A Generalization of Zalcman's Lemma on Complex Lie Groups

Theorem 1.1. Let G be an m-dimensional complex Lie group with holomorphic Lie algebra $\mathfrak{g}$ , and $\Omega$ a domain in G. Fix a basis $\varepsilon = (\varepsilon_1, \dots, \varepsilon_m)$ of $\mathfrak{g}$ . Let N be a compact Hermitian manifold. Let $\mathscr{F}: \Omega \to N$ be a family of holomorphic mappings. Then, $\mathscr{F}$ is not normal on $\Omega$ if and only if there exist - (a) a sequence $\{f_j\}_{j=1}^{+\infty} \subset \mathscr{F};$ - (b) a sequence $\{p_j\}_{j=1}^{+\infty} \subset \Omega$ with $p_j \to p_0 \in \Omega$ as $j \to +\infty$ ; - (c) a sequence $\{\rho_j\}_{j=1}^{+\infty} \subset \mathbb{R}$ with $0 < \rho_j \to 0$ as $j \to +\infty$ such that the sequence $$\phi_j(z) := f_j \circ \exp_{p_j} \left( \rho_j(dL_{p_j})_e(\varepsilon z^T) \right), \quad j = 1, 2, \cdots$$ converges uniformly on compact subsets of $\mathbb{C}^m$ to a nonconstant holomorphic mapping $\phi: \mathbb{C}^m \to N$ . Here, $z^T$ is the transpose of $z = (z_1, \dots, z_m)$ . In further, we obtain the following local version of Theorem 1.1.

Corollary 1.2. Let G be an m-dimensional complex Lie group with holomorphic Lie algebra $\mathfrak{g}$ , and $\Omega$ a domain in G. Fix a basis $\varepsilon = (\varepsilon_1, \dots, \varepsilon_m)$ of $\mathfrak{g}$ . Let N be a compact Hermitian manifold. Let $\mathscr{F}: \Omega \to N$ be a family of holomorphic mappings. Then, $\mathscr{F}$ is not normal at $p_0 \in \Omega$ if and only if there exist - (a) a sequence $\{f_j\}_{j=1}^{+\infty} \subset \mathscr{F};$ - (b) a sequence $\{p_j\}_{j=1}^{+\infty} \subset \Omega$ with $p_j \to p_0$ as $j \to +\infty$ ; - (c) a sequence $\{\rho_j\}_{j=1}^{+\infty} \subset \mathbb{R}$ with $0 < \rho_j \to 0$ as $j \to +\infty$ such that the sequence $$\phi_j(z) := f_j \circ \exp_{p_j} \left( \rho_j(dL_{p_j})_e(\varepsilon z^T) \right), \quad j = 1, 2, \cdots$$ converges uniformly on compact subsets of $\mathbb{C}^m$ to a nonconstant holomorphic mapping $\phi: \mathbb{C}^m \to N$ . Here, $z^T$ is the transpose of $z = (z_1, \dots, z_m)$ . To see how Theorem B can be covered by our result, we consider a special case where $G = \mathbb{C}^m$ . We regard $\mathbb{C}^m$ as an additive group with zero element $\mathbf{0}$ , and define the left translation $L_p(z) = p + z$ for $p, z \in \mathbb{C}^m$ . Put $$\varepsilon = \left(\frac{\partial}{\partial \zeta_1}, \cdots, \frac{\partial}{\partial \zeta_m}\right)\Big|_{\mathbf{0}},$$ where $\zeta_1, \dots, \zeta_m$ stand for complex coordinates of $\mathbb{C}^m$ . Since $T_z^{1,0}\mathbb{C}^m \cong \mathbb{C}^m$ for all $z \in \mathbb{C}^m$ , it is immediate that $$(dL_{p_j})_{\mathbf{0}}(\varepsilon z^T) = \left(z_1 \frac{\partial}{\partial \zeta_1} + \dots + z_m \frac{\partial}{\partial \zeta_m}\right)\Big|_{p_j} \in T_{p_j}^{1,0} \mathbb{C}^m.$$ By the definition of exp, we obtain $\exp_{p_j}(\rho_j(dL_{p_j})_{\mathbf{0}}(\varepsilon z^T)) = p_j + \rho_j z$ , which indicates that Theorem 1.1 extends Theorem B. Afterwards, we shall state a Robinson-Zalcman type heuristic principle of holomorphic families of a Picard type property on a complex Lie group. Let g be a holomorphic mapping on a domain D in a complex manifold. Denote by P a property. Write $(g,D) \in P$ , if g is of P on D; $(g,x) \in \neg P$ , if g is not of P in any neighborhood of x in D; and $(g,D) \in \neg P$ , if $(g,x) \in \neg P$ for all $x \in D$ . Let $\Omega$ be a domain in a complex Lie group G, and N a compact Hermitian manifold. Let $f, f_j : \Omega \to N$ be holomorphic mappings. For $p, p_j \in G$ , put $$\psi(z) = \exp_p(\theta z^T); \quad \phi_j(z) = f_j \circ \exp_{p_j}(\theta^j z^T), \quad j = 1, 2, \dots$$ where $\theta, \theta^j$ are basis of $T_p^{1,0}G, T_{p_j}^{1,0}G$ , respectively. With these notations, we obtain a Robinson-Zalcman type heuristic principle as follows.

Theorem 1.3. Let G be an m-dimensional complex Lie group, and $\Delta, \Omega$ domains in G with $\Delta \subset \Omega$ . Let N be a compact Hermitian manifold. Assume that P is a property of holomorphic mappings into N such that - (a) if $(f, \Omega) \in P$ , then $(f, \Delta) \in P$ for any $\Delta$ ; - (b) if $(f, \Delta) \in P$ , then $(f \circ \psi, \psi^{-1}(\Delta)) \in P$ for any $p, \theta$ ; - (c) if $\{\phi_j; (\phi_j, D_j) \in P\}_{j=1}^{+\infty}$ converges uniformly on compact subsets of $\mathbb{C}^m$ to a holomorphic mapping $\phi : \mathbb{C}^m \to N$ for domains $D_1, D_2, \cdots$ in $\mathbb{C}^m$ such that $D_1 \subset D_2 \subset \cdots$ , then $(\phi, \mathbb{C}^m) \in P$ or $(\phi, \mathbb{C}^m) \in \neg P$ ; - (d) if $(\phi, \mathbb{C}^m) \in P$ or $(\phi, \mathbb{C}^m) \in \neg P$ , then $\phi$ is constant. Then, the family $\mathscr{F} = \{f : (f, \Omega) \in P\}$ is normal on $\Omega$ .

Corollary 2.7. For any $g \in G$ , we have $$\exp_g = L_g \circ \exp \circ (dL_{g^{-1}})_g.$$ Fix $\xi \in T_g^{1,0}G$ with $g \in G$ . Let $$\iota_{\xi}: T_q^{1,0}G \to T_{\xi}^{1,0}(T_q^{1,0}G)$$ be the canonical linear isomorphism, i.e., $\iota_{\xi}(\eta) = \theta'_{\eta}(0)$ for $\eta \in T_g^{1,0}G$ , where $\theta_{\eta}(z) = \xi + z\eta$ is a holomorphic curve in $T_g^{1,0}G$ . For multiple tangent bundles of G, one regards them as identical. Equip G with a left-invariant Hermitian metric (such a metric is aways existent, see [8, 18]), i.e., $$\langle \xi, \eta \rangle = \langle (dL_h)_g(\xi), (dL_h)_g(\eta) \rangle, \quad \xi, \eta \in T_q^{1,0}G$$ for any $g, h \in G$ . With this metric, $(dL_h)_g$ is a linear isometry and so (2) $$||(dL_h)_g|| = ||(dL_h)_g^{-1}|| = 1, \quad g, h \in G.$$ This metric on $T^{1,0}G$ induces a natural metric on $T^{1,0}(T^{1,0}G)$ defined by $$\langle \zeta, \eta \rangle := \big\langle \iota_{\xi}^{-1}(\zeta), \iota_{\xi}^{-1}(\eta) \big\rangle, \quad \ \zeta, \eta \in T_{\xi}^{1,0}(T_{g}^{1,0}G),$$ which forces $\iota_{\mathcal{E}}$ to be a linear isometry and it is therefore (3) $$\|\iota_{\xi}\| = \|\iota_{\xi}^{-1}\| = 1, \quad \|d\iota_{\xi}\| = \|d\iota_{\xi}^{-1}\| = 1.$$ By (2) and (3), we may regard in order to simplify the formula that $$(d(dL_h)_g)_{\xi} = (dL_h)_g \circ \iota_{\xi}^{-1},$$ up to a linear isometry. Let $$(d\exp_g)_{\xi}: T_{\xi}^{1,0}(T_g^{1,0}G) \to T_{\exp_g(\xi)}^{1,0}G$$ be the differential of $\exp_q: T_g^{1,0}G \to G$ at $\xi$ , which can be represented as $$(d\exp_g)_{\xi}(\eta) = \frac{d}{dz}\Big|_{z=0} \exp_g \left(\xi + z\iota_{\xi}^{-1}(\eta)\right), \quad \eta \in T_{\xi}^{1,0}(T_g^{1,0}G).$$ For $d \exp_X$ , the well-known Duhamel formula (see, e.g., Theorem 5 in Section 1.2, [15]) gives an explicit expression (5) $$d\exp_X = \left(dL_{\exp(X)}\right)_e \circ \frac{Id - e^{-\operatorname{ad}_X}}{\operatorname{ad}_Y} \circ \iota_X^{-1}, \quad X \in \mathfrak{g},$$ where $$\frac{Id - e^{-\text{ad}_X}}{\text{ad}_X} = \sum_{k=0}^{+\infty} \frac{(-1)^k}{(k+1)!} (\text{ad}_X)^k$$ with $\operatorname{ad}_X(Y) = [X, Y] = XY - YX$ for $Y \in \mathfrak{g}$ . Set $X = (dL_{g^{-1}})_g(\xi)$ . Using Corollary 2.7, (4) and (5), we derive that (6) $$(d \exp_g)_{\xi} = (dL_g)_{\exp(X)} \circ (dL_{\exp(X)})_e \circ \frac{Id - e^{-\operatorname{ad}_X}}{\operatorname{ad}_X}$$ $$\circ \iota_X^{-1} \circ (dL_{g^{-1}})_g \circ \iota_{\xi}^{-1}.$$ Property 2.8. For any $g \in G$ , $(d \exp_g)_0$ is a linear isometry.

Theorem 2.9. For any $\xi \in T_q^{1,0}G$ with $g \in G$ , we have $$\left\| (d \exp_g)_{\xi} \right\| \leq \frac{e^{C_{\mathfrak{g}} \|\xi\|} - 1}{C_{\mathfrak{g}} \|\xi\|}.$$

Lemma 3.1. Let M, N be Hermitian manifolds with that N is compact. Let $\mathscr{F}: M \to N$ be a family of holomorphic mappings. Then, $\mathscr{F}$ is normal on M if and only if $\{\|df\|\}_{f \in \mathscr{F}}$ is locally uniformly bounded on M.

Theorem 3.2. Let G be a connected complex Lie group, and N a Brody hyperbolic complex manifold. Then, there exist no nonconstant holomorphic mappings from G into N.

Corollary 3.3. Let G be a connected complex Lie group. Then, any meromorphic function on G reduces to a constant if it avoids three distinct values.

Corollary 3.4. There exist no Brody hyperbolic complex Lie groups.