Zalcman's lemma, Pinchuk's rescaling method, and Catlin's estimates revisited

Lemma 1.1. Let (fn)ⁿ be a sequence of holomorphic maps from the unit disc D of C to the Riemann sphere which is not normal at 0. Then there exists a sequence of affine contractions (rn)n, converging to 0 and such that the renormalized sequence (fⁿ ◦ rn)ⁿ is converging to a non constant entire map ϕ, after taking a subsequence. Moreover, the spherical derivative of ϕ is uniformly bounded:|ϕ ′ |^σ ≤ |ϕ ′ (0)|^σ = 1. According to this result, if a property P of meromorphic functions is stable both by affine change of coordinates at the source and local uniform convergence, then very few meromorphic functions on ^D satisfy ^P if there are no non constant entire meromorphic functions which satisfy it: ``` {f : meromorphic on C and satisfying P} ⊂ C ∪ {∞} ⇒ {f : meromorphic on D and satisfying P} is normal. ``` This implication actually establishes the Robinson version of Bloch's principle. For more material on Bloch's principle and a detailed description of its applications, we refer to Walter Bergweiler's review article [\[Berg\]](#page-11-3). Our aim is to prove a quite general renormalization lemma, working for maps defined on the unit disc of C, with values in a metric space, and satisfying some Schwarz type property. As we shall see, this lemma covers Zalcman's result and its extension to quasiregular maps due to Miniowitz. We shall also show that it leads to a general normality statement for the Pinchuk's rescaling method. This new approach considerably simplifies our previous one developped in [\[Bert1\]](#page-11-4), it relies neither on the existence of good peak functions nor on estimates of invariant metrics, but only on the basic geometric properties of finite type hypersurfaces, via an asymptotic Bloch principle. We shall work with metric spaces (X,d) endowed with specific families $(J_\eta)_{\eta \in X}$ of positive real valued functions. We have two main examples in mind. On one hand $J_\eta(\eta') = d(\eta,\eta')$ where X is a compact manifold with a metric d, which corresponds to the Zalcman or Miniowitz cases, and $J_\eta(\eta') = \|A_\eta(\eta')\|$ where $X \subset \mathbb{C}^2$ and $(A_\eta)_{\eta \in X}$ is a family of holomorphic automorphisms of $\mathbb{C}^2$ on the other hand, which corresponds to the Pinchuk's rescaling method. Before stating our result, we must define the Schwarz property on which it is based. We denote by $\mathbb{D}_r$ the disc of radius r centred at the origin in $\mathbb{C}$ .

Lemma 1.3. Let (X,d) be a metric space and $(J_{\eta})_{\eta \in X}$ be a family of positive real valued functions on X such that: - i) $J_{\eta}(\eta) = 0$ for every $\eta \in X$ , - ii) $\lim_{\tau \to 0} \sup_{n \in K} \left( \sup_{d(n,n') \le \tau} J_n(\eta') \right) = 0$ for every compact subset K of X. Let $f_n : \overline{\mathbb{D}} \to X$ be a sequence of continuous maps such that: - 1- $f_n$ satisfies the Schwarz-type property $S(J, \alpha^{\pm}, c, s)$ on $\mathbb D$ for every $n \in \mathbb N$ , - 2- there exist a constant $k \in ]0,1]$ and sequences $(t'_n)_n$ , $(\epsilon'_n)_n$ in $\mathbb C$ such that $\lim_n \epsilon'_n = 0$ , $2|t'_n| + |\epsilon'_n| < 1$ and $J_{f_n(t'_n)}\left(f_n(t'_n + \epsilon'_n)\right) \ge k$ for every $n \in \mathbb N$ . Then there exists a sequence of affine contractions $(r_n(t) := t_n + \epsilon_n t))_n$ and a sequence of positive real numbers $(R_n)_n$ such that $\lim_n \epsilon_n = 0$ , $\lim_n R_n = +\infty$ , $|t_n| + |\epsilon_n| < 1$ for every $n \in \mathbb{N}$ and - 1'- $g_n := f_n \circ r_n$ is defined on $\mathbb{D}_{R_n + \alpha^+}$ , - 2'- $J_{g_n(0)}\left(g_n(1)\right) \geq k$ and $J_{g_n(t)}\left(g_n(t+u)\right) \leq s(|u|)$ for every $n \in \mathbb{N}$ , every $t \in D_{R_n}$ and every $u \in \mathbb{D}_{\alpha^-}$ . If moreover $\lim_n t'_n = 0$ , then the contractions $r_n$ can be chosen so that $\lim_n r_n(0) = 0$ .

Lemma 2.2. There exists a function $s(m, K, \cdot)$ which is continuous and increasing from [0,1] to [0,1] such that $|f(x)| \leq s(m,K,|x|)$ for every K-quasi-regular map $f: \mathbb{B}^m \to \mathbb{B}^m$ satisfying f(0) = 0.

Proposition 3.1. If $U_0$ and $\alpha_0$ are small enough, then there exists constants $C_2, C_3, C_4 > 1$ such that, for any $\eta, \eta' \in U_0$ and any $0 < \epsilon < \alpha_0$ , the following estimates occur if $\eta \in Q[\eta', \epsilon]$ : $$(i) \ \epsilon(\eta) \le C_2(\epsilon(\eta') + \epsilon), \ (ii) \ Q[\eta, \epsilon] \subset Q[\eta', C_3 \epsilon], \ (iii) \ Q[\eta', \epsilon] \subset Q[\eta, C_3 \epsilon],$$ $$(iv) \ \frac{1}{C_4} \tau(\eta, \epsilon) \le \tau(\eta', \epsilon) \le C_4 \tau(\eta, \epsilon).$$ <span id="page-5-3"></span>We now introduce a family of functionals $(J_n)_{n\in U_0}$ on $\mathbb{C}^2$ by setting: (3.6) $$J_{\eta}(w) := \|\Delta_{\hat{\eta}}^{\epsilon(\eta)} \circ \phi_{\hat{\eta}}(w) - \Delta_{\hat{\eta}}^{\epsilon(\eta)} \circ \phi_{\hat{\eta}}(\eta)\|_{\infty}.$$ The next proposition summarizes how the finite type geometry will be used in the remaining of the paper, it crucially relies on the estimates given by Proposition 3.1.

Proposition 3.2. If $U_0$ and $0 < \epsilon_0 < 1$ are small enough, there exists a constant $C_5 > 1$ such that: - i) if $0 < \epsilon < \epsilon_0$ and $\eta, \eta' \in U_0$ then: $\eta' \in Q[\hat{\eta}, \epsilon] \Rightarrow Q[\hat{\eta}', 3\epsilon(\eta')] \subset Q[\hat{\eta}, C_5\epsilon]$ . - ii) If $q_0, q_1, \dots, q_p$ is a chain of points in $U_0^-$ such that $3C_5^{p-1}\epsilon(q_0) < \epsilon_0$ and $J_{q_i}(q_{i+1}) < 1$ for $0 \le i \le p-1$ , then $q_p \in Q[\hat{q}_0, C(p)\epsilon(q_0)]$ . In particular $\Delta_{\hat{q}_0}^{\epsilon(q_0)} \circ \phi_{\hat{q}_0}(q_p) \in C(p)\mathbb{D}^2$ , where $C(p) := 3C_5^{p-1}$ . Proof. Let us prove assertion i). By Proposition 3.1 i) we have $\epsilon(\eta') \leq C_2(\epsilon(\hat{\eta}) + \epsilon) = C_2\epsilon$ . Since $\hat{\eta}' = \eta' + (0, \epsilon(\eta'))$ , we get from the explicit form of the automorphism $\phi_{\hat{\eta}}$ that $\phi_{\hat{\eta}}(\hat{\eta}') = \phi_{\hat{\eta}}(\eta') + (0, \frac{1}{d_0(\hat{\eta})}\epsilon(\eta'))$ . After shrinking $U_0$ we may assume that $|d_0(\hat{\eta})| \geq \frac{1}{2}$ (recall that $\phi_{(0,0)} = Id$ ) and, since $\eta' \in Q[\hat{\eta}, \epsilon]$ , we thus have $$\phi_{\hat{\eta}}(\hat{\eta}') \in \mathbb{D}_{\tau(\hat{\eta},\epsilon)} \times \mathbb{D}_{\epsilon} + (0, \frac{1}{d_0(\hat{\eta})} \epsilon(\eta')) \subset \mathbb{D}_{\tau(\hat{\eta},\epsilon)} \times \mathbb{D}_{\epsilon+2\epsilon(\eta')} \subset \mathbb{D}_{\tau(\hat{\eta},K\epsilon)} \times \mathbb{D}_{K\epsilon},$$ where we have set $K:=1+2C_2$ . This means that $\hat{\eta}' \in Q[\hat{\eta}, K\epsilon]$ . Now, if $2K\epsilon_0 < \alpha_0$ , Proposition 3.1 ii) implies that $Q[\hat{\eta}', 2K\epsilon] \subset Q[\hat{\eta}, 2KC_3\epsilon]$ . Finally, since $3\epsilon(\eta') \leq 3C_2\epsilon \leq 2K\epsilon$ , we get $Q[\hat{\eta}', 3\epsilon(\eta')] \subset Q[\hat{\eta}, C_5\epsilon]$ where $C_5:=2KC_3=(2+4C_2)C_3$ . Let us now prove the second assertion. We first observe that (3.7) $$J_{\eta}(w) < 1 \Rightarrow w \in Q[\hat{\eta}, 3\epsilon(\eta)].$$ Indeed, by definition, $J_n(w) < 1$ implies that $$\phi_{\hat{\eta}}(w) \in \phi_{\hat{\eta}}(\eta) + (\Delta_{\hat{\eta}}^{\epsilon(\eta)})^{-1}(\mathbb{D}^2) = (0, -\tilde{\epsilon}(\eta)) + \mathbb{D}_{\tau(\hat{\eta}, \epsilon(\eta))} \times \mathbb{D}_{\epsilon(\eta)} \subset \mathbb{D}_{\tau(\hat{\eta}, 3\epsilon(\eta))} \times \mathbb{D}_{3\epsilon(\eta)},$$ where the last inclusion comes from (3.3). By (3.7) one has $q_i \in Q[\hat{q}_{i-1}, 3\epsilon(q_{i-1})]$ for every $1 \le i \le p$ and, since $3C_5^i\epsilon(q_0) \le \epsilon_0$ for $1 \le i \le p-1$ , we may use the first assertion inductively to get <span id="page-6-0"></span> $$q_{1} \in Q \left[ \hat{q}_{0}, 3\epsilon(q_{0}) \right]$$ $$q_{2} \in Q \left[ \hat{q}_{1}, 3\epsilon(q_{1}) \right] \subset Q \left[ \hat{q}_{0}, 3C_{5}\epsilon(q_{0}) \right]$$ $$q_{3} \in Q \left[ \hat{q}_{2}, 3\epsilon(q_{2}) \right] \subset Q \left[ \hat{q}_{0}, 3C_{5}^{2}\epsilon(q_{0}) \right]$$ $$\vdots$$ $$q_{p-1} \in Q \left[ \hat{q}_{p-2}, 3\epsilon(q_{p-2}) \right] \subset Q \left[ \hat{q}_{0}, 3C_{5}^{p-2}\epsilon(q_{0}) \right]$$ $$q_{p} \in Q \left[ \hat{q}_{p-1}, 3\epsilon(q_{p-1}) \right] \subset Q \left[ \hat{q}_{0}, 3C_{5}^{p-1}\epsilon(q_{0}) \right].$$ To conclude, one observe that the definition of pseudo-balls, the estimates (3.5) and the fact that $C(p) := 3C_5^{p-1}\epsilon(q_0) \le \epsilon_0 < 1$ imply that $$\begin{split} &\Delta_{\hat{q}_0}^{\epsilon(q_0)} \circ \phi_{\hat{q}_0}\left(Q[\hat{q}_0, C(p)\epsilon(q_0)]\right) \subset \frac{\tau(\hat{q}_0, C(p)\epsilon(q_0))}{\tau(\hat{q}_0, \epsilon(q_0))} \mathbb{D} \times C(p) \mathbb{D} \subset \sqrt{C(p)} \mathbb{D} \times C(p) \mathbb{D} \subset C(p) \mathbb{D}^2, \\ &\text{and thus get } \Delta_{\hat{q}_0}^{\epsilon(q_0)} \circ \phi_{\hat{q}_0}(q_p) \in C(p) \mathbb{D}^2. \end{split}$$ The Bedford-Pinchuk rescaling method. Let $\Omega$ be a domain in $\mathbb{C}^2$ whose boundary is smooth, pseudonconvex and of finite type near (0,0) and $(A_n)_n$ be a sequence of analytic objects which is converging (0,0) in $\Omega$ . Let $U_0$ is a sufficiently small ball centered at (0,0) in $\mathbb{C}^2$ , then we may assume that $\Omega \cap U_0$ is a domain of the form $U_0^- = \{w \in U_0 : \rho(w) < 0\}$ like those that we just studied. Note that $U_0^-$ is pseudoconvex. Let $(\eta_n)_n$ be a sequence converging to (0,0) in $U_0^-$ . When n is big enough, the point $\eta_n$ is sufficently close to (0,0) and thus there exists $\epsilon_n := \epsilon(\eta_n) > 0$ such that (3.8) $$\eta_n + (0, \epsilon_n) =: \hat{\eta}_n \in \{\rho = 0\}.$$ One sees from (3.2), (3.4), and (3.5) that (3.9) $$\rho \circ (\phi_{\hat{\eta}_n})^{-1} \circ (\Delta_{\hat{\eta}_n}^{\epsilon_n})^{-1} = 2\operatorname{Re} w_2 + P_n(w_1, \bar{w}_1) + O(\tau(\hat{\eta}_n, \epsilon_n)),$$ where $P_n$ is the polynomial of degree at most m satisfying $||P_n||_{\infty} = 1$ given by <span id="page-6-1"></span> $$P_n(w_1, \bar{w}_1) := \frac{1}{\epsilon_n} P_{\hat{\eta}_n}(\tau(\hat{\eta}_n, \epsilon_n) w_1, \tau(\hat{\eta}_n, \epsilon_n) \bar{w}_1).$$ Consider the sequence $(S_n)_n$ of rescaling automorphisms of $\mathbb{C}^2$ defined by $S_n := \Delta_{\hat{\eta}_n}^{\epsilon_n} \circ \phi_{\hat{\eta}_n}$ . Then <span id="page-6-2"></span>(3.10) $$S_n: U_0^- \to U_n^- := S_n(U_0^-), \ S_n(\hat{\eta}_n) = (0,0) \ \text{and} \ \lim_n S_n(\eta_n) = (0,-1).$$ After taking a subsequence, we may assume that the sequence of polynomials $(P_n)_n$ is converging to some polynomial P such that $||P||_{\infty} = 1$ . Then, it follows from (3.9) that the sequence of bounded domains $(U_n^-)_n$ is converging in the Hausdorff sense to some unbounded rigid polynomial domain $D_P$ associated to P: <span id="page-7-2"></span>(3.11) $$U_n^- \to \{(w_1, w_2) \in \mathbb{C}^2 : 2\text{Re } w_2 + P(w_1, \bar{w}_1) < 0\} =: D_P.$$ The domains $U_n$ are pseudoconvex and, using the tomato-can principle, one sees that the limit domain $D_P$ is pseudoconvex too. The polynomial P is therefore subharmonic and, like the $P_n$ , does not contain any constant or harmonic term. The rescaling method requires to produce limits of $(S_n(A_n))_n$ . Although the limit domain $D_P$ is taut, this is not obvious because $S_n(A_n)$ is generally not contained in $D_P$ or in any similar domain. However, our next theorem explains why this is always possible.

Theorem 3.3. Let $\Omega$ be a domain in $\mathbb{C}^2$ whose boundary is smooth, pseudonconvex and of finite type m near (0,0). Let $\omega$ be a domain in $\mathbb{C}^k$ and $(f_n)_n$ be a sequence of holomorphic maps from $\omega$ to $\Omega$ such that $\lim_n f_n(a_0) = (0,0)$ for some fixed point $a_0 \in \omega$ . Let $(S_n)_n$ be the sequence of automorphisms of $\mathbb{C}^2$ associated by the above described rescaling method to the sequence of points $(\eta_n)_n := (f_n(a_0))_n$ . Then $(S_n \circ f_n)_n$ is normal and its limits are holomorphic maps from $\omega$ to some rigid domain $D_P := \{(w_1, w_2) \in \mathbb{C}^2 : 2 \text{Re } w_2 + P(w_1, \bar{w}_1) < 0 \}$ where P is a subharmonic and non harmonic polynomial of degree at most m. As we shall see, this result once again illustrates the Bloch principle : as all entire holomorphic curve in the limit domain $D_P = \lim_n S_n(\Omega)$ are constant, the sequence $(S_n \circ f_n)_n$ is normal for any sequence $(f_n)_n$ of holomorphic discs in $\Omega$ whose centers $f_n(0)$ converge to (0,0). The proof we gave in [Bert1] was based on some renormalization technique very similar to Zalcman's one (see Lemma 3.1 in [Bert1] and required some delicate integration arguments for specific pseudo-metrics. The proof we present here avoids all these technical difficulties, it entirely relies on our 1.3 lemma combined with a version of Bloch's principle. The Lemma 1.3 will be applied with $X=U_0^-$ endowed by the metric d induced by $\|\ \|_{\infty}$ , the functionals $J_{\eta}$ being defined by (3.6). The required assumptions are clearly satisfied, in particular the classical Schwarz lemma implies that any $f\in\mathcal{O}(\mathbb{D},U_0^-)$ satisfies the property S(J,1,1,s) with s(u)=|u|. Note that $\alpha^\pm=c=1$ . From now on, for any map $\varphi$ to $U_0^-$ , we shall denote by $\hat{\varphi}(t)$ the point $\widehat{\varphi(t)}$ . The heart of the proof of Theorem 3.3 lies in Lemma 3.4 below. As we will explain in the next section, this lemma implicitely contains Catlin's estimates on the Kobayashi infinitesimal metric.

Lemma 3.4. For every $0 < k \le 1$ there exists $0 < r_0 < 1$ and c > 0 such that $$\left( f \in \mathcal{O}(\mathbb{D}, U_0^-) \text{ and } \|f(0)\|_{\infty} \le c \right) \Rightarrow J_{f(0)}(f(t)) < k, \ \forall t \in \mathbb{D}_{r_0}.$$

Lemma 3.5. The sequence $(f_n)$ is locally uniformly converging to (0,0) on $\omega$ .

Lemma 3.6. If P is a subharmonic and non harmonic function on $\mathbb{C}$ then the domain $D_P := \{(w_1, w_2) \in \mathbb{C}^2 : \text{Re } w_2 + P(w_1) < 0\}$ is Brody-hyperbolic.

Theorem 4.1. Let $\Omega$ be a domain in $\mathbb{C}^2$ whose boundary is smooth, pseudonconvex and of finite type m near (0,0). Let $U_0$ be a sufficiently small ball centered at the origin in $\mathbb{C}^2$ and $\rho$ be a smooth defining function for $b\Omega \cap U_0$ . Assume that $\rho_{w_1}^{(r)}(0,0) = 0$ for $1 \leq r \leq m$ . Then there exists a constant A > 1 such that the Kobayashi pseudo-metric $K_\Omega$ of $\Omega$ satisfies the following estimates: $\frac{1}{A} \max \left( \frac{|X_1|}{\tau(\eta)}, \frac{|\rho_{w_1}(\eta)X_1 + \rho_{w_2}(\eta)X_2|}{|\rho(\eta)|} \right) \leq K_\Omega(\eta, X), \ \forall \eta \in \Omega \cap U_0, \ \forall X \in \mathbb{C}^2$ where $\tau(\eta) := \min \{ \left( \frac{(j+k)!|\rho(\eta)|}{|(\rho_{w_1}^{(j)})_{w_1}^{(k)}(\eta)|} \right)^{\frac{1}{j+k}} : j, k \geq 1, \ j+k \leq m \}.$

Definition 1.2. Let $(J_{\eta})_{\eta \in X}$ be a family of of positive functions on a metric space (X,d) such that $J_{\eta}(\eta) = 0$ for every $\eta \in X$ . Let $0 < \alpha^{-} < \alpha^{+} < 1$ and c > 0 be some constants, and $s : [0, \alpha^{+}[ \to \mathbb{R}^{+} \text{ be a function which is vanishing and continuous at } 0. We shall say that a map <math>f : \mathbb{D}_{\rho} \to X$ satisfies the Schwarz-type property $S(J, \alpha^{\pm}, c, s)$ on $\mathbb{D}_{\rho}$ if the following estimates occur for any $t_{0}, \epsilon_{0} \in \mathbb{C}$ such that $|t_{0}| + |\epsilon_{0}| < \rho$ : $$\sup_{t \in \mathbb{D}_{\alpha^+}} J_{f(t_0)}(f(t_0 + t\epsilon_0)) \le c \implies J_{f(t_0)}(f(t_0 + t\epsilon_0)) \le s(|t|), \quad \forall t \in \mathbb{D}_{\alpha^-}.$$ It might be useful to note that if $f:\mathbb{D}\to X$ satisfies the Schwarz-type property $S(J,\alpha^\pm,c,s)$ on $\mathbb{D}$ and $0<\rho$ , then $t\mapsto f(\rho t)$ satisfies it on $\mathbb{D}_{\frac{1}{\rho}}$ . <span id="page-1-0"></span>We may now state our generalized version of the Zalcman Lemma.