1 Introduction
In [Reference Burns and Krantz1], Burns and Krantz proved several fundamental boundary rigidity results for holomorphic self-maps on bounded domains, often referred to as the Burns–Krantz rigidity nowadays. Here, we state a localized version of the Burns–Krantz rigidity by Huang.
Theorem 1.1 [Reference Huang5, Theorem 2.5]
Let
$\Omega $
be a bounded domain, and let
$p\in \partial \Omega $
be a strongly pseudoconvex boundary point. If f is a holomorphic self-map of
$\Omega $
such that
$f(z)=z+o(\|z-p\|^3)$
as
$z\rightarrow p$
, then
$f(z)\equiv z$
.
In the same article [Reference Huang5], Huang also proved the following two boundary rigidity results for holomorphic self-maps of bounded domains with an interior fixed point.
Theorem 1.2 [Reference Huang5, Corollary 1.5]
Let
$\Omega $
be a bounded smooth domain in
${\mathbb {C}}$
,
${z_0\in \Omega} $
, and
$p\in \partial \Omega $
. If f is a holomorphic self-map of
$\Omega $
such that
$f(z_0)=z_0$
and
${f(z)=z+o(|z-p|)}$
as
$z\rightarrow p$
, then
$f(z)\equiv z$
.
Theorem 1.3 [Reference Huang5, Corollary 2.7]
Let
$\Omega $
be a bounded strongly convex domain in
${\mathbb {C}}^n$
(
$n\ge 2$
),
$z_0\in \Omega $
, and
$p\in \partial \Omega $
. If f is a holomorphic self-map of
$\Omega $
such that
$f(z_0)=z_0$
and
$f(z)=z+o(\|z-p\|^2)$
as
$z\rightarrow p$
, then
$f(z)\equiv z$
.
In [Reference Huang5, Remark 2.9], Huang conjectured that Theorem 1.3 also holds for bounded strongly pseudoconvex domains. Indeed, in [Reference Huang4], Huang already proved this conjecture in
${\mathbb {C}}^2$
, with an extra assumption of
$\Omega $
being “simply connected.” In [Reference Fornæss and Rong2], we were able to remove this extra assumption. The conjecture in any dimension was later proven by the author in [Reference Rong9].
In [Reference Fornæss and Rong3], this Huang-type rigidity was proven on “fibered domains with boundary size zero” (e.g., complex ellipsoids). Recently, in [Reference Wang, Liu and Tang10], Wang et al. proved Huang-type rigidity on polydisks as follows.
Theorem 1.4 [Reference Wang, Liu and Tang10, Theorem 5.2]
Let
$\Delta ^n$
be the unit polydisk in
${\mathbb {C}}^n$
(
$n\ge 2$
),
$z_0\in \Delta ^n$
, and set
$p:=(1,0,\dots ,0)\in \partial \Delta ^n$
. If f is a holomorphic self-map of
$\Delta ^n$
such that
$f(z_0)=z_0$
and
$f(z)=z+o(\|z-p\|^2)$
as
$z\rightarrow p$
, then
$f(z)\equiv z$
.
Since the Huang-type rigidity holds on both the unit ball and the unit polydisk, it is natural to ask whether it holds on any bounded symmetric domain. The main purpose of this article is to show that this is indeed the case, using ideas from [Reference Ng and Rong7]. More precisely, we have the following.
Theorem 1.5 Let
$\Omega \subset {\mathbb {C}}^n$
(
$n\ge 2$
) be a bounded symmetric domain in its Harish-Chandra realization,
$z_0\in \Omega $
and
$p\in \partial \Omega $
a smooth boundary point. If f is a holomorphic self-map of
$\Omega $
such that
$f(z_0)=z_0$
and
$f(z)=z+o(\|z-p\|^2)$
as
$z\rightarrow p$
, then
$f(z)\equiv z$
.
In fact, Theorem 1.5 follows from the Huang-type rigidity on some fibered domains, which we present in Section 2. In Section 3, we prove Theorem 1.5.
2 Fibered domains
First, we slightly modify the definition of the fibered domains introduced in [Reference Ng and Rong7].
Let D be a bounded domain in
${\mathbb {C}}^r$
(
$r\ge 1$
). Let
$\Omega \subset {\mathbb {C}}^{r+s}$
(
$s\ge 1$
) be a fibered domain over D, that is,
where
$\Omega _w\subset \{w\}\times {\mathbb {C}}^s\cong {\mathbb {C}}^s$
is a bounded domain containing the origin
$\{w\}\times \{0\}$
for each
$w\in D$
. We will identify D with
$(D,0)\subset \Omega $
and denote
$\pi :\Omega \rightarrow D$
the orthogonal projection.
Let
$p\in \partial D$
be a smooth boundary point of D. Suppose that the Huang-type rigidity holds on D at p, that is, if g is a holomorphic self-map of D such that
$g(w_0)=w_0$
for some
$w_0\in D$
and
$g(w)=w+o(\|w-p\|^2)$
as
$w\rightarrow p$
, then
$g(w)\equiv w$
. Furthermore, suppose that there exists a proper holomorphic mapping
$\phi :\Delta \rightarrow D$
, which is Lipschitz continuous up to the boundary with
$\phi (1)=p$
. With the above properties, we call
$(D,p)$
a good pointed base.
Denote by
$\partial _S D$
the Shilov boundary of D. We say that D satisfies the strong Shilov maximum principle if for any holomorphic function f on D, we have
$f\equiv 0$
if and only if
$\lim _{z\rightarrow p} f(z)=0$
for every
$p\in \partial _S D$
. Note, in particular, that a polydisk
$\Delta ^r$
satisfies this property.
We say that
$\Omega $
is a fibered domain over D with Shilov boundary size zero, if for every sequence
$\{w_n\}\subset D$
converging to a point
$q\in \partial _S D$
and every sequence
$\{z_n\}\subset \Omega $
such that
$z_n\in \Omega _{w_n}$
for every n, we have
$\lim _{n\rightarrow \infty } \|z_n-(w_n,0)\|=0$
.
The following is the Huang-type rigidity on these special fibered domains.
Theorem 2.1 Let
$\Omega $
be a fibered domain with Shilov boundary size zero over a good pointed base
$(D,p)$
with D satisfying the strong Shilov maximum principle. Let
${\phi :\Delta \rightarrow D}$
be a proper holomorphic map which is Lipschitz continuous up to the boundary with
$\phi (1)=p$
. Suppose that there exists a defining function
$\rho $
of
$\Omega $
which is smooth over
$\overline {\phi (\Delta )}$
with
$d\rho \neq 0$
on
$\phi (\partial \Delta )$
and that
$\rho \circ \phi $
is subharmonic. Let f be a holomorphic self-map of
$\Omega $
such that
$f(w_0,0)=(w_0,0)$
for some
$w_0\in D$
and
$f(z)=z+o(\|z-p\|^2)$
as
$z\rightarrow p$
. Then
$f(z)\equiv z$
.
Proof Set
$g:=\pi \circ f|_D$
. Then
$g:D\rightarrow D$
is a holomorphic self-map of D such that
$g(w_0)=w_0$
and
$g(w)=w+o(\|w-p\|^2)$
as
$w\rightarrow p$
. Since the Huang-type rigidity holds on D at p,
$g(w)\equiv w$
. Therefore, there exists a holomorphic mapping
$h:D\rightarrow \mathbb {C}^s$
, such that
$f|_D=(w,h(w))$
.
Write
$h(w)=(h_1(w),\dots ,h_s(w))$
. By the assumption of Shilov boundary size zero, we have
$|h_j(w)|\rightarrow 0$
as w goes to
$\partial _S D$
for all
$1\le j\le s$
. Since D is also assumed to satisfy the strong Shilov maximum principle, we deduce that
$h(w)\equiv 0$
, that is, f fixes D. In particular, f fixes
$\phi (\Delta )$
.
The theorem then follows from [Reference Huang5, Theorem 2.2].
Remark 2.2 As a primary example, consider
$D=\Delta ^r$
(
$r\ge 2$
) and
$p\in \partial \Delta ^r$
a smooth boundary point. Let
$\phi :\Delta \rightarrow \Delta ^r$
be a linear embedding with
$\phi (0)=0$
and
$\phi (1)=p$
. Together with Theorem 1.4, we see that
$(\Delta ^r,p)$
is a good pointed base. Note that
$\partial _S \Delta ^r=(\partial \Delta )^r$
.
Remark 2.3 Theorem 2.1 is still valid if we replace “as
$z\rightarrow p$
” by “as
$z\rightarrow p$
non-tangentially.” For a discussion in this direction, see, for example, [Reference Rong8].
3 Bounded symmetric domains
Let
$\Omega $
be a bounded symmetric domain in
${\mathbb {C}}^n$
of rank
$r\ge 1$
. By Harish-Chandra embedding theorem (see, for example, [Reference Wolf and Boothby-Weiss11, p. 286]),
$\Omega $
can be realized as a bounded convex domain in
${\mathbb {C}}^n$
.
Choose Harish-Chandra coordinates
$(z_1,\dots ,z_n)$
on
$\Omega $
such that
$\Omega $
is embedded into
${\mathbb {C}}^n$
as a bounded domain and
$({\mathbb {C}}^r\times \{0\})\cap \Omega =\Delta ^r\times \{0\}$
is a maximal totally geodesic r-disk (with respect to the Bergman metric of
$\Omega $
). Identify
$\Delta ^r$
with
$\Delta ^r\times \{0\}$
, and consider the canonical projection
$\pi :\mathbb C^n\rightarrow \mathbb C^r$
, given by
$(z_1,\dots ,z_n)\mapsto (z_1,\dots ,z_r)$
. Then, by [Reference Mok6, Theorem 1.1],
$\pi (\Omega )=\Delta ^r$
. Thus, under the Harish-Chandra realization described above,
$\Omega $
is a fibered domain over
$\Delta ^r$
, when
$1\le r\le n-1$
. Moreover, by [Reference Ng and Rong7, Lemma 3.1],
$\Omega $
is a fibered domain over
$\Delta ^r$
with Shilov boundary size zero.
Now we can prove the Huang-type rigidity on bounded symmetric domains.
Proof of Theorem 1.5
Since
$\Omega $
is homogeneous, we can assume without loss of generality that
$z_0$
is the origin, that is,
$f(0)=0$
. By the well-known Polydisk Theorem (see, for example, [Reference Wolf and Boothby-Weiss11, p. 280]), we may assume that
$p=(1,0,\dots ,0)$
under the Harish-Chandra coordinates described above.
Suppose that
$\Omega $
has rank r. If
$r=n$
, then
$\Omega =\Delta ^n$
and the theorem is just Theorem 1.4. Thus, we assume that
$1\le r\le n-1$
.
By Remark 2.2, we know that
$(\Delta ^r, p)$
is a good pointed base. By [Reference Ng and Rong7, Lemma 3.1],
$\Omega $
is a fibered domain with Shilov boundary size zero over the good pointed base
$(\Delta ^r, p)$
. Furthermore, as mentioned before,
$\Delta ^r$
satisfies the strong Shilov maximum principle. Finally, as in the proof of [Reference Ng and Rong7, Theorem 1.3], we know that there is a defining function for
$\Omega $
which satisfies the hypotheses of Theorem 2.1.
