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On holomorphic mappings of strictly pseudoconvex domains A. B. Sukhovab a Laboratoire Paul Painlevé, Université de Lille, Villeneuve d'Ascq, France b Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, Russia
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     § 1. IntroductionThis paper consists of two parts. In the first (§§ 2–4) we study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with $C^2$ boundaries. In the second part (§§ 5 and 6) we obtain a generalization of the Wong-Rosay theorem to piecewise smooth strictly pseudoconvex domains with noncompact automorphism groups. To date the old problem of boundary regularity of proper or biholomorphic mappings between strictly pseudoconvex domains in $\mathbb{C}^n$ has almost completely been clarified through contributions of several authors. A theorem of Fefferman [10] asserts that a biholomorphic mapping between strictly pseudoconvex domains with boundaries of class $C^\infty$ extends to a $C^\infty$-diffeomorphism between their closures. The original proof was based on the study of the asymptotic behaviour of the Bergman kernel near the boundary. It is difficult to adapt this method to the case of finite smoothness. Subsequently, several authors developed other approaches. One of them is due to Nirenberg, Webster and Yang [24] and uses a smooth version of the reflection principle. Pushing further their approach, Pinchuk and Khasanov [25] proved that a proper holomorphic mapping between strictly pseudoconvex domains with $C^s$-boundaries, where $s > 2$ is real, extends to the boundary as a map of class $C^{s-1}$ if $s$ is not an integer, and of class $C^{s-1-\varepsilon}$ for any $\varepsilon > 0$ when $s$ is an integer. Similar results were obtained by Lempert (see [20] and [21]) by quite different techniques of extremal discs for the Kobayashi metric. Khurumov [18] proved that even a better result still remains true with a loss of regularity of $1/2$. However, the natural question on the precise regularity in the case when the boundaries are exactly of class $C^2$ (which is the minimum possible regularity of the boundaries in the strictly pseudoconvex case) remains open. Khurumov announced without further details that his result is also true in this case but, to the best of my knowledge, a detailed proof is not available. A well-known result in this case was obtained independently by Henkin [16] and Pinchuk [26]; it states that the mapping extends to the boundary as a Hölder $1/2$-continuous mapping. Our first main result is the following. Theorem 1.1. Let $f\colon \Omega_1 \to \Omega_2$ be a proper holomorphic mapping between bounded strictly pseudoconvex domains $\Omega_j \subset \mathbb{C}^n$, $j = 1,2$, with boundaries $b\Omega_j$ of class $C^2$. Then $f$ extends to a mapping of Hölder class $C^{\alpha}(\overline{\Omega}_1)$ for each $\alpha \in [0,1[$ . In particular, this result means that the theorem of Lempert and Pinchuk- Khasanov mentioned above still remains true for boundaries of class $C^2$. This result was also obtained in [34] under the additional assumption that the boundary $b\Omega_1$ is of class $C^{2+\varepsilon}$, $\varepsilon > 0$. From this point of view the first part of our paper is a continuation of and a complement to [34]. The proof of Theorem 1.1 consists of two main steps. The first is based on estimates for the Kobayashi metric in a tubular neighbourhood of a totally real manifold (see [9]). These estimates allow us to obtain uniform Hölder estimates on analytic discs glued along an arc to a prescribed totally real manifold. This leads to the Hölder regularity of holomorphic mappings between wedge-type domains with totally real edges. This argument was presented in details in [34]. Here we consider the second main step of the proof of Theorem 1.1, and so we complete its proof. We study the geometric properties of analytic discs attached to a totally real manifold of class $C^1$ along an arc. This disc-gluing argument, which is often quite useful in the study of totally real submanifolds, was introduced in [27]. In the case when the regularity of a totally real manifold is higher than $C^1$, this construction was developed by several authors (see, for example, [35]). The $C^1$-case was considered by Chirka [17] and Khurumov [19]. However, for our goals we need some additional properties of such discs which were not explicitly stated in the papers [27], [17], [19] and [35] mentioned above. This is the reason why some details of this argument are presented here for completeness of exposition. It is worth stressing that I do not claim any originality: only applications of this construction are new. In order to present §§ 5 and 6 of our paper let us recall some definitions. Let $\Omega$ be a domain with nonempty boundary $b\Omega$ in a complex manifold $M$ of complex dimension $n > 1$. Definition 1.2. (a) We say that $p \in b\Omega$ is a piecewise smooth generic strictly pseudoconvex (boundary) point if the following assumptions hold: (b) A point $p$ is called smooth if $m=1$. Of course, in this case $p$ is a usual $C^2$-smooth strictly pseudoconvex boundary point. Clearly, condition i) can be stated in an equivalent form: the hypersufaces ${\Gamma_j = \{\rho_j = 0\}}$ (the local faces of $b\Omega$) are strictly pseudoconvex, that is, the Levi form of each $\Gamma_j$ is positive definite on the holomorphic tangent bundle of $\Gamma_j$. Condition ii) ensures that the real submanifold $\{\rho_j = 0,\, j=1,\dots,m\}$ (the corner) is generic. We denote by $\operatorname{Aut}(\Omega)$ the holomorphic automorphism group of $\Omega$ equipped with the standard compact-open topology. The limit set of $\operatorname{Aut}(\Omega)$ is the set of points $p \in \overline\Omega$ such that there exists a point $q \in \Omega$ and a sequence of automorphisms $(f^k)_k$ in $\operatorname{Aut}(\Omega)$ satisfying $\lim_{k \to \infty} f^k(q) = p$. If $z = (z_1,\dots,z_n)$ are the standard coordinates of $\mathbb{C}^n$, then we write $z = (z_1,z')$, where $z' = (z_2,\dots,z_n)$ and $z_j = x_j + iy_j$ for $x_j,y_j \in \mathbb R$. Also, $\|z\|^2 = \sum |z_j|^2$ denotes the Euclidean norm. In what follows we use the notation
$$
\begin{equation*}
\mathbb B^n=\bigl\{z \in \mathbb C\colon \|z\|^2<1\bigr\}
\end{equation*}
\notag
$$
for the Euclidean unit ball of $\mathbb{C}^n$ and we denote by
$$
\begin{equation*}
\mathbb H=\bigl\{z \in \mathbb C^n\colon \operatorname{Re} z_1+\|z'\|^2<0\bigr\}
\end{equation*}
\notag
$$
the unbounded realization of $\mathbb{B}^n$ (recall that the domain $\mathbb{H}$ is biholomorphic to $\mathbb{B}^n$ via the Caley transform). In the second part of this paper (§§ 5 and 6) we prove the following result. Theorem 1.3. Assume that the limit set of $\operatorname{Aut}(\Omega)$ contains a piecewise smooth generic strictly pseudoconvex point $p \in b\Omega$. Then $\Omega$ is biholomorphic to $\mathbb{B}^n$ and $p$ is a smooth strictly pseudoconvex point. We stress that in the hypotheses of the theorem we only assume a priori that $p$ is a piecewise smooth generic strictly pseudoconvex point, but the conclusion of the theorem says that in fact $p$ is necessarily a smooth point. Hence we have the following rigidity phenomenon. Corollary 1.4. A piecewise smooth, but not smooth (so that $m > 1$ in (1)) generic strictly pseudoconvex point cannot belong to the limit set of $\operatorname{Aut}(\Omega)$. The results of Theorem 1.3 and Corollary 1.4 are definitive: neither condition i) nor condition ii) in Definition 1.2 can be dropped as the following examples show. First, consider the domain
$$
\begin{equation*}
\Omega_1=\bigl\{\rho_1=\operatorname{Re} z_1+|z_2 |^2<0, \,\rho_2=\operatorname{Re} z_2<0 \bigr\},
\end{equation*}
\notag
$$
which is invariant with respect to the 1-parameter family of dilations $d_t\colon (z_1, z_2) \mapsto (t z_1, \sqrt{t} z_2)$, $t > 0$. This family is noncompact because it degenerates when $t = 0$, and the origin belongs to the limit set of $\operatorname{Aut}(\Omega_1)$. However, $\Omega_1$ is not biholomorphic to $\mathbb{B}^2$. The domain $\Omega_1$ satisfies ii) but does not satisfy i): one of its faces is not strictly pseudoconvex. Second, consider the domain
$$
\begin{equation*}
\Omega_2=\bigl\{z \in \mathbb C^2\colon \rho_1=\operatorname{Re} z_1+|z_2 |^2<0,\, \rho_2= \operatorname{Im} z_1+|z_2 |^2<0 \bigr\},
\end{equation*}
\notag
$$
which is invariant with respect to the same family of dilations $(d_t)$. Of course, $\Omega_2$ is not biholomorphic to $\mathbb{B}^2$ either. Here assumption i) holds, but ii) fails at the origin (though $d \rho_1 \wedge d \rho_2 \neq 0$). Note also that when the assumption of the strict pseudoconvexity of the faces is dropped, the situation becomes more complicated and several questions remain open. I mention here a recent result of Zimmer [37], who proved that if the faces are Levi flat, then $\Omega$ is biholomorphic to the product of a polydisc and a complex manifold with a compact automorphism group. Theorem 1.3 belongs to a series of results which are often called Wong-Rosay type theorems. This paper is not an expository one and so I do not present a detailed history and the state of the art of this direction of research (see [30]). In particular, I skip a discussion of (many beautiful) results dealing with the nonstrictly pseudoconvex case. The fact that a bounded strictly pseudoconvex domain in $\mathbb{C}^2$ with the maximum possible dimension (equal to $8$) of $\operatorname{Aut}(\Omega)$ (which a real Lie group ) is biholomorphic to $\mathbb{B}^2$ was already known to É. Cartan [4]. One can view this phenomenon as a special case of the general differential-geometric principle stating that structures with rich automorphism groups are usually flat. In [3] Burns and Shnider proved that a bounded strictly pseudoconvex domain $\Omega$ in $\mathbb{C}^n$ with noncompact automorphism group is biholomorphic to the unit ball. This was a striking and surprising result, because the assumption of noncompactness of the group $\operatorname{Aut}(\Omega)$ does not a priori impose restrictions on the dimension of $\operatorname{Aut}(\Omega)$. Their proof was based on the Chern-Moser theory [5] (more precisely, they used the part due to Chern, which extends É. Cartan’s approach to higher dimensions; the approach due to Moser is very different) and requires a relatively high regularity (at least $C^6$) of $b\Omega$. The group $\operatorname{Aut}(\Omega)$ is noncompact if and only if its limit set on the boundary of $\Omega$ is not empty. Wong [36] and Rosay [31] discovered that the result of Burns and Shnider can be localized: under the assumption that $\Omega$ is bounded it suffices to assume that the limit set of $\operatorname{Aut}(\Omega)$ contains a strictly pseudoconvex point. Perhaps, their most important observation was that the phenomenon discovered by Burns and Shnider can in fact be treated without the Cartan-Chern-Moser approach. It turned out that other geometric tools (such as biholomorphically invariant metrics and normal families of holomorphic maps) are more efficient and lead to more general results. Subsequently, their approach was considerably simplified by Pinchuk [29], who used his version of the so-called scaling method. The first purely local version of this phenomenon was obtained by Efimov [11]; he established Theorem 1.3 in the special case when $p$ is a smooth strictly pseudoconvex point (that is, $m = 1$ in Definition 1.2). In the nonsmooth case Coupet and Sukhov [7] proved that a bounded piecewise smooth strictly pseudoconvex domain with generic corners in $\mathbb{C}^n$ is biholomorphic to the unit ball if $\operatorname{Aut}(\Omega)$ is noncompact (and therefore the boundary of $\Omega$ is necessarily smooth). Theorem 1.3 generalizes all results mentioned above, beginning with the works of Wong and Rosay. We stress that these results are not used in our proof, which consists of two parts. The first concerns a localization of the Kobayashi- Royden metric. The second part (the principal one) is based on the scaling method. Notice that the proof of Efimov is based on the version of this method that is due to Pinchuk [29]; this version works well only near smooth boundary points. In the present paper we use the approach due to Frankel [12], which seems to be better adapted to the nonsmooth case. Frankel’s approach was also used in [7]. In our paper we simplify the arguments from [7] by reducing them to known estimates for the Kobayashi-Royden metric; this allows us to avoid general arguments of Frankel. This simplified approach works here because we deal with a special type of boundary point, while Frankel’s theory concerns general convex domains without making assumptions on their boundary regularity. § 2. Terminology and notationWe recall briefly some well-known definitions and basic notation. Let $\Omega$ be a domain in $\mathbb{C}^n$. For a positive integer $k$ we denote by $C^k(\Omega)$ the space of $C^k$-smooth complex-valued functions in $\Omega$. Also, $C^k(\overline\Omega)$ denotes the class of functions whose partial derivatives up to order $k$ extend to $\overline\Omega$ as continuous functions. Let $s > 0$ be a real noninteger, and let $k$ be its integer part. Then $C^s(\Omega)$ denotes the space of functions of class $C^k(\overline\Omega)$ such that their partial derivatives of order $k$ are (globally) $(s-k)$-Hölder continuous in $\Omega$; these derivatives satisfy automatically the $(s-k)$-Hölder condition on $\overline\Omega$, so the notation $C^s(\overline\Omega)$ for the same space of functions is also appropriate. We also use the space $L^p(\Omega)$, $p > 1$, of Lebesgue $p$-integrable functions with the usual norm $\|f\|_{L^p(\Omega)}$; if $f = (f_1,\dots,f_m)$ is a vector-valued function, then we set $\|f\|_{L^p(\Omega)} = \sum_{j=1}^m \|f_j\|_{L^p(\Omega)}$. In what follows we denote by $W^{k,p}(\Omega)$ (respectively, $W^{k,p}_{\mathrm{loc}}(\Omega)$) the Sobolev spaces of (vector-valued) functions with generalized derivatives of order up to $k$ which are Lebesgue $p$-integrable in $\Omega$ (respectively, locally). Denote by $\mathbb{D} = \{\zeta \in \mathbb{C}\colon |\zeta |< 1 \}$ the unit disc of $\mathbb{C}$. Recall that in this special case the Sobolev embedding theorem asserts that for $p > 2$ and $\alpha = 1 - 2/p$ the natural inclusion $W^{1,p}(\mathbb{D}) \to C^{\alpha}(\mathbb{D})$ is a compact bounded linear operator. A (closed) real submanifold $E$ of a domain $\Omega \subset \mathbb{C}^n$ is of class $C^s$ (with real ${s \geqslant 1}$) if for every point $p \in E$ there exists an open neighbourhood $U$ of $p$ and a map $\rho\colon U\to \mathbb{R}^d$ of maximum possible rank $d<2n$ and of class $C^s$ such that ${E \cap U = \rho^{-1}(0)}$; then $\rho$ is called a local defining (vector-valued) function of $E$. The positive integer $d$ is the real codimension of $E$. In the most important special case $d=1$ we obtain the class of real hypersurfaces. Let $J$ denote the standard complex structure of $\mathbb{C}^n$. In other words, $J$ acts on a vector $v$ by multiplication by $i$, that is, $J v = i v$. For every $p \in E$ the holomorphic tangent space $H_pE:= T_pE \cap J(T_pE)$ is the maximal complex subspace of the tangent space $T_pE$ of $E$ at $p$. Clearly $H_pE = \{v \in \mathbb{C}^n\colon \partial \rho(p) v = 0\}$. The complex dimension of $H_pE$ is called the CR-dimension of $E$ at $p$; a manifold $E$ is called a CR-manifold (Cauchy-Riemann manifold) if its CR-dimension is independent of ${p \in E}$. A real submanifold $E \subset \Omega$ is called generic (or generating) if the complex span of $T_pE$ coincides with $\mathbb{C}^n$ for all $p \in E$. Note that every generic manifold of real codimension $d$ is a CR-manifold of CR-dimension ${n-d}$. A function $\rho=(\rho_1,\dots,\rho_d)$ defines a generic manifold if $\partial\rho_1 \wedge \dots \wedge \rho_d \neq 0$. Of special importance are so-called totally real manifolds, that is, submanifolds $E$ for which $H_pE=\{0\}$ at every ${p \in E}$. A totally real manifold in $\mathbb{C}^n$ is generic if and only if its real dimension is equal to $n$; this is the maximum possible value of the dimension of a totally real manifold. Let $\Omega$ be a bounded domain in $\mathbb{C}^n$. Assume that its boundary $b\Omega$ is a (compact) real hypersurface of class $C^s$ in $\mathbb{C}^n$. Then there exists a $C^s$-smooth real function $\rho$ in a neighbourhood $U$ of the closure $\overline\Omega$ such that $\Omega = \{\rho<0\}$ and $d\rho|_{b\Omega} \ne 0$. We call such a function $\rho$ a global defining function. If $s \geqslant 2$, then we can consider the Levi form of $\rho$,
$$
\begin{equation}
L(\rho,p,v)=\sum_{j,k=1}^n \frac{\partial^2\rho}{\partial z_j\,\partial\overline{z}_k}(p)v_j \overline v_k.
\end{equation}
\tag{2}
$$
A bounded domain $\Omega$ with $C^2$-boundary is called strictly pseudoconvex if $L(\rho,p,v) > 0$ for every nonzero vector $v \in H_p(b\Omega)$. As usual, by a wedge-type domain we mean a domain
$$
\begin{equation}
W=\bigl\{z \in \mathbb C^n\colon \phi_j(z)<0, \,j=1,\dots ,n \bigr\}
\end{equation}
\tag{3}
$$
with edge (or corner)
$$
\begin{equation}
E=\bigl\{z \in \mathbb C^n\colon \phi_j(z)=0, \,j=1,\dots ,n \bigr\}.
\end{equation}
\tag{4}
$$
We assume that the defining functions $\phi_j$ are of class $C^{1}$. Furthermore, as usual, we assume that $E$ is a generic manifold, that is, $\partial \phi_1 \wedge \dots \wedge \partial \phi_n \neq 0$ in a neighbourhood of $E$. Given $\delta > 0$ (which is supposed to be small enough), we also define a shrunk wedge
$$
\begin{equation}
W_{\delta}=\biggl\{z \in \mathbb C^n\colon \phi_j-\delta \sum_{l \neq j} \phi_l<0,\, j=1,\dots ,n\biggr \} \subset W.
\end{equation}
\tag{5}
$$
It has the same edge $E$. Note that there exists a constant $C > 0$ such that for every point $z \in W_\delta$ one has
$$
\begin{equation}
C^{-1} \operatorname{dist}(z, b W) \leqslant \operatorname{dist}(z, E) \leqslant C \operatorname{dist} (z,b W).
\end{equation}
\tag{6}
$$
In what follows we often use the notation $C,C_1,C_2,\dots$ for positive constants which can change from line to line. § 3. Gluing complex discs to $C^1$ totally real manifoldsIn this section we present the main technical tool for the proof of Theorem 1.1. Consider a wedge-type domain (3) with edge (4). We need the well-known construction of filling a wedge $W$ (or, more precisely, the wedge $W_\delta$) by complex discs glued to $E$ along open arcs. A complex (or analytic, or holomorphic) disc is a holomorphic map $h\colon \mathbb{D} \to \mathbb{C}^n$ that is at least continuous on the closed disc $\overline{\mathbb{D}}$. We say that such a disc is glued (or attached) to a subset $K$ of $\mathbb{C}^n$ along an (open, nonempty) arc $\gamma \subset b\mathbb{D}$ if ${f(\gamma) \subset K}$. Our presentation consists of several steps, which are described in §§ 3.1 and 3.2. 3.1. The generalized Bishop equation; the existence and the regularity of discsLet $E$ be an $n$-dimensional totally real manifold of class $C^1$ in a neighbourhood of $0$ in $\mathbb{C}^n$; we assume that $0 \in E$. After a linear change of coordinates, using the implicit function theorem we may also assume that in a neighbourhood $\Omega$ of the origin the manifold $E$ is defined by the (vector) equation where the vector function $h = (h_1,\dots,h_n)$ is of class $C^1$ in a neighbourhood of $0$ in $\mathbb{R}^n$ and satisfies the conditions
$$
\begin{equation}
h_j(0)=0\quad\text{and} \quad\nabla h_j(0)=0, \qquad j=1,\dots ,n.
\end{equation}
\tag{8}
$$
Here and below $\nabla$ denotes the gradient. Fix a positive noninteger $s$. Consider the Hilbert transform $T\colon u \to Tu$, associating with a real function $u \in C^s(b\mathbb{D})$ its harmonic conjugate function vanishing at the origin. In other words, $u + iTu$ is the trace on $b\mathbb{D}$ of a function of class $C^s(\mathbb{D})$ which is holomorphic on $\mathbb{D}$ and satisfies $Tu(0) = 0$. Recall that the Hilbert transform is explicitly given by
$$
\begin{equation*}
Tu (e^{i\theta})=\frac{1}{2\pi} \operatorname{\text{v.p.}} \int_{-\pi}^{\pi} u(e^{it}) \cot \frac{\theta-t}{2}\, dt.
\end{equation*}
\notag
$$
This is a classical linear singular integral operator; it is bounded on the space $C^s(b\mathbb{D})$ for any noninteger $s > 0$. Furthermore, for $p > 1$ the operator $T$: $L^p(b\mathbb{D}) \to L^p(b\mathbb{D})$ is also a bounded linear operator; we denote its norm by $\|T\|_p$. Let $b\mathbb{D}^+ = \{e^{i\theta}\colon \theta\in [0,\pi]\}$ and $b\mathbb{D}^- = \{e^{i\theta}\colon \theta\in {]}\pi,2\pi[\,\}$ denote the upper and lower semicircles, respectively. Fix a $C^\infty$-smooth real functions $\psi_j$ on $b\mathbb{D}$ such that $\psi_j|b\mathbb{D}^+ = 0$ and $\psi_j|b\mathbb{D}^- < 0$, $j=1,\dots,n$ (we can take the same function for all $j$). Set $\psi = (\psi_1,\dots,\psi_n)$. Consider the generalized Bishop equation
$$
\begin{equation}
u(\zeta)=-Th(u(\zeta))-tT\psi(\zeta)+c, \qquad \zeta \in b\mathbb D,
\end{equation}
\tag{9}
$$
where $c\in\mathbb{R}^n$ and $t=(t_1,\dots,t_n)\in\mathbb{R}^n$, $t_j\geqslant0$, are real parameters; here and below we use the notation $tT\psi = (t_1 T\psi_1,\dots,t_n T\psi_n)$. We will prove that for any $p > 2$ and any $c$ and $t$ that are close enough to the origin this singular integral equation admits a unique solution $u(c,t)(\zeta)$ in the Sobolev class $W^{1,p}(b\mathbb{D})$ of vector functions. Such a solution is of class $C^{\alpha}(b\mathbb{D})$, $\alpha = 1 - 2/p$, by the Sobolev embedding theorem. Before we proceed to solving Bishop’s equation, we explain how such a solution is related to complex discs glued to $E$ along $b\mathbb{D}^+$. Indeed, consider the function
$$
\begin{equation*}
U(c,t)\zeta)=u(c,t)(\zeta)+i h(u(c,t)(\zeta))+i t\psi(\zeta).
\end{equation*}
\notag
$$
Since $T^2 = -\mathrm{Id}$ and $u$ is a solution of (9), the function $U$ extends holomorphically to $\mathbb{D}$ as a function of class $C^\alpha(\mathbb{D})$. Here $P$ denotes the Poisson operator of harmonic extension to $\mathbb{D}$:
$$
\begin{equation}
PU(c,t)(\zeta)=\frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1- |\zeta |^2}{|e^{it}-\zeta |^2} U(c,t)(e^{it})\,dt.
\end{equation}
\tag{11}
$$
The function $\psi$ vanishes on $b\mathbb{D}^+$, so by (7) we have $H(c,t)(b\mathbb{D}^+) \subset E$ for all $(c,t)$. It is convenient to extend equation (9) to the whole of $\mathbb{C}^n$. Fix a $C^\infty$-smooth function $\lambda\colon \mathbb{R}^{n} \to \mathbb{R}^+ = [0, + \infty[$ which is equal to $1$ on the unit ball $\mathbb{B}^{n}$ and vanishes on $\mathbb{R}^{n} \setminus 2\mathbb{B}^{n}$. For $\delta > 0$ small enough the function $h_\delta(x) = \lambda(x/\delta)h(x)$ extends naturally by $0$ to the whole of $\mathbb{R}^{n}$. Fix $\tau > 0$ small enough, which will be specified below. Then in view of (8) we can choose $\delta = \delta(\tau) > 0$ so that the gradient $\nabla h_\delta(x)$ is small on the whole of $\mathbb{R}^{n}$:
$$
\begin{equation}
\|\nabla h_\delta\|_{L^\infty(\mathbb R^{n})} \leqslant \tau.
\end{equation}
\tag{12}
$$
First we study the global equation
$$
\begin{equation}
u(\zeta)=-Th_\delta(u(\zeta))-tT\psi(\zeta)+c, \qquad \zeta \in b\mathbb D.
\end{equation}
\tag{13}
$$
We prove that its solutions depend continuously on the parameters $(c,t)$; this allows us to localize the solutions and make conclusions about the initial equation (9). Let $V$ be a domain in $\mathbb{R}^m$ and $f \in L^p(V \times b\mathbb{D})$, $p > 1$. Then by Fubini’s theorem $Tf \in L^p(V \times b\mathbb{D})$ (in the action of $T$ the variables in $V$ are viewed as parameters). Hence, keeping the same notation we obtain a bounded linear operator $T\colon L^p(V \times b\mathbb{D}) \to L^p(V \times b\mathbb{D})$ with the same norm as in $L^p(b\mathbb{D})$. Again, we denote its norm by $\|T\|_p$. Fix a domain $V\subset \mathbb{R}^{2n}$ of the parameters $(c,t)$. Lemma 3.1. Under the above assumptions, for any $p > 1$ one can choose $\tau > 0$ in (12) and $\delta = \delta(\tau) > 0$ so that equation (13) admits a unique solution $u(c,t)(\zeta) \in L^p(V \times b\mathbb{D})$. Proof. Consider the operator $\Phi\colon L^p(V \times b\mathbb{D}) \to L^p(V \times b\mathbb{D})$ defined by
$$
\begin{equation*}
\Phi\colon u \mapsto -Th_\delta(u(\zeta))-tT\psi(\zeta)+c.
\end{equation*}
\notag
$$
In view of (12) it follows from Lagrange’s mean value theorem that for all $u^1$ and $u^2$ in $L^p(V \times b\mathbb{D})$ we have
$$
\begin{equation*}
\begin{aligned} \, \|\Phi(u^1)-\Phi^2(u^2)\|_{L^p(V \times b\mathbb D)} &\leqslant \|T\|_p \, \| h_\delta(u^1)-h_\delta(u^2)\|_{L^p(V \times b\mathbb D)} \\ &\leqslant \frac12 \|u^1- u^2\|_{L^p(V \times b\mathbb D)}, \end{aligned}
\end{equation*}
\notag
$$
provided that a small enough $\tau$ in (12) is fixed. Hence $\Phi$ is a contracting map and the lemma is proved. Since $V$ is arbitrary, we conclude that (13) admits a unique solution $u \in L^p_{\mathrm{loc}}(\mathbb{R}^{2n} \times b\mathbb{D})$. (By this space we mean the space of functions belonging to $L^p$ on $K \times b\mathbb{D}$ for each (Lebesgue) measurable compact subset $K \subset \mathbb{R}^{2n}$. Next we study the regularity of solutions of (13) on the Sobolev scale. Let $\Omega$ be a domain in $\mathbb{R}^k$ and $f $ be a function in $\Omega$. Denote by $e_j$, $j=1,\dots,k$, the canonical basis of $\mathbb{R}^k$. Given $j = 1,\dots,k$ and $\Delta x_j \in \mathbb{R}^*$, consider the finite differences
$$
\begin{equation*}
\frac{\Delta f}{\Delta x_j}=\frac{f(x+e_j \Delta x_j)-f(x)}{\Delta x_j}.
\end{equation*}
\notag
$$
Recall two well-known properties of Sobolev spaces (see, for example, [23]):
Lemma 3.2. Every solution $u$ of (13) is of class Proof. Let $x_j$ denote one of the variables $c_j$, $t_j$ or $\zeta \in b\mathbb{D}$. We estimate the finite difference $\Delta u / \Delta x_j$. It follows from (12) that $h_\delta$ satisfies the Lipschitz condition with Lipschitz constant $\tau$. Hence from (13) we obtain where the $C_j$ are positive constants. When $\tau > 0$ is small enough, we obtain for some constant $C_3 > 0$, and therefore $u\in W^{1,p}_{\mathrm{loc}}(\mathbb{R}^{2n} \times b\mathbb{D})$. The lemma is proved. It follows from the Sobolev embedding theorem that a solution $u$ belongs to $C^{1-(2n+1)/p}(V\times b\mathbb{D})$, where $V$ is an open subset in $\mathbb{R}^{2n}$. In particular, the family of discs we have constructed is continuous in all variables for $p$ large enough. Now we note that for $t = 0$ equation (13) admits the constant solution $u(c,0)(\zeta) = c$. When $c$ is close enough to the origin in $\mathbb{R}^n$, this solution gives a point $c + ih(c) \in E$. By continuity and the uniqueness of solutions there exists a neighbourhood $V$ of the origin in $\mathbb{R}^{2n}$ such that for $(c,t) \in V$ any solution of (13) is a solution of (9). We have proved the following. Lemma 3.3. Given $p > 2$, there exists a neighbourhood $V$ of the origin in $\mathbb{R}^{2n}$ such that Bishop’s equation (9) admits a unique solution Since $p$ is arbitrary, we obtain that our equation admits solutions in the Hölder class $C^{\alpha}(V \times b\mathbb{D})$ for $\alpha = 1-(2n+1)/p$. Note that here $V$ depends on $p$ (and therefore on $\alpha$). Nevertheless, it follows from [9] that for each fixed point $(c,t)$ the map $\zeta \mapsto u(c,t)(\zeta)$ is of class $C^\alpha(b\mathbb{D})$ for every $\alpha < 1$. 3.2. The stability of discsUntil now we have not studied the geometric properties of the family (10). Here we consider some of them which will be useful for our applications. We represent the family (10) as a small perturbation of some model family in the $W^{1,p}$-norm. The model case arises when $E = \mathbb{R}^n$, that is, $h = 0$ in (4). Then the general solution of equation (9) has the form
$$
\begin{equation}
u(\zeta)=- tT\psi(\zeta)+c, \qquad \zeta \in b\mathbb D ,
\end{equation}
\tag{14}
$$
where, as usual, $c \in \mathbb{R}^n$ and $t = (t_1,\dots,t_n)$, $t_j \geqslant 0$, are real parameters. In this case the family (10) becomes where Geometrically, this family of discs arises from the family of complex lines intersecting $\mathbb{R}^n$ along real lines; discs are obtained by a mere biholomorphic reparametrization of the corresponding half-lines by the unit disc. These lines are given by $l(c,t)\colon \zeta \mapsto t\zeta + c$, $\zeta \in \mathbb{C}$. The conformal map $-T \psi + i\psi$ takes the unit disc to a smoothly bounded domain in the lower half-plane, gluing $b\mathbb{D}^+$ to the real axes. One can view the parameter $t$ as a directing vector of $l$. In what follows we refer to this as the flat case, and we call the discs (16) flat discs. Their geometric properties are very simple; their detailed description (in a more general case) is contained, for example, in [35]. Let $E$ be a totally real manifold given by (7), (8). Given $d \in I \setminus \{0 \}$, where $I \ni 0$ is a small enough open interval in $\mathbb{R}$, consider the manifolds $E_d$ given by Note that for every $d \neq 0$ the manifold $E_d$ is biholomorphic to $E$ via the isotropic dilation $z \mapsto d^{-1}z$.Set $h(x,d) = d^{-1}h(dx)$ for $d \neq 0$ and $h(x,0) = 0$. In the last case, when $d = 0$, we have $E_0 = \{y = 0 \} = \mathbb{R}^n = T_0(E)$, which is the flat case. Note that the function $h(x,d)$ and its first-order partial derivatives with respect to $x$ are continuous in ${d \in I}$. Thus, we consider the $1$-parameter family $E_d$ of totally real manifolds defined by the equation
$$
\begin{equation}
h_j(0,d)=0, \quad \nabla_x h_j(0,d)=0, \qquad d \in I, \quad j=1,\dots ,n
\end{equation}
\tag{19}
$$
(we consider the gradient $\nabla_x$ with respect to $x$). Hence for each tuple $(c,t,d)$ we have the discs $H(c,t,d)$ defined by (10). By the uniqueness of the solution of Bishop’s equation the family $H(c,t,0)(\zeta)$ coincides with (16). Proof. Let $u^0$ be a solution of the flat Bishop equation (14). For any $d$ let $u(c,t,d)$ be a solution of Bishop’s equation It follows from (19) that the previous estimates on the norm of $u$ are uniform in $d$, that is, $\|u\|_{W^{1,p}(V \times \mathbb{D})} \leqslant C$, where the constant $C > 0$ is independent of $d$. Indeed, in the above estimates $\tau > 0$ and $\delta(\tau) > 0$ can be chosen independently of $d$.
Since $u \in W^{1,p}(V \times \mathbb{D})$ for any $d$, and since $h(\,\bullet\,,d)$ is of class $C^1(\mathbb{R}^n)$ for every $d$, the composition $h(u,d)$ is also of class $W^{1,p}(V \times \mathbb{D})$. Hence the chain rule can be applied to generalized derivatives. Therefore, if $s_j$ denotes one of the variables $c_j$, $t_j$ and $\zeta$, then we have $h(u,d)_{s_j} = (D_xh)(u,d) u_{s_j}$, where $D_x$ denotes the tangent map with respect to $x$. Therefore, where $\|h(\,\bullet\,,d)\|_{C^1_x}$ denotes the $C^1$-norm of $h(x,d)$ with respect to the variables $x$, and $C$ is a positive constant. But $\|h(\,\bullet\,,d)\|_{C^1_x} \to 0$ as $d \to 0$, and the lemma follows.§ 4. Proof of Theorem 1.1The following proposition is the key technical step. Proposition 4.1. Let $M$ be an $m$-dimensional totally real manifold of class $C^1$ in $\mathbb{C}^m$. Assume that $\Omega \subset \mathbb{C}^n$ is a pseudoconvex domain defined by $\Omega = \{\phi < 0\}$, where $\phi$ is a plurisubharmonic function of class $C^2$ and $d \phi \neq 0$ near $\Gamma=b\Omega$. Also let $W \subset \Omega$ be a wedge (3) with edge $E\subset \Gamma$ of type (4). Consider a holomorphic map $f\colon\Omega \to \mathbb{C}^m$ such that $f$ is continuous on $\Omega \cup E$ and $f(E) \subset M$. Then for every $\delta > 0$ and every $\alpha < 1$ the map $f$ extends to a Hölder $\alpha$-continuous map of ${W_{\delta} \cup E}$. Proof. Everything is local; we may assume that $0 \in E$ and $T_0E = \mathbb{R}$, as in the previous sections. Consider the family of discs constructed in § 3 and attached to $E$ along $b\mathbb{D}^+$. The flat discs fill a prescribed wedge of type (3) with edge $E_0 = \mathbb{R}^n$. More precisely, we can fix an open convex cone $K$ in $W^0 = \{(x,y)\in \mathbb{R}^{2n}\colon {y_j < 0}, j = 1,\dots,n\}$ with vertex at the origin and such that for $r > 0$ small enough $\overline K \cap r\mathbb{B}^n$ is contained in $W^0 \cup \{0\}$. Clearly, the flat discs fill a neighbourhood of $\overline K \cap r \mathbb{B}^n$. The same remains true for the cone $K_z$ obtained by the parallel translation of $K$ to a cone with vertex $z \in \mathbb{R}^n$. Since the family $H(c,t,d)(\zeta)$ is a small perturbation of the flat discs in $C^s(V \times \overline{\mathbb{D}})$ (for any $s$, $0 < s < 1$), we conclude by continuity that for $d$ small enough the family $H(c,t,d)(\zeta)$ also fills a prescribed edge of type (5) with edge $E_d$. By holomorphic equivalence the same is true for the original edge $E$ and the shrunk wedge $W_\delta$ for any $\delta > 0$. Note that in this construction we consider the discs $H(c,t,d)(\zeta)$ for the parameter $t$ separated from the origin, so that these discs do not degenerate to the constant discs $H(c,0,d)(\zeta) \equiv c$. Also note that since $\phi$ is plurisubharmonic, it follows from the maximum principle that all discs belong to $\Omega$.
Applying the strong version of Hopf’s lemma (see [28]) to the subharmonic function $\phi \circ H$ on $\mathbb{D}$ we obtain where $C > 0$ is independent of discs, that is, independent of $(c,t)$ (note that we drop $d$ in the notation since $d$ is fixed). Recall that
$$
\begin{equation*}
C^{-1} \operatorname{dist} (z, \Gamma) \leqslant |\phi(z) |\leqslant C \operatorname{dist}(z,\Gamma).
\end{equation*}
\notag
$$
Since $E \subset \Gamma$, we have
$$
\begin{equation*}
\operatorname{dist}(z, \Gamma) \leqslant \operatorname{dist}(z,E).
\end{equation*}
\notag
$$
This implies the estimate
$$
\begin{equation}
1- |\zeta |\leqslant C \operatorname{dist}(H(c,t)(\zeta),E).
\end{equation}
\tag{21}
$$
Any point $z \in W_\delta$ belongs to some disc $H(c,t)$. Setting $z = H(c,t)(\zeta)$, from (21) we obtain the estimate: On the other hand recall that $M$ can be defined by $M = \rho^{-1}(0)$, where $\rho$ is a nonnegative strictly plurisubharmonic function of class $C^2$ (see [8] and [15]). The following estimate was obtained in [34]: More precisely, in [34] this estimate was obtained for $E$ of class $C^s$, $s > 1$. Using the family of discs $H(c,t)$ constructed here for $E$ of class $C^1$ we repeat the argument in [34] and obtain the same estimate in our case, when $E$ is of class $C^1$.Hence we obtain the key estimate for all $z \in W_\delta$.This estimate established, the argument from [34] (based on estimates for the Kobayashi metric in a tubular neighbourhood of $M$) goes through literally and shows that $f$ is $\alpha$-Hölder on $W_\delta$ for each $\alpha < 1$ (see Lemma 3.6 in [34]). This proves the proposition. Now Theorem 1.1 follows exactly as in [34]. Indeed, the approach in [34] was based on the well-known construction developed in [25]. It reduces the study of boundary regularity of a biholomorphism $f\colon \Omega_1 \to \Omega_1$ between strictly pseudoconvex domains to the study of a suitably defined holomorphic lift of $f$ to $\Omega_1 \times \mathbb{C} \mathbb{P}^{n-1}$. The holomorphic tangent bundle of $b\Omega_1$ can be viewed as a $(2n-1)$-dimensional totally real submanifold $E$ of the boundary of $\Omega_1 \times \mathbb{C}\mathbb{P}^{n-1}$. Furthermore, the lift of $f$ extends to $E$ continuously and takes it to the holomorphic tangent bundle over $b\Omega_2$. Now we apply Proposition 4.1 to the lift of $f$. The proof of Theorem 1.1 is complete. § 5. The Kobayashi-Royden metric and normal familiesFor the convenience of the reader, in this section we recall several results concerning the Kobayashi-Royden metric. We fix an arbitrary Riemannian metric on $M$ inducing the usual topology of $M$ and use it to measure distances on $M$ and the norms of tangent vectors. In the case when $M = \mathbb{C}^n$ we always use the standard Euclidean norm and metric. In what follows we denote by $\mathbb{D} = \{\zeta \in \mathbb{C}\colon |\zeta|< 1\}$ the unit disc in $\mathbb{C}$ (that is, $\mathbb{B}^1$). Also let $\mathcal{O}(\mathbb{D},M)$ denote the space of holomorphic maps from $\mathbb{D}$ to $M$; we call such maps complex or analytic discs in $M$. Recall that the Kobayashi-Royden pseudometric $F_M$ of $M$ is defined at a point $p \in M$ and a tangent vector $v \in T_pM$ by
$$
\begin{equation*}
\begin{aligned} \, F_M(p,v)=\inf\biggl\{ \lambda^{-1} \colon &\text{there exists }f \in \mathcal O(\mathbb D,M) \\ &\text{such that }f(0)=p,\,\frac{df}{d\zeta}(0)=\lambda v, \, \lambda>0 \biggr\}. \end{aligned}
\end{equation*}
\notag
$$
Denote by $K_M(p,q)$ the usual Kobayashi pseudodistance of $M$ between points $p,q \in M$. According to a fundamental result of Royden [32], $F_M$ is an upper semicontinuous function on the tangent bundle of $M$ and $K_M$ is the integration form of $F_M$. We will use the fundamental property of the Kobayashi-Royden pseudometric and the Kobayashi pseudodistance: they are holomorphically decreasing. Namely, if $f\colon M \to N$ is a holomorphic map between two complex manifolds, then $F_N(f(p), df(p) v) \leqslant F_M(p,v)$ and $K_N(f(p),f(q)) \leqslant K_M(p,q)$. Also recall that $M$ is said to be hyperbolic at $p \in M$ if there exists a constant $C > 0$ such that $F_M(p,v) \geqslant C\|v\|$ for every tangent vector $v \in T_pM$, and $M$ is said to be locally hyperbolic if it is hyperbolic at every point. Also, $M$ is called (Kobayashi) hyperbolic if $K_M$ is a distance, that is, $K_M(p,q) > 0$ for $p \neq q$; in this case it induces the usual topology of $M$. According to [32], $M$ is hyperbolic if and only if it is locally hyperbolic. Also recall (see [32]) that $M$ is hyperbolic if and only if the family $\mathcal{O}(\mathbb{D},M)$ is equicontinuous (with respect to the Riemannian metric fixed above). The Kobayashi ball with centre $p$ and radius $\delta > 0$ is defined by
$$
\begin{equation*}
B_{K_M}(p,\delta)=\bigl\{ q \in M \colon K_M(p,q)<\delta \bigr\}.
\end{equation*}
\notag
$$
Recall that a manifold $M$ is complete hyperbolic if it is a complete space with respect to the Kobayashi distance, that is, every Kobayashi ball is compactly contained in $M$. 5.1. Localization and normal familiesHere we discuss some results on the localization and asymptotic behaviour of the Kobayashi-Royden metric. In what follows we use the same convention as above: $C $ denotes a positive constant, which is allowed to change its value from estimate to estimate. We begin with the following localization principle, which follows from Lemma 2.2 in [6]. Lemma 5.1. Let $p \in b\Omega$ be a piecewise smooth strictly pseudoconvex point. Then there exist open neighbourhoods $U$ and $U'$, $U \subset U'$, of $p$ in $M$ and $\delta > 0$ such that for every $q \in \Omega \cap U$ the Kobayashi ball $B_{K_\Omega}(q,\delta) $ is contained in $\Omega \cap U'$. The hypotheses in [6] assume the existence of a negative plurisubharmonic function on $\Omega$ that is strictly plurisubharmonic (in the generalized sense) in a neighbourhood of $p$. Applying the construction from [33] to the functions $\rho_j$ in (1) we can extend each of these functions to a plurisubharmonic function, say $\widetilde \rho_j$, which is globally defined and negative on $\Omega$. Then the function $\sup_j \widetilde \rho_j$ satisfies the assumptions made in [6]. Since the Kobayashi distance is holomorphically decreasing, we obtain the following. Corollary 5.2. There exists $\tau = \tau(\delta) > 0$ such for every point $q \in \Omega \cap U$ and every holomorphic map $h\colon \mathbb{D} \to \Omega$ satisfying $h(0) = q$ one has $h(\tau \mathbb{D}) \subset \Omega \cap U'$. The localization principle in this form was obtained and used by Berteloot; see [2], for example. It follows now from the definition of the Kobayashi-Royden metric that there exists a constant $C > 0$ such that for all $z \in \Omega \cap U$ and $v \in T_z\Omega$.As a consequence of these localization results, we have the following. Lemma 5.3. In the hypotheses of Theorem 1.3 there exists a subsequence of the sequence $(f^k)$ that converges to the constant map $f \equiv p$ uniformly on compact subsets of $\Omega$. Choose a coordinate neighbourhood $U \subset U'$ of $p$ small enough so that Corollary 5.2 can be applied. Let $K$ be a compact subset of $\Omega$ containing the point $q$. We claim that $f^k(K) \subset \Omega \cap U'$ for $k$ large enough. Consider two finite coverings of $K$ by open coordinate neighbourhoods $V_j$ and $W_j$, $j = 1,\dots,N$, such that $V_j \subset W_j \subset \Omega$, and the following holds:
For $k$ large enough $f^k(q) \in U$. Given a unit vector $v \in \mathbb{C}^n$, we apply Corollary 5.2 to the discs $h^k\colon \mathbb{D} \to \Omega$, $h^k\colon \mathbb{D} \ni \zeta \mapsto f^k \circ \phi_1^{-1}(\zeta v)$. This implies that $f^k(V_1) \subset U'$. Hence there exists a subsequence, denoted by $(f^k)$ again, that converges uniformly on $\overline V_1$ to a holomorphic map $f$. Since $f(q) = p$, by the maximum principle we have $f \equiv p$. Then by (iii), $f^k(q^2) \in U$ for $k$ large enough, and a similar argument shows that $f^k(V_2) \subset U'$. Repeating this argument for all $j$ we conclude the following. Indeed, let $z^0$ be an arbitrary point of $\Omega$. Then $f^k(z^0) \in \Omega \cap U$ for $k$ large enough. But the domain $\Omega \cap U$ is biholomorphic to a bounded domain in $\mathbb{C}^n$, and so it is hyperbolic. Therefore,
$$
\begin{equation*}
F_\Omega(z^0,v)=F_\Omega(f^k(z^0),df^k(z^0)v) \geqslant C F_{\Omega \cap U}(f^k(z^0),df^k(z^0)v) \geqslant C \|v \|
\end{equation*}
\notag
$$
by (23). Here we have used the fact that $\Omega \cap U$ is hyperbolic. Hence $\Omega$ is locally hyperbolic and so $\Omega$ is hyperbolic. 5.2. EstimatesWe assume that $\Omega$ satisfies the assumptions of Theorem 1.3. The following upper bound for the Kobayashi-Royden infinitesimal metric $F_\Omega$ is classical. Lemma 5.5. Given any point $p$ in $M$, there exist a constant $C > 0$ and a (coordinate) neighbourhood $U$ of $p$ such that for each $z \in \Omega$ and a tangent vector $v \in T_z\Omega$ one has Indeed, the ball with centre $z$ and radius $\operatorname{dist}(z, b\Omega)$ lies in $\Omega$, so the estimate follows from the holomorphic decreasing property of the Kobayashi-Royden metric. For a bound from below, recall some results from [33]. Lemma 5.6. There exists a neighbourhood $U$ of $p$ in $M$ and a constant $C > 0$ such that for every $z \in \Omega \cap U$ and $v \in T_z\Omega$. Finally, we need estimates for the Kobayashi-Royden metric on convex domains. Let $G \subset \mathbb{C}^n$ be a convex domain, $p \in G$ be a point and $v$ be a vector in $\mathbb{C}^n$. Consider a complex line $A$ through $p$ in the direction $v$ and set
$$
\begin{equation*}
L_G(p,v)=\sup \bigl\{ \delta>0\colon \mathbb B^n(p,\delta) \cap A \subset G \bigr\}.
\end{equation*}
\notag
$$
In other words, $L(p,v)$ is the supremum of the radii of discs with centre $p$ that are contained in $A \cap G$. The following result is due to Graham [14] and Frankel [13]; a short geometric proof was obtained by Bedford and Pinchuk in [1]. Lemma 5.7. Let $G$ be a convex domain in $\mathbb{C}^n$. Then for all $p \in \Omega$ and $v \in \mathbb{C}^n$ This result implies many useful consequences. For example, $G$ becomes convex after a biholomorphic change of coordinates near a smooth strictly pseudoconvex boundary point. Then Lemma 5.7 implies that $F_G(z,v) \geqslant C/\operatorname{dist} (z, b\Omega)$ for vectors $v$ which are transverse (for instance, orthogonal) to the holomorphic tangent space to $bG$ at $p$. This implies the classical fact that a smoothly bounded strictly pseudoconvex domain is complete hyperbolic. § 6. Proof of the Theorem 1.3Our approach is based on [7] and employs the scaling method due to Frankel [12]. However, in contrast to [7] we do not use Frankel’s general results on the convergence of dilated families. Our proof is self-contained and uses only Lemma 5.7. Assume that we are under the hypotheses of Theorem 1.3. 6.1. ScalingAssume that $\Omega$ is of the form (1) in a coordinate neighbourhood $U$ of $p$. Recall that the strictly pseudoconvex hypersurfaces $\Gamma_j = \{\rho_j = 0\}$ are called faces of $b\Omega \cap U$. Lemma 6.1. There exists a local biholomorphic change of coordinates $\Phi$ such that $\Phi(p) = 0$ and $\Phi(\Omega \cap U)$ is convex. For the proof, see Proposition 1.1 in [7]. In what follows we assume that local coordinates are fixed in accordance with Lemma 6.1, so that $\Omega \cap \varepsilon \mathbb{B}^n$ is convex for $\varepsilon > 0$ small enough (slightly abusing the notation, we identify $\Phi(\Omega \cap U)$ with $\Omega \cap \varepsilon \mathbb{B}^n$). One can also choose these coordinates so that, in addition, every local defining function of $\Omega$ near the origin has an expansion
$$
\begin{equation}
\rho_j(z)=\operatorname{Re} z_j+H_j(z,\overline z)+S_j(z ), \qquad j=1,\dots ,m,
\end{equation}
\tag{24}
$$
where each $H_j$ is a positive definite Hermitian quadratic form and $S _j(z) = o(|z|^2)$. Let $\Omega_k = (f^k)^{-1}(\Omega \cap \varepsilon \mathbb{B}^n)$. Since the sequence $(f^k)$ converges to $0$ uniformly on compact subsets of $\Omega$, each compact subset of $\Omega$ is contained in all domains $\Omega_k$ for $k$ large enough. Fix a point $q$ belonging to all the $\Omega_k$ for $k$ large enough. Set $p^k := f^k(q)$ and consider the affine linear maps Define the new sequence of maps Note that
$$
\begin{equation}
g^k(q)=0 \quad\text{and}\quad dg^k(q)=\mathrm{Id} \quad \text{for all } k.
\end{equation}
\tag{26}
$$
Consider the images $G_k = g^k(\Omega_k) = A^k(\Omega \cap \varepsilon \mathbb{B}^n)$. Our ultimate goal is to prove that the sequence of convex domains $(G^k)$ converges to a domain $G$ in the Hausdorff distance and to determine this limit domain $G$. 6.2. Convergence of domainsFirst we note that the tangent maps converge to $0$; therefore, the domains $(df^k(q))^{-1}(\varepsilon \mathbb{B}^n - p^k)$ converge to the whole space $\mathbb{C}^n$. For this reason they do not affect our argument, and we do not mention them anymore. Every domain $G_k$ is defined by
$$
\begin{equation*}
\bigl\{ z \colon \rho_j(p^k+R^k z)<0, \,j=1,\dots ,m \bigr\}.
\end{equation*}
\notag
$$
Set $\tau_k^j:= |\rho_j(p^k)|$ and $\delta_k:= \inf_j \tau^j_k$. Consider the functions
$$
\begin{equation*}
\phi_j^k(z)=(\tau_k^j)^{-1} \rho_j(p^k+R^k z), \qquad j=1,\dots ,m.
\end{equation*}
\notag
$$
Their expansions at the origin have the from
$$
\begin{equation}
\begin{aligned} \, \notag \phi_j^k(z) &=-1+\operatorname{Re} \lambda_j^k(z)+(\tau_k^j)^{-1}\operatorname{Re} Q_j^k(R^k z, R^k z) \\ &\qquad +(\tau_k^j)^{-1}H^k_j(R^k z, \overline{R^k z})+S_j^k(z). \end{aligned}
\end{equation}
\tag{27}
$$
Here the $\lambda_j^k$ are complex linear forms, $Q_j^k(w,w)$ are holomorphic quadratic forms and in view of (24) one has $Q_j^k \to 0$ as $k \to \infty$; the $H^k_j(w, \overline w)$ are positive definite quadratic forms converging to $H_j$ from (24). Finally, $S^k_j(z) = o(|z|^2)$ uniformly in $k$. Lemma 6.2. For every $j$ the sequence $(\phi_j^k)_k$ converges (after passing to a subsequence) uniformly on compact subsets of $\mathbb{C}^n$ as $k \to \infty$ to a function Here every $\lambda_j$ is a complex linear form and every $H_j'$ is a nonnegative Hermitian quadratic form. The domains $G_k$ converge in the Hausdorff distance to the domain which is hyperbolic at the origin. Proof. There exists $C > 0$ such that for any $k$ and $j$
$$
\begin{equation*}
C^{-1} \operatorname{dist} (p^k,\Gamma_j) \leqslant \tau_j^k \leqslant C \operatorname{dist}(p^k, \Gamma_j).
\end{equation*}
\notag
$$
Since $\operatorname{dist}(p^k,b\Omega) = \inf_j \operatorname{dist}(p^k,\Gamma_j)$, we have
$$
\begin{equation*}
C^{-1} \operatorname{dist} (p^k, b\Omega) \leqslant \delta_k \leqslant C \operatorname{dist}(p^k, b\Omega).
\end{equation*}
\notag
$$
Lemmas 5.6 and 5.5 imply the following estimates (for each $v \in \mathbb{C}^n$):
$$
\begin{equation*}
C^{-1} \|v \|\geqslant F_{\Omega_k}(q,v) \geqslant F_{\Omega \cap \varepsilon \mathbb B^n}(p^k, R^kv) \geqslant \frac{C \|R^k v\|}{\delta_k^{1/2}},
\end{equation*}
\notag
$$
which give
$$
\begin{equation}
\|R^k \|\leqslant C (\delta_k)^{1/2} \leqslant C (\tau_k^j)^{1/2}
\end{equation}
\tag{29}
$$
for all $j$. As a consequence, we obtain that the sequence $(\tau_k^j)^{-1}\operatorname{Re} Q_j^k(R^k z, R^k z)$ in (27) converges to $0$ uniformly on compact subsets of $\mathbb{C}^n$ as $k\to \infty$ and $(\tau_k^j)^{-1}H^k_j(R^k z, \overline{R^k z})$ converges uniformly on compact subsets of $\mathbb{C}^n$. It is also easy to see that $S^k_j$ converges to $0$ uniformly on compact subsets of $\mathbb{C}^n$.
Next, it follows from (26) that $F_{\Omega_k}(q,v) = F_{G_k}(0,v)$ for all $k$ and $v \in \mathbb{C}^n$. Therefore, there exists $C > 0$ such that
$$
\begin{equation*}
C^{-1} \|v \|\leqslant F_{G_k}(0,v) \leqslant C \|v\|.
\end{equation*}
\notag
$$
Since the domains $G_k = \{\phi_j^k < 0,\, j=1,\dots,m\}$ are convex, by Lemma 5.7 we have for all $k$ and $v$. Arguing ad absurdum, assume that the norms $\alpha_j^k$ of the forms $\lambda^k_j$ are not bounded in $k$; we can assume that $\alpha_j^k \to \infty$. Then the functions $(\alpha_j^k)^{-1}\phi_j^k$ converge to functions $\operatorname{Re} \theta_j(z)$, where $\theta_j$ is a nontrivial complex linear form. This means that the boundaries of the convex domains $G_k$ approach the origin as $k \to \infty$ and for some nonzero vector $v$ we have $L_{G_k}(0,v) \to 0$ as $k \to \infty$. This contradiction proves that the sequence of norms of the forms $\lambda_j^k$ is bounded and completes the proof of the lemma. Now our goal is to prove that $m=1$ and $G$ is biholomorphic to $\mathbb{B}^n$. 6.3. The identification of $G$: the simplest case $m=1$First we consider the simplest case when $m = 1$. Then
$$
\begin{equation*}
G =\bigl\{ z\colon {-}1+\operatorname{Re}\lambda(z)+H(z,\overline z)<0 \bigr\}.
\end{equation*}
\notag
$$
If a nonzero vector $v$ lies in the intersection $\ker \lambda \cap \ker H = \{0 \}$, then the complex line through the origin in the direction of $v$ is contained in $G$ and $L_{G}(0,v) = \infty$, which contradicts (30). Hence the restriction of $H$ to $\ker \lambda$ is positive definite and $G$ is biholomorphic to $\mathbb{B}^n$. In order to complete the proof of Theorem 1.3 in this case, we need the following. Lemma 6.3. The sequence of maps (25) converges (after passing to a subsequence) uniformly on compact subsets of $\Omega$ to a biholomorphism between $\Omega$ and $G$. Proof. Fix a compact subset $K \subset G$. Since the sequence of convex domains $(G_k$) converges to $G$, there exists $k_0$ such that $K \subset G_k$ for all $k \geqslant k_0$. It follows from Lemma 5.7 that there exists $C > 0$ such that
$$
\begin{equation*}
F_{G_k}(z,v) \geqslant C\|v \| \quad \text{for all } z \in K, \ v \in \mathbb C^n\text{ and } k \geqslant k_0.
\end{equation*}
\notag
$$
Then the classical argument (see [32]) shows that the family $(g^k)$ is normal. Since $g^k(q) = 0$ for all $k$, the sequence $(g^k)$ contains a subsequence converging uniformly on compact subsets of $\Omega$ to a holomorphic map $g$. On the other hand the domain $\Omega$ is hyperbolic (Corollary 5.4). Hence a similar argument implies the convergence of the family of inverse maps $((g^k)^{-1})$. Now a classical theorem of H. Cartan (see [22]) shows that $g\colon \Omega \to G$ is biholomorphic. The lemma is proved. 6.4. Identification of $G$: the case $m> 1$Now we consider the case $m > 1$. More precisely, our goal is to prove by arguing ad absurdum that such a situation never occurs in fact. In this case it is appropriate to modify slightly the scaling sequence $(g^k)$. Namely, consider the linear maps
$$
\begin{equation*}
\begin{gathered} \, B^k\colon z \mapsto w, \\ w_j=\sum_{l=1}^n \frac{\partial \rho_j}{\partial z_l}(p^k) z_l, \qquad j=1,\dots ,m, \\ w_j=z_j, \qquad j=m+1,\dots ,n. \end{gathered}
\end{equation*}
\notag
$$
Note that the sequence of $(B^k)_k$ converges to the identity map as $k$ tends to $\infty$. Consider the sequence of maps and consider the domains Then
$$
\begin{equation*}
\widetilde G^k=\bigl\{ z\colon \rho_j(p^k+(B^k)^{-1} \circ R^k z)<0, \,j=1,\dots, m \bigr\}.
\end{equation*}
\notag
$$
Precisely as above, consider the functions
$$
\begin{equation*}
\phi_j^k(z)=(\tau_k^j)^{-1} \rho_j(p^k+(B^k)^{-1} \circ R^k z).
\end{equation*}
\notag
$$
Let where the $R^k_j$ are complex linear forms. Then the expansions at the origin have the form
$$
\begin{equation}
\begin{aligned} \, \notag \phi_j^k(z) &=-1+(\tau_k^j)^{-1} \operatorname{Re} R_j^k(z)+(\tau_k^j)^{-1}\operatorname{Re} Q_j^k(R^k z, R^k z) \\ &\qquad +(\tau_k^j)^{-1}H^k_j(R^k z, \overline{R^k z})+S_j^k(z). \end{aligned}
\end{equation}
\tag{31}
$$
Here the $R_j^k$ are complex linear forms as above (namely, the components of $R^k$; the additional maps $B^k$ have been introduced in the scaling process in order to make the $\tau^k_j$ appear explicitly in this expansion), the $Q_j^k(w,w)$ are holomorphic quadratic forms such that, in view of (24), $Q_j^k \to 0$ as $k \to \infty$; the $H^k_j(w, \overline w)$ are positive definite quadratic forms converging to $H_j$ from (24). Note that we have used the fact that the sequence $(B^k)$ converges to the identity map. Finally, $S^k_j(z) = o(|z|^2)$ and tends to $0$ as $k$ goes to $\infty$. Now Lemma 6.2 can be applied for every $j$. We obtain that the sequence $(\phi_j^k)_k$ converges (after passing to a subsequence) to
$$
\begin{equation}
\phi_j (z)=-1+\operatorname{Re} \lambda_j(z)+H_j'(z,\overline z), \qquad j=1,\dots, m.
\end{equation}
\tag{32}
$$
Here every $\lambda_j$ is a complex linear form, and every $H_j'$ is a Hermitian quadratic form. Notice that the forms $H_j'$ are nonnegative, hence the functions $\phi_j$ are plurisubharmonic. The key observation is that we can obtain more information about the limit functions (32). We simplify their expressions step by step. First it follows from (29) that (after passing to a subsequence) the sequence $R^{k}/{\delta_k}^{1/2}$ converges to a $\mathbb{C}$-linear map
$$
\begin{equation*}
L=(L_1,\dots ,L_n)\colon \mathbb C^n \to \mathbb C^n.
\end{equation*}
\notag
$$
Furthermore, for each $j$ the sequence $(\delta_k/\tau_k^j)_k$ is bounded. Passing to a subsequence we assume that it converges to some $\kappa_j \geqslant 0$. Since $\delta_k = \min_{j=1,\dots,m} \tau^j_k$, at least one $\kappa_j$ is different from zero (in fact, it is equal to $1$). Then the sequence
$$
\begin{equation*}
(\tau_k^j)^{-1} H^k_j (R^z,\overline{R^k z})=\frac{\delta^k}{\tau_k^j} H^k_j ((\delta^k)^{-1/2}R^kz, (\delta^k)^{-1/2}\overline{R^kz})
\end{equation*}
\notag
$$
converges to $\kappa_j H_j(L z, \overline{Lz})$, where $H_j$ is the Levi form from (24); recall that the forms $H_j(w,\overline w)$ are positive definite on $\mathbb{C}^n$. Thus,
$$
\begin{equation}
\phi_j(z)=-1+\operatorname{Re} \lambda_j(z)+\kappa_j H_j(Lz, \overline{Lz}), \qquad j=1,\dots ,m.
\end{equation}
\tag{33}
$$
We can assume that for some integer $s$, $1 \leqslant s \leqslant m$, we have $\kappa_j > 0$ for $j = 1,\dots,s$ and $\kappa_j = 0$ for $j = s+1,\dots,m$. Thus, the limit functions $\phi_j$ in (32) are affine linear for $j \geqslant s+1$. Proof. Each $L_j$ is the limit of the sequence $R^k_j/(\delta^{k})^{1/2} = (R^k_j/\tau_k^j) (\tau^j_k/(\delta^k)^{1/2})$. The sequence $(R^k_j/\tau_k^j)$ converges to $\lambda_j(z)$ from (32). On the other hand, for $j = 1,\dots,s$ the sequence $\tau^j_k/\delta^k$ converges to $1/\kappa_j$. Hence $\tau^j_k/(\delta^k)^{1/2}$ converges to $0$ and the lemma is proved. Note again that for each $j = 1,\dots,m$ the forms $R^k_j/\tau^j_k$ converge to $\lambda_j$. On the other hand $R^k_j/(\delta^k)^{1/2}$ converges to $L_j$. Since for every $k$ the forms $R^k_j/\tau^j_k$ and $R^k_j/(\delta^k)^{1/2}$ are linearly dependent, their limits $\lambda_j$ and $L_j$ are linearly dependent too. Consider an integer $t$, $s+1\leqslant t\leqslant m$, such that $\lambda_j \neq 0$ for $j = s+1,\dots,t$ and $\lambda_j = 0$ for $j = t+1,\dots,m$ (after a permutation of the defining functions). If ${\lambda_j = 0}$ for all $j = s+1,\dots,m$, then we set ${t=s}$. Then $L_j = a_j \lambda_j$ for some real $a_j$, $j = s+1,\dots,t$. Note that some of the $a_j$ can be equal to $0$. Furthermore, the forms $\lambda_j$, $j = 1,\dots,t$, and $L_j$, $j = t+1,\dots,n$, are linearly independent. Indeed, if a nonzero vector $v$ is contained in the intersection of their kernels, then the complex line through the origin in the direction of $v$ is contained in the limit domain $G = \{\phi_j(z) < 0,\, j= 1,\dots,t\}$. But this contradicts the hyperbolicity of the convex domain $G$ at the origin, which we established in Lemma 6.2. After a $\mathbb{C}$-linear change of coordinates we have $\lambda_j(z) = z_j$, $j=1,\dots,t$, and $L_j(z) = z_j$, $j = t + 1,\dots,m$. Thus, in the new coordinates the defining functions (32) can now be written in the form
$$
\begin{equation*}
\begin{aligned} \, \phi_j (z) &=-1+\operatorname{Re} z_j \\ &\qquad +\kappa_j H_j(a_{s+1}z_{s+1},\dots ,a_t z_t, z_{t+1},\dots ,z_n, \overline{a_{s+1}z_{s+1}},\dots ,\overline{a_t z_t}, \overline{z_{t+1}},\dots ,\overline{z_n}), \end{aligned}
\end{equation*}
\notag
$$
$j = 1,\dots,t$. For $j > t$ the functions $\phi_j$ are constants: $\phi_j\!=\!-1$, and we remove them. Set
$$
\begin{equation*}
\begin{aligned} \, & \widetilde H_j(z_{s+1},\dots,z_n, \overline{z_{s+1}},\dots ,\overline{z_n}) \\ &\qquad=\kappa_j H_j(a_{s+1}z_{s+1},\dots ,a_t z_t, z_{t+1},\dots ,z_n, \overline{a_{s+1}z_{s+1}},\dots ,\overline{a_t z_t}, \overline{z_{t+1}},\dots ,\overline{z_n}). \end{aligned}
\end{equation*}
\notag
$$
Note that $\widetilde H_j = 0$ for $j= s+1,\dots,t$ and the other forms $\widetilde H_j$ are nonnegative on $\mathbb{C}^{n-s}(z_{s+1},\dots,z_n)$. Furthermore, they are positive definite on $\mathbb{C}^{n-t}(z_{t+1},\dots,z_n)$. (In general they are not positive definite on $\mathbb{C}^{n-s}(z_{s+1},\dots,z_n)$ because some of the $a_j$ can be equal to $0$.) Thus, the limit domain $G$ is defined by
$$
\begin{equation*}
\begin{gathered} \, \widetilde \phi_j(z)=-1+\operatorname{Re} z_j+\widetilde H_j(z_{s+1},\dots,z_n, \overline{z_{s+1}},\dots ,\overline{z_n})<0, \qquad j=1,\dots ,s, \\ \widetilde \phi_j(z)=-1+\operatorname{Re} z_j<0, \qquad j=s+1,\dots ,t \end{gathered}
\end{equation*}
\notag
$$
(there are no affine linear defining functions for $t=s$). Clearly, $G$ is biholomorphically equivalent to a bounded domain. Therefore, the domain $G$ is hyperbolic. Now, similarly to the proof of Lemma 6.3 we conclude that the family $(\widetilde g^k)$ is normal. Hence $\Omega$ is biholomorphic to $G$ as above. The last step of the proof is to show that $m=1$. Assume that $t > 1$. Let $f\colon G \to \Omega$ be a biholomorphic map. Consider a piecewise smooth strictly pseudoconvex point $p \in b\Omega$. Assume that $p$ is not smooth, that is, $m > 1$. Since $f^{-1}\colon \Omega \to G$ is a biholomorphism, it follows by the boundary uniqueness theorem that the cluster set of $f$ on an open piece of $b\Omega$ containing the point $p$ cannot contain only the point at infinity. Hence, moving $p$ slightly if necessary, we may assume that a finite boundary point $a \in bG$ is contained in the cluster set of $f^{-1}(p)$ This means that there exists a sequence $(a^k)$ in $ G$ converging to $a$ such that $f(a^k)$ converges to $p$. Then the map $f$ extends to the boundary $b{G}$ in a neighbourhood of $a$ as a Hölder continuous map by Theorem 1.1 in [33]. Note that in [33] a boundary point in the source domain ($ G$ in our case) was assumed to be piecewise smooth strictly pseudoconvex. However, this assumption was imposed there because the maps under consideration in [33] were only locally proper. In our case $f$ is biholomorphic, the domain $ G$ admits a global defining plurisubharmonic function $\sup_j \widetilde \phi_j$ and Step 2 of the proof in [33] (based on Hopf’s lemma) goes through directly. The remaining part of the proof in [33] goes through literally, which establishes the Hölder continuity up to the boundary. Now, when $s > 1$, every face of $bG$ is foliated (locally) by complex discs. Then the argument in Theorem 1.2 of [33] (or in [7]) shows that the Jacobian determinant of $f$ vanishes identically on a one-sided neighbourhood of $a$ in $G$, and therefore everywhere on $G$. This is a contradiction because $f$ is a biholomorphic map. Hence, $s = 1$. Now, if $t > 1$, then the defining functions $\widetilde \phi_j$, $j = 2,\dots,t$, are affine linear and the corresponding faces are foliated by complex discs. If $a$ belongs to one of these faces (even if $a$ lies on a corner of a face), then we conclude as above that the Jacobian determinant of $f$ vanishes identically, which is a contradiction. It remains to consider the case when the cluster set of $f^{-1}$ on an open neighbourhood of $p$ in $b\Omega$ is contained in the face $\Gamma_1 = \{\widetilde \phi_1 = 0 \}$. If $\Gamma_1$ is foliated by complex discs, then we argue as above. If $\Gamma_1$ is not foliated by complex discs, then it is strictly pseudoconvex and biholomorphic to the unit sphere. But then, according to [33], $\Gamma_1$ can contain the cluster set of an open neighbourhood of $p$ in $b\Omega$ only when $m = 1$, that is, $p$ is a smooth point. The only remaining possibility is $s = t = 1$. In this case $G$ is biholomorphic to the unit ball $\mathbb{B}^n$. This means that $\Omega$ is also biholomorphic to $\mathbb{B}^n$, which implies that $m = 1$ (see, for example, [33]; of course, there are many other references). The proof of Theorem 1.3 is complete. 
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\Bibitem{Suk22} |
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