In this paper, we study the Br\"{ u}ck conjecture \cite{ Bruck-1996} by interpreting it through solutions of first-order partial differential equations in several complex variables. Our results show that the Br\"{ u}ck 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.
Categories: math.CV math.AP