\setminus P$, where $P$ is a
relatively closed polar set and $z=z_{0}$.

As an application of Theorem 1.2, we show that this sharp example is in fact
the only possible domain where the equality can be achieved. Corollary 3.1
below, which is the main result of [15], was originally proved by the first
and second named authors using the equality part of the Suita Conjecture. Here
we provide a new proof based on our Theorem 1.2 and Suita’s Theorem.

######  Corollary 3.1 (Originally proved in [15]).

Let $\Omega\subset\mathbb{C}$ be a domain. Then there exists a
$z_{0}\in\Omega$ such that

$K(z_{0})={1\over v(\Omega)},$ (3.1)

if and only if either of the following holds true:

  1. $1$.

$v(\Omega)<\infty$ and $\Omega=\mathbb{D}(z_{0},r)\setminus P$, where $P$ is a
relatively closed polar set and with $r^{2}=\pi^{-1}v(\Omega)$.

  2. $2$.

$v(\Omega)=\infty$ and $\Omega=\mathbb{C}\setminus P$, where $P$ is a possibly
empty, closed polar set.

###### Proof.

First assume $v(\Omega)<\infty$. By Suita’s Theorem (Theorem 1.1) and Theorem
1.2,

$\pi K(z_{0})\geq c_{B}^{2}(z_{0})\geq{\pi\over v(\Omega)}=\pi K(z_{0}).$

By the equality part of Theorem 1.2,

$\Omega=\mathbb{D}(z_{0},r)\setminus P,\quad
P\cap\mathbb{D}(z_{0},s)\in\mathcal{N}_{B},\quad
s<r=\sqrt{\pi^{-1}v(\Omega)}.$

The equality part of Suita’s Theorem (Theorem 1.1) implies additionally that
$P$ is polar. The case when $v(\Omega)=\infty$ is already known (see [8,
Theorem 4]).

∎

######  Remark 3.2.

The case $v(\Omega)<\infty$ in the preceding proof used Suita’s Theorem and
not the Suita Conjecture. The proof of Suita’s Theorem, which was proved using
Riemann surface theory, is much simpler than the proof of the Suita
Conjecture. Thus, the proof given here is simpler than the original proof in
[15].

######  Remark 3.3.

Boas [10] has kindly pointed out to us that Corollary 3.1 fails for domains in
$\mathbb{C}^{n},n\geq 2$, as indicated by the following examples. Let $\Omega$
be a domain in $\mathbb{C}^{n},n\geq 2$, satisfying (3.1) (for instance, a
ball centered at $z_{0}$), and consider images $F(\Omega)$ under maps of the
form $F:\mathbb{C}^{n}\to\mathbb{C}^{n}$ defined by

$F(z)=(z^{\prime},z_{n}+f_{0}(z^{\prime})),\quad
z^{\prime}=(z_{1},\ldots,z_{n-1}),$ (3.2)

where $f_{0}:\mathbb{C}^{n-1}\to\mathbb{C}$ is any holomorphic function. Such
maps are called shears and were considered extensively by Rosay and Rudin
[30]. $F$ is biholomorphic, and the determinant $\det J_{\mathbb{C}}F$ of its
complex Jacobian is constantly 1. Hence $F$ is volume preserving with
$v(F(\Omega))=v(\Omega)$. On the other hand, by the transformation rule of the
Bergman kernel, $K_{F(\Omega)}(F(z_{0}))=K_{\Omega}(z_{0})$, so (3.1) holds
for $F(\Omega)$ at $F(z_{0})$. Due to the arbitrariness of $f_{0}$, a large
degree of freedom is afforded to the geometries of such $F(\Omega)$, in stark
contrast to the situation in $\mathbb{C}$. Moreover, the domains $\Omega$ that
satisfy (3.1) include the bounded complete Reinhardt domains, bounded complete
circular domains, bounded quasi-circular domains containing the origin, and
bounded quasi-Reinhardt domains containing the origin [25], which form a
strictly increasing sequence under the set containment relation in
$\mathbb{C}^{n},n\geq 2$. By selecting $\Omega$ from one of these classes, we
can produce additional domains $F(\Omega)$ satisfying (3.1) that are not
biholomorphically equivalent to the ball. Lastly, for a bounded quasi-
Reinhardt $\Omega$ containing the origin, by choosing a non-polynomial mapping
$F$ in $\eqref{shear}$ such that $F(\Omega)$ is bounded and F(0) = 0, we may
get a minimal domain centered at 0 that is not quasi-Reinhardt (see [13, 25]).

For domains in $\mathbb{C}$, we also have the following more precise estimate
on the on-diagonal Bergman kernel. Recall that given $z\in\Omega$,
$j_{z}={1\over\cdot-z}$.

######  Corollary 3.4.

Let $\Omega\subset\mathbb{C}$ be a domain with $v(\Omega)<\infty$. Then for
all $z\in\Omega$,

$K(z)\geq{v(\mathbb{C}_{\infty}\setminus j_{z}(\Omega))\over\pi^{2}},$

and equality holds at some $z_{0}\in\Omega$ if and only if $\Omega$ is a disk
less a relatively closed polar set.

###### Proof.

By Suita’s Theorem (Theorem 1.1), the Ahlfors-Beurling Inequality (2.4), and
Theorem 2.4,

$\pi K(z)\geq c_{B}^{2}(z)\geq{v(\mathbb{C}_{\infty}\setminus
j_{z}(\Omega))\over\pi}$

for $z\in\Omega$. The equality part is a consequence of Theorems 1.1 and 2.4.

∎

For $j=1,2,\ldots$, let

$K^{(j)}(z)=\sup\\{|f^{(j)}(z)|^{2}:f\in A^{2}(\Omega),\|f\|_{\Omega}\leq
1,f^{(k)}(z)=0,k=0,\ldots,j-1\\}$

denote the Bergman kernels for higher order derivatives, and set $K^{(0)}=K$.
Błocki and Zwonek [8] established that for $z\in\Omega\subset\mathbb{C}$,

$K^{(j)}(z)\geq{j!(j+1)!\over\pi}(c_{\beta}(z))^{2j+2},$ (3.3)

which is sharp for a disk less a relatively closed polar set
$\mathbb{D}(z_{0},r)\setminus P$ with $z=z_{0}$.

###### Proof of Corollary 1.3 (Rigidity theorem of higher-order Bergman
kernels).

The case $v(\Omega)=\infty$ is already known, cf. [8, Theorem 4]. For
$v(\Omega)<\infty$, this follows from (3.3) and Theorem 1.2.

∎

One important property of the analytic capacity $c_{B}$ is that it is distance
decreasing with respect to any given holomorphic map $f:X\to Y$, where $X,Y$
are hyperbolic Riemann surfaces, cf. [23, Chapter 2]:

$f^{*}\left(c_{B;Y}(z)|dz|\right)\leq c_{B;X}(z)|dz|,$ (3.4)

where $f^{*}\left(c_{B;Y}(z)|dz|\right)$ denotes the pull-back to $X$ via $f$
of the analytic capacity on $Y$.

######  Remark 3.5.

For a hyperbolic Riemann surface $X$ with a local coordinate $z$, if there
exists $z_{0}\in X$ and a non-constant holomorphic function $f:X\to\mathbb{D}$
such that

$K(z_{0})|dz|^{2}=\frac{|df(z_{0})|^{2}}{\pi(1-|f(z_{0})|^{2})^{2}},$ (3.5)

then by (1.1) and (3.4), we have

$\sqrt{\pi K(z_{0})}|dz|\geq c_{B;X}(z_{0})|dz|\geq
f^{*}\left(c_{B;\mathbb{D}}(z_{0})|dz|\right)={|df(z_{0})|\over
1-|f(z_{0})|^{2}}.$

Without loss of generality, assume that $df(z_{0})>0$. Therefore, (3.5) forces
the equality condition in Theorem 1.1 to hold true, and there exists a
biholomorphism $h:X\to\mathbb{D}\setminus P$ such that $h(z_{0})=0$ and
$\frac{\partial h}{\partial z}(z_{0})>0$, where $P$ is a relatively closed
polar subset. By the transformation rule of the Bergman kernel, $\sqrt{\pi
K(z_{0})}=\frac{\partial h}{\partial z}(z_{0}).$ Take
$\varphi(z):=\frac{z-f(z_{0})}{1-\overline{f(z_{0})}z}\in\text{Aut}(\mathbb{D}).$
Then $\varphi\circ f\circ h^{-1}$ is bounded, so it extends to a holomorphic
function $F:\mathbb{D}\to\mathbb{D}$ such that $F(0)=0$. Moreover,
$F^{\prime}(0)=\sqrt{\pi K(z_{0})}^{-1}\cdot\sqrt{\pi
K(z_{0})}(1-|f(z_{0})|^{2})\cdot(1-|f(z_{0})|^{2})^{-1}=1$. The Schwarz lemma
implies that $F(z)\equiv z$ on $\mathbb{D}$, so $f$ is in fact biholomorphic.
Consequently, the identity (3.5) extends to all of $X$, and $f$ is a
holomorphic isometry with respect to the Bergman kernel.

In the remaining part of the section, we focus on a bounded domain $\Omega$ in
$\mathbb{C}$ with Lipschitz boundary. The Hardy space $H^{2}(\partial\Omega)$
is defined to be the set of holomorphic functions on $\Omega$ which admit a
non-tangential boundary limit function and a maximal function on the boundary
which belong to $L^{2}(\partial\Omega)$. See Appendix B or [24] for more
details. Let $\|\cdot\|_{\partial\Omega}$ denote the $L^{2}(\partial\Omega)$
norm and $S(\cdot,\cdot)$ denote the Szegö kernel. When restricted to the
diagonal, it satisfies

$S(z,z):=S(z)=\sup\\{|f(z)|^{2}:f\in
H^{2}(\partial\Omega),\|f\|_{\partial\Omega}\leq 1\\}.$

By considering constant functions in the defining set of the kernel, for any
$z\in\Omega$, we get

$S(z)\geq{1\over\sigma(\partial\Omega)},$

where $\sigma(\partial\Omega)$ denotes the arclength of $\partial\Omega$. The
lower bound is sharp for $\Omega=\mathbb{D}(z_{0},r)$ and $z=z_{0}$. As
applications of the Rigidity theorem of $c_{B}$ and $c_{\beta}$, Theorem 1.2,
we will show that these are the only possible domains where the equality can
be achieved.

Let $\\{\Omega_{j}\\}_{j=1}^{\infty}$ be a family of exhausting subdomains of
$\Omega$, and $S_{j}(\cdot,\cdot)$ be the corresponding Szegö kernels. Boas
showed in [9, Theorem 2.1] that if in addition $\Omega$ has
$C^{\infty}$-smooth boundary and is exhausted by sublevel sets $\Omega_{j}$ of
its defining function, then for $a,z\in\Omega_{j}$,

$\lim_{j\to\infty}S_{j}(z,a)=S(z,a).$

For a bounded domain $\Omega\subset\mathbb{C}$ with Lipschitz boundary, there
are subdomains $\\{\Omega_{j}\\}_{j=1}^{\infty}$ with $C^{\infty}$ boundary
that approximate $\Omega$ well uniformly and non-tangentially in the sense of
Nečas (see [24, p. 539] or Appendix B for more details). In Proposition B.3,
we extend Boas’ stability result of the Szegö kernel to Lipschitz domains in
$\mathbb{C}$ with respect to the Nečas approximation.

It is known for a finitely connected domain with $C^{\infty}$ boundary that
the analytic capacity and (on-diagonal) Szegö kernel satisfy the relation
$c_{B}(z)=2\pi S(z)$. For $\\{\Omega_{j}\\}_{j=1}^{\infty}$ given above, as
shown by Ahlfors and Beurling [3, Theorem 1], the analytic capacities
$c_{j;B}$ and $c_{B}$ of these domains, respectively, similarly satisfy

$\lim_{j\to\infty}c_{j;B}(z)=c_{B}(z),\quad z\in\Omega.$

Consequently, we can conclude

######  Proposition 3.6.

If $\Omega\subset\mathbb{C}$ is bounded with Lipschitz boundary, then

$c_{B}(z)=2\pi S(z),\quad z\in\Omega.$ (3.6)

The Rigidity theorem of $c_{B}$ and $c_{\beta}$, Theorem 1.2, together with
Proposition 3.6, enables us to prove the rigidity phenomenon of the Szegö
kernel.

###### Proof of Theorem 1.4 (Rigidity theorem of the Szegö kernel).

The equality conditions in Suita’s Theorem (Theorem 1.1) and in $c_{\beta}\geq
c_{B}$ (Theorem A.1) imply that the equality condition in Case 1 holds if and
only if $\Omega$ is biholomorphic to a disk less a relatively closed polar
set. Since $\Omega$ is Lipschitz, it is a regular domain for the Dirichlet
problem. Thus, the polar set is empty, cf. [29].

For Case 2, it suffices to prove the ‘only if’ direction. By the Rigidity
theorem of $c_{B}$ and $c_{\beta}$ (Theorem 1.2), $2\pi S(z_{0})=\sqrt{\pi
v^{-1}(\Omega)}$ if and only if

$\Omega=\mathbb{D}(z_{0},r)\setminus Q,\quad
Q\cap\overline{\mathbb{D}(z_{0},s)}\in\mathcal{N}_{B},\quad\hbox{for all
}s<r=\sqrt{\pi^{-1}v(\Omega)}.$

Since $\Omega$ has Lipschitz boundary, its boundary has no singleton connected
components. Thus $Q=\emptyset.$ The equality case $2\pi
S(z_{0})=2\pi\sigma^{-1}(\partial\Omega)$ follows now as well.

By (3.6), $2\pi S(z_{0})=\delta^{-1}(z_{0})$ is equivalent to
$c_{B}(z_{0})=\delta^{-1}(z_{0})$. Let $f_{0}$ be an extremal function for
$c_{B}(z_{0})$. By the Schwarz lemma,
$f_{0}|_{\mathbb{D}(z_{0},\delta(z_{0}))}:\mathbb{D}(z_{0},\delta(z_{0}))\to\mathbb{D}$
is extremal for $\mathbb{D}(z_{0},\delta(z_{0}))$, and thus equals
$e^{i\theta}\delta^{-1}(z_{0})(z-z_{0})$ for some $\theta\in[0,2\pi)$. Since
$|f_{0}|<1$, $\Omega=\mathbb{D}(z_{0},\delta(z_{0})).$

∎

Theorem 2.4 combined with Proposition 3.6 also gives another rigidity result
concerning the Szegö kernel below.

######  Corollary 3.7.

Suppose $\Omega\subset\mathbb{C}$ is a bounded domain with Lipschitz boundary.
Then for $z\in\Omega$,

$S^{2}(z)\geq{v(\mathbb{C}_{\infty}\setminus j_{z}(\Omega))\over 4\pi^{3}}.$

Moreover, equality holds at some $z_{0}\in\Omega$ if and only if $\Omega$ is a
disk.

######  Remark 3.8.

In comparison with (1.4) and Theorem 1.4, concerning the inequality between
the on-diagonal Bergman and Szegö kernels and the equality conditions, we
would like to point out that when the domain is bounded and simply-connected
with $C^{\infty}$-boundary,

$K(z,a)=4\pi S^{2}(z,a),\quad z,a\in\Omega,$

cf. [4, Theorem 25.1].

######  Remark 3.9.

Let $f:\Omega_{1}\to\Omega_{2}$ be a holomorphic map, where
$\Omega_{1},\Omega_{2}\subset\mathbb{C}$ are bounded domains with Lipschitz
boundaries. Then, by Proposition 3.6 and (3.4), the Szegö kernel is decreasing
with respect to $f$, namely

$|f^{\prime}(z)|S_{\Omega_{2}}(f(z))\leq S_{\Omega_{1}}(z),$

and equality holds if $f$ is a biholomorphism.

##  4 Rigidity of Sublevel Sets of Green’s Function

For a fixed $z_{0}\in\Omega$, let $G_{z_{0}}(\cdot)=G(\cdot,z_{0})$ and

$\Omega_{t}=\\{z\in\Omega:G_{z_{0}}(z)<t\\},\quad t\in(-\infty,0]$

denote the sublevel sets of the Green’s function. The following monotonic
property was proved by Błocki and Zwonek.

######  Theorem 4.1.

[7] Let $\Omega$ be a bounded domain in $\mathbb{C}$. Then for any
$z_{0}\in\Omega$,

$f(t):=\frac{\pi e^{2{t}}}{v(\Omega_{t})}$

is non-increasing in $t\in(-\infty,0)$.

######  Remark 4.2.

Note that

$\lim_{t\rightarrow 0^{-}}\frac{e^{2t}}{v{(\Omega_{t})}}=\frac{1}{v(\Omega)}.$

As $t\rightarrow-\infty$, $\Omega_{t}$ is approximable by the set
$\\{\log\left(c_{\beta}(z_{0})|z-z_{0}|\right)<t\\}$ (see [5, 7]). This
implies

$\lim_{t\rightarrow-\infty}\frac{e^{2t}}{v{(\Omega_{t})}}=\lim_{t\rightarrow-\infty}\frac{e^{2t}}{v{(\\{|z-z_{0}|<{e^{t}c_{\beta}^{-1}(z_{0})}\\})}}=\frac{c_{\beta}^{2}(z_{0})}{\pi}.$

This gives the inequalities (1.5) between the logarithmic capacity and the
sublevel sets of the Green’s function.

The next theorem discusses the second inequality in (1.5), which states that
either $f$ in Theoerem 4.1 is strictly decreasing, or the domain has to be
rigid.

######  Theorem 4.3.

Let $\Omega$ be a bounded domain in $\mathbb{C}$. If there exist
$z_{0}\in\Omega$ and $t_{1}<t_{2}$ in $(-\infty,0)$ such that

$\frac{\pi e^{2{t_{1}}}}{v{(\Omega_{t_{1}})}}=\frac{\pi
e^{2{t_{2}}}}{v(\Omega_{t_{2}})},$

then $G_{z_{0}}(z)=\log|z-z_{0}|/R$ for some constant $R>0$. Consequently,
$\Omega$ is a disk centered at $z_{0}$ with radius $R$ possibly less a
relatively closed polar subset.

###### Proof.

By scaling and translating if necessary, we may assume that $z_{0}=0$ and the
diameter of $\Omega$ is $2$. By the inequality part of Theorem 4.1,
$f(t_{1})=f(t_{2})$ implies that there exists a constant $C$ such that

$v{(\Omega_{t})}=Ce^{2t},$

for $t\in(t_{1},t_{2})$. Thus

$\frac{d}{dt}v{(\Omega_{t})}=2v{(\Omega_{t})}.$

Following [7, 8], we see from the Cauchy-Schwarz inequality that for almost
every $t\in(-\infty,0)$,

$\sigma^{2}(\partial\Omega_{t})\leq\int_{\partial\Omega_{t}}\frac{1}{|\nabla{G_{0}}|}d\sigma\int_{\partial\Omega_{t}}{|\nabla{G_{0}}|}d\sigma.$
(4.1)

Note that due to the harmonicity of $G_{0}$ away from $0$,

$\int_{\partial\Omega_{t}}{|\nabla{G_{0}}|}d\sigma=\int_{\partial\Omega_{t}}\frac{\partial{G_{0}}}{\partial\nu}d\sigma=2\pi.$
(4.2)

On the other hand, by the co-area formula
$v(\Omega_{t})=\int_{-\infty}^{t}\int_{\partial\Omega_{s}}\frac{d\sigma}{|\nabla{G_{0}}|}ds$,
which further leads to

$\int_{\partial\Omega_{t}}\frac{1}{|\nabla{G_{0}}|}d\sigma=\frac{d}{dt}v(\Omega_{t}).$
(4.3)

for almost every $t\in(-\infty,0)$. Pick up a point $t_{0}\in(t_{1},t_{2})$
such that (4.1-4.3) hold. Then,

$\sigma^{2}(\partial\Omega_{t_{0}})\leq 4\pi v(\Omega_{t_{0}}).$

According to the classical isoperimetric inequality (see [12] and the
references therein), $\Omega_{t_{0}}$ is equivalent (two sets $E_{1}$ and
$E_{2}$ are equivalent if and only if $v(E_{1}\cup E_{2}\setminus E_{1}\cap
E_{2})=0$ ) to a disk centered at a point $a\in\Omega$ with radius
$re^{t_{0}}$ for some $r>0$, and in particular, the involved Cauchy-Schwarz
inequality (4.1) attains the equality. This means $\frac{1}{|\nabla{G_{0}}|}$
is necessarily a constant multiple of $|\nabla{G_{0}}|$, or equivalently,
$|\nabla{G_{0}}|$ is a constant on $\partial\Omega_{t_{0}}$. Combining this
with (4.2), one obtains
$\frac{\partial}{\partial\nu}G_{0}=|\nabla{G_{0}}|=r^{-1}e^{-t_{0}}$ on
$\partial\Omega_{t_{0}}$.

Now $G_{0}$ is harmonic on $\Omega\setminus\Omega_{t_{0}}$ and satisfies the
following Cauchy data

$G_{0}=t_{0},\ \ \frac{\partial G_{0}}{\partial\nu}=\frac{1}{re^{t_{0}}}$

on some smooth piece in $\partial\Omega_{t_{0}}$ contained in
$\partial\mathbb{D}(a,re^{t_{0}})$. As a consequence of the unique
continuation property of harmonic functions for local Cauchy data (see [34]),
we get $G_{0}(z)=\log\frac{|z-a|}{r}$ on $\Omega\setminus\Omega_{t_{0}}$, and
further on $\Omega$ by the uniqueness. Since $G_{0}$ has a pole $0$ and the
diameter of $\Omega$ is $2$, we see that $a=0$ and $r=1$, with
$G_{0}(z)=\log{|z|}$ on $\Omega$. Then we complete the proof using Lemma A.2.

∎

In particular, the corollary below follows from Theorem 4.3 directly. Here we
provide an alternative proof adopting the property of the Bergman kernel,
instead of using the unique continuation property of harmonic functions as in
Theorem 4.3.

######  Corollary 4.4.

Let $\Omega$ be a bounded domain in $\mathbb{C}$. Then, there exist
$z_{0}\in\Omega$ and $t_{0}\in(-\infty,0]$ such that

$c_{\beta}^{2}(z_{0})=\frac{\pi e^{2{t_{0}}}}{v{(\Omega_{t_{0}})}},$

if and only if $\Omega$ is a disk centered at $z_{0}$ possibly less a
relatively closed polar subset. In particular,

$c_{\beta}^{2}(z_{0})=\frac{\pi}{v(\Omega)}$

for some $z_{0}\in\Omega$ if and only if $\Omega$ is a disk centered at
$z_{0}$ possibly less a relatively closed polar subset.

###### Proof.

It suffices to prove the necessity. By scaling and translating, we may assume
$z_{0}=0$ and $c_{\beta}(0)=1$. By the isoperimetric inequality as in the
first part of the proof of Theorem 4.3, $\Omega_{t}$ is equivalent to a disk
of radius $e^{t}$ for almost every $t\in(-\infty,t_{0})$. Let
$t^{\sharp}<t_{0}$ be a negatively large constant such that when
$t<t^{\sharp}$, $\partial\Omega_{t}$ is smooth. Hence $\Omega_{t}$ is
precisely a disk of radius $e^{t}$ for each such arbitrarily fixed $t$. Denote
by $a_{t}$ the center of $\Omega_{t}$.

We claim that $a_{t}=0$. To see this, let $K_{t}$ and $c_{t}$ stand for the
corresponding Bergman kernel and logarithmic capacity of $\Omega_{t}$,
respectively. Then for $z\in\Omega_{t}$,

$K_{t}(z)=\frac{e^{2t}}{\pi(e^{2t}-|z-a_{t}|^{2})^{2}}.$

Since the Green’s function on $\Omega_{t}$ with a pole 0 is $G_{0}(z)-t$, by
definition,

$c_{t}(0)=\exp\lim_{z\to{0}}\left\\{(G_{0}(z)-t)-\log|z|\right\\}=e^{-t}c(0)=e^{-t}.$

Making use of the fact that $\pi K_{t}=c_{t}^{2}$ on the disk $\Omega_{t}$, we
further have at $z=0$ that

$\frac{e^{2t}}{(e^{2t}-|a_{t}|^{2})^{2}}=\pi K_{t}(0)=c_{t}^{2}(0)=e^{-2t}.$

It immediately tells us that $a_{t}=0$, so

$\Omega_{t}=\mathbb{D}(0,e^{t}),\ \ \text{for}\ \ t<t^{\sharp}.$ (4.4)

Lastly let $\rho(z):=G_{0}(z)-\log|z|$ on $\Omega$. Then
$\Omega_{t}=\\{z\in\mathbb{C}:|z|<\frac{e^{t}}{e^{\rho(z)}}\\},\ t<0$.
Comparing this with (4.4) for $t<t^{\sharp}$, we have
$\rho|_{\partial\Omega_{t}}=0$. Since $\rho$ is harmonic on $\Omega$,
$\rho\equiv 0$ on $\Omega_{t}$. By the uniqueness again, $\rho\equiv 0$ on
$\Omega$, where $G_{0}(z)=\log|z|$. The proof is complete in view of Lemma
A.2.

∎

###### Proof of Theorem 1.5. .

The ‘only if’ direction is straightforward. For the ‘if’ direction, Case 2
follows from Theorem 4.3. Case 1 follows from Corollary 4.4, or alternatively,
from Theorem 4.3. Indeed, if Case 1 holds, then by (1.5) we have for all
$t<t_{1}$,

$\frac{\pi e^{2{t}}}{v(\Omega_{t})}=\frac{\pi
e^{2{t_{1}}}}{v(\Omega_{t_{1}})}.$

Case 1 is thus reduced to Theorem 4.3 and so $\Omega$ is a disk centered at
$z_{0}$ possibly less a relatively closed polar subset. Case 3 can be proved
similarly from Theorem 4.3.

∎

##  5 Rigidity Properties of the Distance Function in $\mathbb{C}^{n}$

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ with $C^{2}$-boundary. It
is known that there exists a constant $C$ depending only on $n$ such that

$K(z)\leq{C}{\delta^{-n-1}(z)}$

for all $z\in\Omega$; if in addition $\Omega$ is pseudoconvex, by [28, 18]
there exists a constant $C_{\Omega}$ depending only on $\Omega$ such that

$K(z)\geq\frac{C_{\Omega}}{\delta^{2}(z)}.$

In particular, the Bergman kernel of $\mathbb{B}^{n}(z_{0},r)$, the ball in
$\mathbb{C}^{n}$ centered at $z_{0}$ with radius $r$, is given by

$K_{\mathbb{B}^{n}(z_{0},r)}(z)=\frac{n!r^{2}}{\pi^{n}(r^{2}-|z-z_{0}|^{2})^{n+1}}.$
(5.1)

Denote by $PSH^{-}(\Omega)$ the space of negative plurisubharmonic functions
on $\Omega$. Let $G_{z}(\cdot)$ be the pluricomplex Green’s function of
$\Omega$ with pole $z\in\Omega$ given by

$G_{z}(w)=\sup\\{u(w):u\in PSH^{-}(\Omega),\limsup_{\zeta\to
z}u(\zeta)-\log|\zeta-z|<\infty\\}.$

The Azukawa indicatrix for $z\in\Omega\subset\mathbb{C}^{n}$ is defined as

$I_{\Omega}^{A}(z):=\\{X\in\mathbb{C}^{n}:\limsup_{\zeta\rightarrow
0}\left(G_{z}(z+\zeta X)-\log|\zeta|\right)<0\\}.$ (5.2)

Straightforward calculations show
$I_{\mathbb{B}^{n}(z,r)}^{A}(z)=\mathbb{B}^{n}(0,r),$ and when $n=1$,
$I_{\Omega}^{A}(z)=\mathbb{D}(0,c_{\beta;\Omega}^{-1}(z))$.

###### Proof of Theorem 1.6. .

Firstly, we deal with the relation between the Bergman kernel and the distance
function. Let $z\in\Omega$ be fixed and consider
$\mathbb{B}^{n}(z,\delta(z))$. Then
$\mathbb{B}^{n}(z,\delta(z))\subset\Omega$. By (5.1) and the monotonic
decreasing property of the Bergman kernels with respect to domains,

$K(z)\leq
K_{\mathbb{B}^{n}(z,\delta(z))}(z)\leq\frac{n!}{\pi^{n}\delta^{2n}(z)},$

which yields (1.7).

Then we prove the rigidity part in (1.7) and without loss of generality assume
equality is attained at $z_{0}=0$, namely
$K(0)={n!}{\pi^{-n}\delta^{-2n}(0)}$. Let $f$ be an extremal holomorphic
function on $\Omega$ such that $\|f\|_{L^{2}(\Omega)}=1$ and
$K(0)=|f(0)|^{2}$. Then the restriction of $f$ to
$\Omega_{1}:=\mathbb{B}^{n}(0,\delta(0))$ is also holomorphic with
$\|f\|_{L^{2}(\Omega_{1})}\leq\|f\|_{L^{2}(\Omega)}=1$. Therefore, the Bergman
kernel on the ball $\Omega_{1}$ satisfies

$\frac{n!}{\pi^{n}\delta^{2n}(0)}=K_{\Omega_{1}}(0)\geq\frac{|f(0)|^{2}}{\|f\|^{2}_{L^{2}(\Omega_{1})}}\geq|f(0)|^{2}=K(0)=\frac{n!}{\pi^{n}\delta^{2n}(0)},$

which forces both inequalities above to become equalities. In particular,

$\|f\|_{L^{2}(\Omega)}=1=\|f\|_{L^{2}(\Omega_{1})}.$ (5.3)

This implies $\Omega=\Omega_{1}$, i.e., $\Omega$ is a ball. In fact, if
$\Omega\neq\Omega_{1}$, then $f=0$ almost everywhere on the non-empty open set
$\Omega\setminus\overline{\Omega_{1}}$ (of positive Lebesgue measure) by
(5.3). By the holomorphicity we know that $f\equiv 0$ almost everywhere on
$\Omega$, which contradicts the fact that $\|f\|_{L^{2}(\Omega)}=1$.

Secondly, we deal with the relation between the Azukawa indicatrix and the
distance function. Once again, we may suppose $z=0$. Since
$\mathbb{B}^{n}(0,\delta(0))\subset\Omega$,

$\mathbb{B}^{n}(0,\delta(0))=I^{A}_{\mathbb{B}^{n}(0,\delta(0))}(0)\subset
I_{\Omega}^{A}(0),$ (5.4)

which gives the inequality (1.6).

Assume equality in (1.6) holds. Then

$v(I^{A}_{\Omega}(0)\setminus\mathbb{B}^{n}(0,\delta(0)))=0.$ (5.5)

We first claim that for all $X\in\partial\mathbb{B}^{n}(0,\delta(0))$,

$\limsup_{\lambda\to 0}G_{0}(\lambda X)-\log|\lambda|\leq 0.$ (5.6)

Indeed, if $\limsup_{\lambda\to 0}G_{0}(\lambda
X)-\log|\lambda|=\epsilon_{0}>0$, then for $t>0$ sufficiently close to $1^{-}$
(say $t>e^{-\epsilon_{0}})$,

$\begin{split}\limsup_{\lambda\to
0}G_{0}\left(\lambda(tX)\right)-\log|\lambda|&=\limsup_{\lambda\to
0}G_{0}(\lambda tX)-\log|\lambda t|+\log t\\\ &=\left(\limsup_{\lambda t\to
0}G_{0}\left((\lambda t\right)X)-\log|\lambda t|\right)+\log t\\\
&=\epsilon_{0}+\log t>0.\end{split}$

This would mean $tX\notin I_{\Omega}^{A}(0)$, which contradicts (5.4). The
claim is proved. The same argument also shows that if $X\in
I_{\Omega}^{A}(0)$, then so is $tX$ for all $0\leq t\leq 1$.

By (5.6), there is a function
$f:\partial\mathbb{B}^{n}(0,\delta(0))\to[1,\infty)$ such that

$\left(I^{A}_{\Omega}(0)\cup\partial\mathbb{B}^{n}(0,\delta(0))\right)\setminus\mathbb{B}^{n}(0,\delta(0))=\\{r\omega:\omega\in\partial\mathbb{B}^{n}(0,\delta(0)),1\leq
r\leq f(\omega)\\}.$

Then by (5.5),

$0=\int_{\partial\mathbb{B}^{n}(0,\delta(0))}\int_{1}^{f(\omega)}r^{2n-1}drd\sigma(\omega).$

Thus $f(\omega)=1$ almost everywhere with respect to
$\sigma(\partial\mathbb{B}^{n}(0,\delta(0)))$. Thus, for almost every
$X\in\partial\mathbb{B}^{n}(0,\delta(0))$,

$\limsup_{\lambda\to 0}G_{0}(\lambda X)-\log|\lambda|=0.$ (5.7)

For fixed $X\in\partial\mathbb{B}^{n}(0,\delta(0))$ satisfying (5.7), consider
$u:\mathbb{D}(0,1)\setminus\\{0\\}\to\mathbb{R}$ by

$u(\lambda)=G_{0}(\lambda X)-\log|\lambda|.$

By (5.7), $u$ extends subharmonically to equal 0 at 0. On the other hand, by
the monotonicity of the pluri-complex Green’s functions,

$\displaystyle u(\lambda)\leq\log\left|{\lambda
X\over\delta(0)}\right|-\log|\lambda|=0.$

Thus, $u\in SH^{-}(\mathbb{D}(0,1)\setminus\\{0\\})$. By the maximum principle
$u\equiv 0$, or equivalently,

$G_{0}(\lambda X)\equiv\log\left|{\lambda X\over\delta(0)}\right|$

for almost every $X\in\partial\mathbb{B}^{n}(0,\delta(0)))$. Since $G_{0}<0$
on $\Omega$, there is a (possibly empty)
$\sigma(\partial\mathbb{B}^{n}(0,\delta(0)))$-null set $\mathcal{E}$ such that

$\partial\mathbb{B}^{n}(0,\delta(0))\setminus\mathcal{E}\subset\partial\Omega.$

Since $\partial\Omega$ is closed and
$\mathbb{B}^{n}(0,\delta(0))\subset\Omega$,

$\partial\Omega=\partial\mathbb{B}^{n}(0,\delta(0)),\quad\Omega=\mathbb{B}^{n}(0,\delta(0)).$

∎

In the case of pseudoconvex domains, the rigidity result concerning the
Azukawa indicatrix and the distance function follows immediately from the
multi-dimensional Suita conjecture and the first part of our proof of Theorem
1.6. Recently, the first author and Wong [16, 17] used curvature properties of
the Bergman metric to characterize pseudoconvex domains that are biholomorphic
to a ball $\mathbb{B}^{n}$ possibly less a relatively closed pluripolar set.

When $n=1$, since $I^{A}_{\Omega}(z)=\mathbb{D}(0,c_{\beta;\Omega}^{-1}(z))$,
Theorem 1.6 implies the corollary below. Here we provide an alternative proof
using more elementary methods.

######  Corollary 5.1.

Let $\Omega$ be a bounded domain in $\mathbb{C}$. Then for any $z\in\Omega$,

$\delta^{-1}(z)\geq c_{\beta}(z).$ (5.8)

Moreover, equality in (5.8) holds at some $z_{0}\in\Omega$ if and only if
$\Omega$ is a disk centered at $z_{0}$.

###### Proof.

For any $z\in\Omega$, write the Green’s function as
$G_{z}(w)=\log|z-w|+\rho(w)$. Then $\rho$ is harmonic in $\Omega$ and
$\rho(z)=\log c_{\beta}(z)$. For any positive number $R<\delta(z)$, we have

$\frac{1}{2\pi}\int_{0}^{2\pi}G_{z}(z+Re^{it})dt=\frac{1}{2\pi}\int_{0}^{2\pi}\log
R+\rho(z+Re^{it})dt.$

By the mean value theorem for harmonic functions,
$\frac{1}{2\pi}\int_{0}^{2\pi}\rho(z+Re^{it})dt=\rho(z)=\log c_{\beta}(z)$.
Thus

$\frac{1}{2\pi}\int_{0}^{2\pi}G_{z}(z+Re^{it})dt=\log(Rc_{\beta}(z)).$ (5.9)

Recall that for $w\in\bar{\Omega}\setminus\\{z\\}$, $G_{z}(w)\leq 0.$ This
implies from (5.9) that

$c_{\beta}(z)\leq R^{-1}.$

Letting $R\to\delta(z)^{-}$, we obtain (5.8).

Assume $c_{\beta}(z_{0})=\delta^{-1}(z_{0})$ at some point $z_{0}\in\Omega$.
Then by (5.9) and the nonpositivity of $G_{z_{0}}$, $G_{z_{0}}(w)\rightarrow
0^{-}$ for all $w\in\partial\mathbb{D}(z_{0},R)$ as
$R\rightarrow\delta(z_{0})^{-}$. The continuity and nonpositivity of
$G_{z_{0}}$ in $\Omega$ further concludes that this can only happen when
$\partial\Omega$ coincides with $\partial\mathbb{D}(z_{0},\delta(z_{0}))$.

∎

######  Remark 5.2.

When $\Omega$ is a bounded domain in $\mathbb{C}$, Theorem 1.6 reduces to

$\frac{1}{\delta^{2}(z)}\geq\pi K(z)$

for all $z\in\Omega$ and equality holds at some $z_{0}\in\Omega$ if and only
if $\Omega$ is a disk centered at $z_{0}$. Combining this with the previous
inequality chains considered in the paper, we know that there exists some
$z_{0}\in\Omega$ (and some $t_{0}\leq 0$) such that

$\frac{1}{\delta^{2}(z_{0})}=c_{B}^{2}(z_{0})\quad\text{or}\quad\frac{\pi
e^{2{t_{0}}}}{v{(\\{G(\cdot,z_{0})<t_{0}\\})}}\quad\text{or}\quad{v(\mathbb{C}_{\infty}\setminus
j_{z_{0}}(\Omega))\over\pi}$

if and only if $\Omega$ is a disk centered at $z_{0}$.

Appendices

##  Appendix A Rigidity Phenomenon Between $c_{B}$ and $c_{\beta}$

The aim of this appendix is to prove the following rigidity theorem
characterizing the equality conditions for $c_{B}\leq c_{\beta}$.

######  Theorem A.1.

For an open hyperbolic Riemann surface $X$, $c_{\beta}(z_{0})=c_{B}(z_{0})$
for some $z_{0}\in X$ if and only if $X$ is biholomorphic to the unit disk
$\mathbb{D}$ possibly less a relatively closed polar subset $P$; in this case,
the extremal functions of the analytic capacity $c_{B}(z_{0})$ equal
$\varphi\circ f$, where $f:X\to\mathbb{D}\setminus P$ is a biholomorphism and
$\varphi\in\text{Aut}(\mathbb{D})$ such that $\varphi\circ f(z_{0})=0$.

Given two Riemann surfaces $X$ and $Y$ with $X$ admitting a Green’s function,
Minda in [26, Theorem 3] proved that if $f:X\to Y$ is holomorphic, then

$f^{*}(c_{\beta;Y}(w)|dw|)\leq c_{\beta;X}(\zeta)|d\zeta|.$

Moreover, if equality holds at a single point, then $f$ is biholomorphic onto
its image and $Y\setminus f(X)$ is a closed polar set. Although Minda’s
approach does not mention the analytic capacity, it can be used to deduce
Theorem A.1 as follows. For any $f\in\hbox{Hol}(X,\mathbb{D})$,

$|f^{\prime}(z)||dz|\leq{|f^{\prime}(z)|\over
1-|f(z)|^{2}}|dz|=f^{*}(c_{\beta;\mathbb{D}}(z))|dz|\leq c_{\beta}(z)|dz|.$

If we suppose $c_{\beta}(z_{0})=c_{B}(z_{0})$ and set $f=f_{0}$ to be an
extremal function for $c_{B}(z_{0})$, then all inequalities above become
equalities at $z_{0}$. By Minda’s result, $f_{0}$ is a biholomorphism onto
$\mathbb{D}\setminus P$ where $P$ is a relatively closed polar set. Thus,
Theorem A.1 is proved. The key ingredient of Minda’s proof is the “sharp form
of the Lindelöf principle” for the Green’s function. In the following, we
shall reprove Theorem A.1 without resorting to this principle.

Let $f$ be a holomorphic function on an open Riemann surface $X$ such that
$f(X)\subset\mathbb{C}$ admits a Green’s function. Then the Green’s function
satisfies a subordination property:

$G_{X}(z,z^{\prime})\geq G_{f(X)}(f(z),f(z^{\prime}))$ (A.1)

for all $(z,z^{\prime})\in X\times X$. Moreover, if there exists some
$(z_{0},z_{0}^{\prime})\in X\times X$ with $z_{0}\neq z_{0}^{\prime}$ such
that equality in (A.1) holds, then

$G_{X}(z,z^{\prime})\equiv G_{f(X)}(f(z),f(z^{\prime}))$ (A.2)

for all $(z,z^{\prime})\in X\times X$ and $f$ is injective. See for instance
[29, Theorem 4.4.4] for the proof for planar domains. The cases for Riemann
surface can be proved similarly. A consequence of the property is the
following rigidity property Lemma A.2 of the Green’s function. We note that
strengthened forms of the subordination property and Lemma A.2 were proved in
[26, Theorem 1] by Minda using the Lindelöf principle for the Green’s
function.

######  Lemma A.2 (Rigidity lemma of the Green’s function).

On an open Riemann surface $X$, the Green’s function with a pole $z_{0}\in X$
is

$G(z,z_{0})=\log|f(z)|,\quad z\in X$

for some holomorphic function $f$ on $X$ if and only if $f$ is a
biholomorphism from $X$ to the unit disk possibly less a relatively closed
polar subset such that $f(z_{0})=0$.

###### Proof.

Since $G_{X}(z,z_{0})<0$ for $z\in X$, $f(X)\subset\mathbb{D}$. The image
$f(X)$ admits a Green’s function because $f(X)$ is bounded. Also, observe that
$f(z_{0})=0$. Applying the subordination property (A.1) to $f$ and the
identity map, we get

$\log|f(z)|=G_{X}(z,z_{0})\geq G_{f(X)}(f(z),0)\geq
G_{\mathbb{D}}(f(z),0)=\log|f(z)|.$

By the subordination property (A.2), $f$ is injective and
$G_{f(X)}(\zeta,0)=\log|\zeta|$ for $\zeta\in f(X)$. Let $\eta\in\partial
f(X)\cap\mathbb{D}$ and $\zeta_{n}(\in f(X))\to\eta$. Since

$\lim_{n\to\infty}G_{f(X)}(\zeta_{n},0)=\lim_{n\to\infty}\log|\zeta_{n}|=\log|\eta|<0,$

$\eta$ is an irregular boundary point. By Kellogg’s Theorem, cf. [29, Theorem
4.2.5], $P=\partial f(X)\cap\mathbb{D}$ is a relatively closed polar set in
$\mathbb{D}$. Suppose $w_{0}\in\mathbb{D}\setminus\overline{f(X)}$. Then for
some $\epsilon>0$,
$f(X)\subset\mathbb{D}\setminus\overline{\mathbb{D}(w_{0},\epsilon)}$. Let $k$
be the harmonic function defined on the latter set with Dirichlet boundary
data

$k(z)=\begin{cases}0,&z\in\partial\mathbb{D}\\\
-\log(|w_{0}|+\epsilon),&z\in\partial\mathbb{D}(w_{0},\epsilon).\end{cases}$

Since $G_{f(X)}(z,0)\leq G_{f(X)}(z,0)+k(z)$ and $G_{f(X)}(z,0)+k(z)$ is in
the defining set of (2.1) for the domain $f(X)$, we have arrived at a
contradiction unless $\mathbb{D}\setminus\overline{f(X)}$ is empty. Thus,
$\mathbb{D}=f(X)\sqcup P,$ where $\sqcup$ denotes the disjoint union. The
proof of the theorem is complete.

∎

In [21, Theorem 3.1, p. 1196] of the equality part of the Suita conjecture,
Guan and Zhou showed that if $\pi K(z_{0})=c_{\beta}^{2}(z_{0})$, then by
their optimal $L^{2}$ extension theorem, $\exp G(z,z_{0})=|g(z)|$, for some
holomorphic function $g$ on $\Omega$. By proving further
$c_{\beta}^{2}(z_{0})=c_{B}^{2}(z_{0})$ if $\exp G(z,z_{0})=|g(z)|$, they were
able to apply Suita’s Theorem (Theorem 1.1) to yield that $X$ is biholomorphic
to a disk less a relatively closed polar set.

In fact, Lemma A.2 says that $|g(z)|=\exp G(z,z_{0})$ if and only if the
surface is biholomorphic to a disk less a relatively closed polar set.
Consequently, one does not need to involve the analytic capacity or Suita’s
Theorem to prove the equality part of the Suita conjecture as in [21]. See
also [14] for related work.

Next, by relying more explicitly on the analytic capacity, we shall give a
proof of Theorem A.1 by making use of Lemma A.2 and following an idea of Guan
and Zhou in [21, Lemma 4.25].

###### Proof of Theorem A.1. .

If $X$ is biholomorphic to a disk less a relatively closed polar set, then
$c_{\beta}\equiv c_{B}$, since the polar part is negligible.

Conversely, write $G_{z_{0}}(z)$ for $G(z,z_{0})$, and let
$u(z):=\log(|f_{0}(z)|)$ for $z\in X$, where $f_{0}$ is an extremal function
of the analytic capacity at $z_{0}$. Since $|f_{0}|<1$ on $X$, $u\in
SH^{-}(X)$. By definition of the Green’s function, we further see that
$u-G_{z_{0}}\in SH^{-}(X)$. We will show that $u-G_{z_{0}}|_{z=z_{0}}=0$. In
the local coordinate $w(z)$ of $X$ near $z_{0}$, $\lim_{z\rightarrow
z_{0}}\log|f_{0}(z)|-\log|w(z)|=\log|\frac{df_{0}}{dw}(z_{0})|=\log
c_{B}(z_{0})$ by definition of $f_{0}$. Thus for $z\in X$ near $z_{0}$,

$u(z)-G_{z_{0}}(z)=(\log(|f_{0}(z)|)-\log|w(z)|)-(G_{z_{0}}(z)-\log|w(z)|)\rightarrow\log
c_{B}(z_{0})-\log c_{\beta}(z_{0})=0.$

As a consequence of the maximum principle of subharmonic functions, we have

$G_{z_{0}}(z)=\log(|f_{0}(z)|).$

By Lemma A.2, $f_{0}$ is a biholomorphism from $X$ to the unit disk less a
relatively closed polar set, with $f_{0}(z_{0})=0$.

To complete the second part of the theorem, let $f:X\to\mathbb{D}\setminus P$
be any biholomorphism, and let
$\varphi(z):=\frac{z-f(z_{0})}{1-\overline{f(z_{0})}z}\in\text{Aut}(\mathbb{D}).$
Then, $F:=\varphi\circ f:X\to\mathbb{D}\setminus\varphi(P)$ is also a
biholomorphism such that $F(z_{0})=0$, and $G_{z_{0}}=\log|F|$ by Lemma A.2.
We will show that $F$ is an extremal function for $c_{B}(z_{0})$. For any
$h\in\hbox{Hol}(X,\mathbb{D})$ with $h(z_{0})=0$ and $h^{\prime}(z_{0})\neq
0$, we have that $\log|h|\in SH^{-}(X)$. By the definition of the Green’s
function, $\log|h|\leq G_{z_{0}}=\log|F|$ and so
$|h^{\prime}(z_{0})|\leq|F^{\prime}(z_{0})|$. Therefore,

$c_{B}(z_{0})=|F^{\prime}(z_{0})|,$

which means $F$ is extremal.

∎

It is known (see [31, VII.5H]) that if $X$ is a regular region of connectivity
greater than or equal to 2, then

$c_{\beta}(z)>c_{B}(z),\quad\hbox{for all }z\in X.$ (A.3)

Theorem A.1 says that (A.3) in fact holds for any hyperbolic Riemann surface
$X$ which is not biholomorphic to a disk possibly less a relatively closed
polar subset.

##  Appendix B Stability of the Szegö Kernel on Lipschitz Domains

The Szegö projection and its kernel on planar domains with Lipschitz
boundaries were studied extensively by Lanzani [24]. One of the key steps in
extending results from smoothly bounded domains to those with Lipschitz
boundaries is by approximating the bounded Lipschitz domain $\Omega$ by
smoothly bounded subdomains in the sense of Nečas (see [24, Theorem 2.3]).

######  Theorem B.1.

[24] Let $\Omega\subset\mathbb{C}$ be a bounded domain with Lipschitz
boundary. Then there exists a sequence of subdomains
$\\{\Omega_{j}\\}_{j=1}^{\infty}$ with $C^{\infty}$ smooth boundaries such
that

  1. $1.$

There is a sequence of Lipschitz homeomorphisms
$\Lambda_{j}:\partial\Omega\to\partial\Omega_{j}$ such that
$\Lambda_{j}(P)\in\Gamma(P)$ (see Definition B.2) and
$\lim_{j\to\infty}\Lambda_{j}(P)=P$.

  2. $2.$

There exists functions $\omega_{j}:\partial\Omega\to(0,\infty)$ that are
uniformly bounded away from 0 and $\infty$, converge to 1 almost everywhere
and

$\int_{\partial\Omega}h(\Lambda_{j}(w))\omega_{j}(w)d\sigma(w)=\int_{\partial\Omega_{j}}h(\zeta)d\sigma_{j}(\zeta)$

for any $h\in L^{1}(\partial\Omega_{j})$.

  3. $3.$

The unit tangent vector $T_{j}$ for $\partial\Omega_{j}$ and the unit tangent
vector $T$ for $\partial\Omega$ satisfy that $T_{j}(\Lambda_{j}(\cdot))\to
T(\cdot)$ almost everywhere on $\partial\Omega$.

If $\Omega$ has $C^{\infty}$ boundary with a defining function $\rho$ and
Szegö kernel $S$, define the sublevel sets of the defining function for small
$\epsilon>0$

$\Omega_{\epsilon}=\\{z\in\Omega:\rho(z)<-\epsilon\\},$

and consider the Szegö kernels $S_{\epsilon}(\cdot,\cdot)$ of these respective
domains. In the proof of Theorem 2.1 of [9], Boas showed that for
$a,z\in\Omega_{\epsilon}$,

$\lim_{\epsilon\to 0^{+}}S_{\epsilon}(z,a)=S(z,a).$

The purpose of this section is to extend this stability result of the Szegö
kernel to Lipschitz domains in $\mathbb{C}$ with respect to the Nečas
approximation. We begin by giving notations along the lines of [24].

######  Definition B.2.

  1. $1.$

For $\lambda>0$, $P\in\partial\Omega$, the non-tangential approach region to P
is

$\Gamma(P)=\\{\zeta:|\zeta-P|\leq(1+\lambda)dist(\zeta,\partial\Omega)\\}$

  2. $2.$

For a function $f$ defined on a Lipschitz domain, the non-tangential limit
$f^{+}$ and non-tangential maximal function $f^{*}$, if they exist, are
defined respectively as

$f^{*}(P)=\sup_{w\in\Gamma(P)}|f(w)|,\quad
f^{+}(P)=\lim_{\begin{subarray}{c}w\to P\\\
w\in\Gamma(P)\end{subarray}}f(w),\quad P\in\partial\Omega.$

  3. $3.$

The Hardy space of a bounded domain $\Omega$ with Lipschitz boundary is

$H^{2}(\partial\Omega)=\\{f^{+}:f\in\mathcal{O}(\Omega),f^{*}\in
L^{2}(\partial\Omega)\\}.$

A useful characterization of the Szegö kernel for our purposes is

$S(z,a)={f(z)\over\|f^{+}\|^{2}_{\partial\Omega}},$ (B.1)

where $f$ is the unique function with minimal $L^{2}(\partial\Omega)$-norm
among all functions in $H^{2}(\partial\Omega)$ with $f(a)=1$. Using Theorem
B.1 we can adapt the proof of Boas [9, Theorem 2.1].

######  Proposition B.3.

Let $\Omega$ be a bounded domain with Lipschitz boundary and $S(\cdot,\cdot)$
be its Szegö kernel. Let $\\{\Omega_{j}\\}_{j=1}^{\infty}$ be a sequence of
subdomains as in Theorem B.1. Denote by $\\{S_{j}(\cdot,\cdot)\\}$ the
corresponding Szegö kernels for these domains. Then for each $a,z\in\Omega$,

$\lim_{j\to\infty}S_{j}(z,a)\to S(z,a).$

###### Proof.

Let $\\{f_{j}\\}$ and $f$ be the extremal functions for the domains
$\\{\Omega_{j}\\}_{j=1}^{\infty}$ and $\Omega$ as in (B.1). Since
$|f\circ\Lambda_{j}|^{2}\omega_{j}\leq C|f^{*}|^{2}\in L^{1}(\partial\Omega),$

$\|f^{+}_{j}\|^{2}_{\partial\Omega_{j}}\leq\|f\|^{2}_{\partial\Omega_{j}}=\|(f\circ\Lambda_{j})\omega_{j}^{{1\over
2}}\|^{2}_{\partial\Omega}\leq M<\infty$ (B.2)

for some constant $M$, and by the dominated convergence theorem

$\lim_{j\to\infty}\|(f\circ\Lambda_{j})\omega_{j}^{{1\over
2}}\|^{2}_{\partial\Omega}=\|f^{+}\|^{2}_{\partial\Omega}.$ (B.3)

By (B.2), on each $\Omega_{k}$, $\\{f_{j}\\}_{j\geq k}$ is a normal family.
After passing to a subsequence, there exists an $F\in\mathcal{O}(\Omega)$ such
that $f_{j}\to F$ uniformly on compact subsets. Necessarily this implies
$F(a)=1$.

By (B.2), after passing to a subsequence
$(f_{j}^{+}\circ\Lambda_{j})\omega_{j}^{{1/2}}$ converges weakly to
$f_{\infty}$ in $L^{2}(\partial\Omega)$. Let $\mathcal{C}$ denote the Cauchy
transform. We claim that

$F(z)=\mathcal{C}(f_{\infty})(z)\left(:={1\over 2\pi
i}\int_{\partial\Omega}{f_{\infty}(w)\over w-z}dw\in
H^{2}(\partial\Omega)\right).$

In fact for any fixed $z\in\Omega_{j}$,

$\begin{split}|f_{j}(z)-\mathcal{C}(f_{\infty})(z)|=&\left|\int_{\partial\Omega}{f_{j}^{+}(\Lambda_{j}(w))T_{j}(\Lambda_{j}(w))\omega_{j}(w)\over\Lambda_{j}(w)-z}d\sigma-\int_{\partial\Omega}\frac{f_{\infty}(w)T(w)}{w-z}d\sigma\right|\\\
\leq&\int_{\partial\Omega}|f_{j}^{+}(\Lambda_{j}(w))\omega_{j}^{\frac{1}{2}}(w)|\left|{T_{j}(\Lambda_{j}(w))\omega_{j}^{\frac{1}{2}}(w)\over\Lambda_{j}(w)-z}-\frac{T(w)}{w-z}\right|d\sigma\\\
&\quad+\left|\int_{\partial\Omega}\left(f_{j}^{+}(\Lambda_{j}(w))\omega_{j}^{\frac{1}{2}}(w)-f_{\infty}(w)\right)\frac{T(w)}{w-z}d\sigma\right|:=A+B.\end{split}$

Since $\frac{T(w)}{w-z}\in L^{\infty}(\partial\Omega)$ and
$(f_{j}^{+}\circ\Lambda_{j})\omega_{j}^{\frac{1}{2}}$ converges weakly to
$f_{\infty}$ in $L^{2}(\partial\Omega)$, we have $B\rightarrow 0$. For $A$,
notice that by Theorem B.1 and the dominated convergence theorem,
${T_{j}(\Lambda_{j}(w))\omega_{j}^{\frac{1}{2}}(w)\over\Lambda_{j}(w)-z}$
converges to $\frac{T(w)}{w-z}$ in $L^{2}(\partial\Omega)$ norm. Making use of
Hölder inequality and the uniform boundedness of
$\|(f_{j}^{+}\circ\Lambda_{j})\omega_{j}^{1/2}\|_{\partial\Omega}$, we get

$A\leq\|(f_{j}^{+}\circ\Lambda_{j})\omega_{j}^{\frac{1}{2}}\|_{\partial\Omega}\left\|{T_{j}(\Lambda_{j}(w))\omega_{j}^{\frac{1}{2}}(w)\over\Lambda_{j}(w)-z}-\frac{T(w)}{w-z}\right\|_{\partial\Omega}\rightarrow
0.$

Hence $f_{j}\to\mathcal{C}f_{\infty}$ pointwisely, and so
$F=\mathcal{C}(f_{\infty})$.

Using the fact that $\\{\overline{TG}:G\in
H^{2}(\partial\Omega)\\}=H^{2}(\partial\Omega)^{\perp}$, we will show that
$f_{\infty}\in H^{2}(\partial\Omega)$, cf. [24, Theorem 5.1], [4, Theorem
4.3]. Since $\langle
f_{j}^{+},\overline{T_{j}G}\rangle_{\partial\Omega_{j}}=0$,

$\displaystyle|\langle f_{\infty},\overline{TG}\rangle|$ $\displaystyle=$
$\displaystyle|\langle
f_{\infty}-(f_{j}^{+}\circ\Lambda_{j})\omega_{j}^{1\over
2},\overline{TG}\rangle_{\partial\Omega}$
$\displaystyle\quad+\int_{\partial\Omega}f_{j}^{+}(\Lambda_{j})(w))\omega_{j}^{1\over
2}(w)T(w)G(w)-(f_{j}^{+}(\Lambda_{j})(w))T_{j}(\Lambda_{j}(w))G(\Lambda_{j}(w))\omega_{j}(w)d\sigma(w)|$
$\displaystyle\leq$ $\displaystyle|\langle
f_{\infty}-(f_{j}^{+}\circ\Lambda_{j})\omega_{j}^{1\over
2},\overline{TG}\rangle_{\partial\Omega}|+M^{1\over
2}\Big{(}\int_{\partial\Omega}|T(w)G(w)-T_{j}(\Lambda_{j}(w))G(\Lambda_{j}(w))\omega_{j}^{1\over
2}(w)|^{2}d\sigma(w)\Big{)}^{1\over 2}.$

The first term above approaches 0 as $j$ approaches $\infty$ by weak-
convergence. The integrand in the second term is dominated by $C|G^{*}|^{2}$
for some constant $C$. By the dominated convergence theorem, the integral
approaches 0 as well. Thus, $f_{\infty}\in H^{2}(\partial\Omega).$ Since the
Cauchy transform is a projection onto $H^{2}(\partial\Omega)$,
$f_{\infty}=F^{+}$ on $\partial\Omega.$ In particular
$(f_{j}^{+}\circ\Lambda_{j}){\omega_{j}}^{\frac{1}{2}}$ converges weakly to
$F^{+}$ in $L^{2}(\partial\Omega)$ as well. Hence

$\displaystyle\|F^{+}\|^{2}_{\partial\Omega}$ $\displaystyle\leq$
$\displaystyle\liminf_{j\rightarrow\infty}\|(f_{j}\circ\Lambda_{j})\omega_{j}^{{1\over
2}}\|^{2}_{\partial\Omega}=\liminf_{j\rightarrow\infty}\|f_{j}^{+}\|^{2}_{\partial\Omega_{j}}$
(B.4) $\displaystyle\leq$
$\displaystyle\limsup_{j\rightarrow\infty}\|f_{j}^{+}\|^{2}_{\partial\Omega_{j}}\leq\limsup_{j\rightarrow\infty}\|f\|^{2}_{\partial\Omega_{j}}=\|f^{+}\|^{2}_{\partial\Omega}.$

Here the first inequality uses the weakly sequentially lower semicontinuity
for the $L^{2}(\partial\Omega)$ norm, the third inequality applies (B.2) and
the last equality uses (B.3). Since $f$ is unique, $F=f$ on $\Omega$, and all
inequalities in (B.4) become equalities. In particular, $f_{j}$ converges
pointwisely to $f$ on $\Omega$ and
$\|f_{j}^{+}\|_{\partial\Omega_{j}}\rightarrow\|f^{+}\|_{\partial\Omega}$ by
(B.4). By (B.1), $S_{j}(z,a)\to S(z,a).$

∎

### Funding

The research of the first and second author is supported by an AMS-Simons
travel grant. The research of the third author is partially supported by NSF
grant DMS-1501024.

### Acknowledgements

Some of the results in this paper are a part of the second-named author’s
Ph.D. dissertation [35] at the University of California, Irvine. He would like
to thank his thesis advisor Song-Ying Li for the support throughout the years,
and Harold Boas and Emil Straube for useful conversations and encouragement
during the writing process. The authors thank Boas [10] for explaining the
relevance of the shears to our paper. Finally, we thank the anonymous referee
for valuable comments, and Marc Benoit for spotting a few typos in an earlier
draft.

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<EMAIL_ADDRESS>
Department of Mathematics, University of Connecticut, Storrs, CT 06269-1009,
USA

(formerly<EMAIL_ADDRESS>(now<EMAIL_ADDRESS>
Department of Mathematics, Texas A&M University, College Station, TX
77843-3368, USA

<EMAIL_ADDRESS>
Department of Mathematical Sciences, Purdue University Fort Wayne, Fort Wayne,
IN 46805-1499, USA


# Interactive Visual Learning for Stable Diffusion

Seongmin Lee1 Benjamin Hoover1,2 Hendrik Strobelt2 Zijie J. Wang1
ShengYun Peng1 Austin Wright1 Kevin Li1 Haekyu Park1
Haoyang Yang1&Polo Chau1
1Georgia Tech
2IBM Research
{seongmin<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Diffusion-based generative models’ impressive ability to create convincing
images has garnered global attention. However, their complex internal
structures and operations often pose challenges for non-experts to grasp. We
introduce Diffusion Explainer, the first interactive visualization tool
designed to elucidate how Stable Diffusion transforms text prompts into
images. It tightly integrates a visual overview of Stable Diffusion’s complex
components with detailed explanations of their underlying operations. This
integration enables users to fluidly transition between multiple levels of
abstraction through animations and interactive elements. Offering real-time
hands-on experience, Diffusion Explainer allows users to adjust Stable
Diffusion’s hyperparameters and prompts without the need for installation or
specialized hardware. Accessible via users’ web browsers, Diffusion Explainer
is making significant strides in democratizing AI education, fostering broader
public access. More than 7,200 users spanning 113 countries have used our
open-sourced tool at https://poloclub.github.io/diffusion-explainer/. A video
demo is available at https://youtu.be/MbkIADZjPnA.

##  1 Introduction

Figure 1: With Diffusion Explainer, users can examine how (A) a text prompt,
e.g., “a cute and adorable bunny… pixar character”, is encoded by (B) the Text
Representation Generator into vectors to guide (C) the Image Representation
Refiner to iteratively refine the vector representation of the image being
generated. (D) The Timestep Controller enables users to review the incremental
improvements in image quality and adherence to the prompt over timesteps. (E)
The final image representation is upscaled to a high-resolution image.
Diffusion Explainer tightly integrates a visual overview of Stable Diffusion’s
complex components with detailed explanations, enabling users to fluidly
transition between abstraction levels through animations and interactive
elements (see Fig. 2 and Fig. 3).

Diffusion-based generative models Rombach et al. (2022) like Stable Diffusion
Stability AI (2022) have captured global attention for their impressive image
creation abilities, from AI developers, designers, to policymakers. However,
the popularity and progress of generative AI models have sparked ethical
Brusseau (2022) and social concerns, such as accusations of artistic style
theft by AI image generators Sung (2022); Choudhary (2022). Policymakers are
also discussing ways to combat malicious data generation and revise copyright
policies Engler (2023); Ryan-Mosley (2023); U.S. Copyright Office (2023).
There is an urgent need for individuals from many different fields to
understand how generative AI models function and communicate effectively with
AI researchers and developers Dixit (2023); Hendrix (2023).

Key challenges in designing learning tools for Stable Diffusion. Stable
Diffusion iteratively refines noise into a high-resolution image’s vector
representation, guided by a text prompt. Internally, the prompt is tokenized
and encoded into vector representations by the CLIP’s Text Encoder Radford et
al. (2021). With text representations’ guidance, Stable Diffusion improves the
image quality and adherence to the prompt by incrementally denoising the
image’s vector representation using the UNet Ronneberger et al. (2015) and the
Scheduler algorithm Nichol and Dhariwal (2021). The final image representation
is upscaled to a high-resolution image. The crux of learning about Stable
Diffusion tems from the complex interplay between the multiple neural network
subcomponents, their intricate operations, and the iterative nature of image
representation refinements. Such complex interactions are challenging even for
experts to comprehend von Platen (2022). While some articles Alammar (2022)
and video lessons Howard (2023) explain Stable Diffusion, they often focus on
model training and mathematical details.

Contributions. In this demonstration, we contribute:

  * •

Diffusion Explainer, the first interactive visualization tool designed for
non-experts to learn how Stable Diffusion transforms a text prompt into a
high-resolution image (Fig. 1), overcoming design challenges in developing
learning tools for Stable Diffusion. Diffusion Explainer integrates an
overview of Stable Diffusion’s complex structure with explanations of their
underlying operations enabling users to fluidly transition between multiple
abstraction levels through animations and interactive elements.

  * •

Real-time interactive visualization to discover how Stable Diffusion’s
hyperparameters and text prompt affect image generation, empowering users to
experiment with their settings and gain insight into each hyperparameter’s
impact without the need for complex mathematical derivations.

  * •

Open-sourced, web-based implementation that broadens the public’s education
access to modern generative AI without requiring any installation, advanced
computational resources, or coding skills. Diffusion Explainer is open-
sourced111https://github.com/poloclub/diffusion-explainer and available at
https://poloclub.github.io/diffusion-explainer/. A video demo is available at
https://youtu.be/MbkIADZjPnA. With over 7,200 users across 113 countries,
Diffusion Explainer is making significant strides in democratizing AI
education.

##  2 System Design and Implementation

Diffusion Explainer is an interactive visualization tool that explains how
Stable Diffusion generates a high-resolution image from a text prompt. It
incorporates an animation of random noise gradually refined and a Timestep
Controller (Fig. 1D) that enables users to visit each refinement timestep.
From the Prompt Selector (Fig. 1A), users select one out of the 13 prompts
that follow a template and contain popular keywords identified from literature
Smith (2022). Diffusion Explainer provides an overview of Stable Diffusion’s
architecture, which can be expanded into details via user interactions (Fig.
2, Fig. 3). While users can interactively change Stable Diffusion’s two key
hyperparameters, guidance scale and random seed, we fix the number of
timesteps as 50, a commonly chosen value, and consistently use the Linear
Multistep Scheduler Karras et al. (2022), a fundamental and widely adopted
scheduling method. Diffusion Explainer is implemented using a standard web
technology stack (HTML, CSS, JavaScript) and the D3.js Bostock et al. (2011)
visualization library.

###  2.1 Text Representation Generator

The Text Representation Generator converts text prompts into vector
representations. Clicking on the Text Representation Generator expands to the
Text Operation View (Fig. 2A) that explains how the Tokenizer splits the
prompt into tokens and how the Text Encoder encodes the tokens into vector
representations. Clicking on the Text Encoder displays the Text-image Linkage
Explanation (Fig. 2B), which visually explains how Stable Diffusion connects
text and image by utilizing the CLIP Radford et al. (2021) text encoder to
generate text representations with image-related information.

Figure 2:  To understand how Stable Diffusion converts a text prompt into
vector representations, users click on the Text Representation Generator,
which smoothly expands to (A) the Text Operation View that explains how the
prompt is split into tokens and encoded into vector representations. (B) The
Text-image Linkage Explanation demonstrates how Stable Diffusion bridges text
and image, enabling text representations to guide the image generation
process.

###  2.2 Image Representation Refiner

The Image Representation Refiner (Fig. 3) refines random noise into the vector
representation of a high-resolution image that adheres to the text prompt.
Diffusion Explainer visualizes the image representation of each refinement
step in two ways: (1) decoding it as a small image using linear operations
Turner (2022) and (2) upscaling it to the Stable Diffusion’s output resolution
(Fig. 1E). Users expands the Image Representation Refiner to access the Image
Operation View (Fig. 3A), which explains how the UNet neural network
Ronneberger et al. (2015) predicts the noise to be removed from the image
representation.

The guidance scale hyperparameter, which controls the image’s adherence
strength to the text prompt, is described at the bottom, and further explained
in the Interactive Guidance Explanation (Fig. 3B). Using a slider, users can
experiment with different guidance scale values to better understand how
higher values lead to stronger adherence of the generated image to the text
prompt.

Figure 3: Users learn how Stable Diffusion gradually refines noise into a
high-resolution image’s vector representation aligned with the text prompt by
selecting the Image Representation Refiner from the high-level overview. This
smoothly expands to (A) the Image Operation View that demonstrates how the
noise is iteratively predicted and removed from the image representation. (B)
The Interactive Guidance Explanation enables users to interactively experiment
with various guidance scale values (0, 1, 7, 20) to better understand how
higher values lead to stronger adherence.

##  3 Demonstrating Diffusion Explainer

We provide a demonstration scenario both to illustrate how people with limited
experience with Stable Diffusion may benefit from using Diffusion Explainer
and to describe what we will show the audience.

####  3.0.1 Demonstration Scenario

Troy, a government policymaker overseeing AI image creation in the
entertainment and media industries, has recently received concerns from
artists. They express worry that their artwork has been exploited by AI models
to create commercial products without their consent AMELION (2023). Troy is
eager to help these artists in getting compensated for their contributions. He
has found a tool that could potentially address their concerns, which would
attribute AI-generated images to human artists Huber and Troynikov (2023);
anton (2023). However, before proposing any policies, Troy needs to understand
how and if such attribution may work.

Troy launches Diffusion Explainer which illustrates how Stable Diffusion
transforms a text prompt into a high-resolution image through an iterative
process (Fig. 1). He identifies two controllable hyperparameters: random seed
and guidance scale. Adjusting the random seed from 1 to 2 and 3, Troy observes
substantial changes in the generated image. Intrigued by these variations, he
examines timestep 1 using the Timestep Controller (Fig. 1D) and discovers that
different random seeds yield different initial noises, thus generating diverse
images. Continuing his exploration, Troy experiments with different guidance
scale values. He notes that a guidance scale value of 7 produces a realistic
image closely aligned with the text prompt, while values of 1 or 20 result in
images that are hard to interpret or exaggerated.

To delve into the details of how the text prompt is processed, Troy clicks on
the Text Representation Generator to expand it into the Text Operation View
(Fig. 2A). Here, he discovers that the prompt is tokenized and converted into
vector representations. Seeking clarity on how text is connected to the image,
he then displays the Text-image Linkage Explanation (Fig. 2B) and learns that
Stable Diffusion’s text representations contain image-related information.
Troy proceeds to understand the refinement of image representation by
examining the Image Operation View (Fig. 3A). He discovers that each
refinement step involves UNet’s noise prediction and removal, with the
guidance scale hyperparameter controlling the adherence of the generated image
to the prompt. Intrigued, Troy accesses the Interactive Guidance Explanation
(Fig. 3B) and learns that the model predicts two types of noise, each of which
is generic and prompt-specific. The final noise is a weighted sum of these
noises, with the weight being controlled by the guidance scale.

With an improved understanding of the image generation process of Stable
Diffusion, Troy recognizes that image analysis alone, without considering text
prompts, will not suffice to discern how an artist’s creations contributed to
AI-generated images. He asserts that further research is imperative to
accurately attribute AI-generated images.

##  4 Conclusion

Diffusion Explainer, the first interactive visualization for non-experts,
explains how Stable Diffusion generates high-resolution images from text
prompts. It tightly integrates a visual overview of Stable Diffusion’s
components with detailed explanations of their underlying operations and
provides real-time hands-on experience to change Stable Diffusion’s
hyperparameters and prompts on the browser without any installation or
hardware requirements. More than 7,200 users spanning 113 countries have used
Diffusion Explainer.

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