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

Sujoy Majumder, Nabadwip Sarkar, Debabrata Pramanik

TL;DR

The paper addresses the Brück conjecture for solutions of first-order PDEs in several complex variables by recasting the problem in a multi-variable Nevanlinna framework. It proves rigidity results: if $f$ satisfies $\partial_{z_i}(f)-a=e^{\alpha(z)}(f-a)$ for all $i$ and the pair $f,\partial_{z_i}(f)$ share $a$ CM, then $\alpha$ must be constant, yielding $f(z)=c_1 e^{A(z_1+\cdots+z_m)}+a-\frac{a}{A}$ (and in the $a=0$ case, $f(z)=c_1 e^{A(z_1+\cdots+z_m)}$). The work also establishes a Borel–Carathéodory-type bound in $\mathbb{C}^m$ and analyzes the order and hyper-order of $f$ to relate growth with derivative behavior. Together, these results extend Brück-type uniqueness from one complex variable to higher dimensions under mild growth and sharing hypotheses, contributing to Nevanlinna theory in several complex variables and providing a rigidity phenomenon for entire PDE solutions.

Abstract

In this paper, we study the Brück conjecture \cite{Bruck-1996} by interpreting it through solutions of first-order partial differential equations in several complex variables. Our results show that the Brück conjecture \cite{Bruck-1996} in $\mathbb{C}^m$ holds under certain additional conditions. In pursuit of this objective, we also establish a Borel-Caratheodory theorem in $\mathbb{C}^m$ and derive several fundamental results on the order and hyper-order of entire functions in higher dimensions.

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

TL;DR

The paper addresses the Brück conjecture for solutions of first-order PDEs in several complex variables by recasting the problem in a multi-variable Nevanlinna framework. It proves rigidity results: if satisfies for all and the pair share CM, then must be constant, yielding (and in the case, ). The work also establishes a Borel–Carathéodory-type bound in and analyzes the order and hyper-order of to relate growth with derivative behavior. Together, these results extend Brück-type uniqueness from one complex variable to higher dimensions under mild growth and sharing hypotheses, contributing to Nevanlinna theory in several complex variables and providing a rigidity phenomenon for entire PDE solutions.

Abstract

In this paper, we study the Brück conjecture \cite{Bruck-1996} by interpreting it through solutions of first-order partial differential equations in several complex variables. Our results show that the Brück conjecture \cite{Bruck-1996} in holds under certain additional conditions. In pursuit of this objective, we also establish a Borel-Caratheodory theorem in and derive several fundamental results on the order and hyper-order of entire functions in higher dimensions.

Paper Structure

This paper contains 9 sections, 11 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Auxiliary Lemmas
  3. Maximum principle
  4. The function $A(r,f)$
  5. The function $M(r,f)$
  6. Schwarz's Lemma
  7. Borel-Caratheodery Lemma in several complex variables
  8. Proof of Theorem \ref{['t3.1']}
  9. Proof of Theorem \ref{['t3.2']}

Key Result

Theorem 1.1

Let $f$ be a non-constant entire function in $\mathbb{C}^m$ such that $\rho_1(f)\not\in\mathbb{N}\cup\{\infty\}$. If for all $i\in\mathbb{Z}[1,m]$, where $\alpha(z)$ is an entire function in $\mathbb{C}^m$, then $\alpha(z)$ reduces to a constant, $c$ say and where $A=e^c$ and $c_1$ are non-zero constant.

Theorems & Definitions (20)

  • Conjecture 1.1
  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 10 more