Bruck conjecture for solutions of first-order partial differential equations in Cm

Technical Summary: Brück Conjecture for Solutions of First-Order Partial Differential Equations in $\mathbb{C}^m$

Problem Statement
This paper investigates the extension of the Brück conjecture from the theory of entire functions in one complex variable to the setting of several complex variables ($\mathbb{C}^m$). The classical Brück conjecture concerns the relationship between a non-constant entire function $f$ and its derivative $f'$ when they share a value $a$ counting multiplicities (CM). Specifically, it posits that if the hyper-order $\rho_1(f)$ is not a non-negative integer or infinity, and $f$ and $f'$ share $a$ CM, then $f' - a = c(f - a)$ for some non-zero constant $c$.

The authors generalize this problem to first-order partial differential equations. They consider a non-constant entire function $f$ in $\mathbb{C}^m$ and investigate the scenario where the partial derivatives $\partial_{z_i} f$ share a value $a$ with $f$ for all $i \in \{1, \dots, m\}$. The central question is whether the relation $\partial_{z_i} f - a = e^{\alpha(z)}(f - a)$ implies that the entire function $\alpha(z)$ must reduce to a constant, thereby forcing $f$ into a specific exponential form. The paper notes that this conjecture fails if the growth condition on the hyper-order is removed or if the exponential term is replaced by an arbitrary entire function with zeros.

Methodology
The authors employ Nevanlinna theory in several complex variables to analyze the value distribution of meromorphic functions and their partial derivatives. The methodology involves:

1. Establishing Fundamental Lemmas: The paper derives and utilizes several auxiliary results, including:

   • A generalized Borel-Carathéodory theorem in $\mathbb{C}^m$ (Lemma 2.5), which bounds the maximum modulus $M(r, f)$ in terms of the maximum real part $A(R, f)$.
   • Relationships between the order of a function and its partial derivatives (Lemma 2.9).
   • Estimates for the proximity and counting functions of derivatives (Lemmas 2.1, 2.3).
   • A result stating that if the second partial derivatives of a function vanish identically, the function must be a polynomial (Lemma 2.4).

2. Analysis of Shared Values: The core of the proof involves analyzing the condition where $f$ and $\partial_{z_i} f$ share a value $a$ CM. This leads to the functional equation $\partial_{z_i} f - a = e^{\alpha(z)}(f - a)$.

3. Case Distinction and Contradiction:

   • Case $a=0$: The authors prove that if $f$ and $\partial_{z_i} f$ share 0 CM, then $f$ has no zeros. This allows the representation $f = e^\beta$, reducing the problem to analyzing the derivatives of $\beta$. By examining the growth orders and using the condition $\rho_1(f) \notin \mathbb{N} \cup \{\infty\}$, they show that the derivatives of $\beta$ must be constants.
   • Case $a \neq 0$: The authors consider two sub-cases based on the nature of $\alpha(z)$. If $\alpha(z)$ is constant, the result follows directly. If $\alpha(z)$ is non-constant, they assume the contrary and construct auxiliary functions $F$ and $G$ involving second and third-order derivatives. By analyzing the Wronskian and using the assumption that the counting function of the zeros of the derivatives is small ($o(T(r, f))$), they derive a contradiction ($T(r, f) = o(T(r, f))$), thereby proving $\alpha(z)$ must be constant.

Key Contributions and Results
The paper establishes two main theorems confirming the Brück conjecture in $\mathbb{C}^m$ under specific conditions:

• Theorem 1.1: If $f$ is a non-constant entire function in $\mathbb{C}^m$ with hyper-order $\rho_1(f) \notin \mathbb{N} \cup \{\infty\}$, and if $\partial_{z_i} f = e^{\alpha(z)} f$ for all $i$, then $\alpha(z)$ reduces to a constant $c$. Consequently, $f(z) = c_1 e^{A(z_1 + \dots + z_m)}$, where $A = e^c$.
• Theorem 1.2: If $f$ is a non-constant entire function in $\mathbb{C}^m$ such that the counting function of the zeros of its partial derivatives satisfies $\|N(r, 0; \partial_{z_i} f) = o(T(r, f))$ for all $i$, and if $\partial_{z_i} f - a = e^{\alpha(z)}(f - a)$ for a non-zero constant $a$, then $\alpha(z)$ reduces to a constant. The solution takes the form $f(z) = c_1 e^{A(z_1 + \dots + z_m)} + a - a/A$.

The authors also provide counterexamples (Examples 1.1–1.3) demonstrating that the conjecture does not hold if the hyper-order condition is violated (i.e., if $\rho_1(f) \in \mathbb{N} \cup \{\infty\}$) or if the exponential term is replaced by an entire function with zeros.

Significance and Claims
The paper claims to extend the scope of the Brück conjecture from one complex variable to higher dimensions ($\mathbb{C}^m$) within the framework of first-order partial differential equations. The authors state that their results show the conjecture holds in $\mathbb{C}^m$ under the additional conditions of restricted hyper-order growth or specific smallness conditions on the counting functions of the derivatives.

In pursuit of these results, the authors also establish a Borel-Carathéodory theorem in $\mathbb{C}^m$ and derive fundamental results regarding the order and hyper-order of entire functions in higher dimensions. The work contributes to the growing field of Nevanlinna value distribution theory in several complex variables, offering a deeper understanding of the uniqueness and growth properties of solutions to partial differential equations in multidimensional complex spaces.