In the paper, we extend the results of Chen and Yi [4] and Fang et al. [6] concerning sharing three distinct values for meromorphic functions in \(\mathbb {C}^m\) . We demonstrate that if f and \(\Delta _cf\) share three distinct values \(a_1\) , \(a_2\) and \(\infty \) CM, then either \(f(z) = e^{b_{1} z_{1} + \cdots + b_{m} z_{m}} h(z), \text { where h is c-periodic meromorphic function in } \mathbb {C}^m, (b_1, \dots , b_m) \in \mathbb {C}^m \setminus \{0\} \text { such that } b_{1} c_{1} + \cdots + b_{m} c_{m} = \ln 2 + 2k\pi i, k \in \mathbb {Z}\) or \(f(z)\equiv f(z+2c)\) for all \(z\in \mathbb {C}^m\) , with the latter case arising when \(\limsup \limits _{r\rightarrow \infty }\frac{\log T(r,f)}{r}>0\) . Also in the paper, we improve the recent result of Huang and Zhang [13] for entire function in \(\mathbb {C}\) to meromorphic function into higher dimensions. Moreover, we show by plenty of examples that our results are best possible.
