This paper deals with the finiteness problem of meromorphic funtions on an annulus sharing four values regardless of multiplicity. We prove that if three admissible meromorphic functions $f_1$, $f_2$, $f_3$ on an annulus $𝔸\left(R_0\right)$ share four distinct values regardless of multiplicity and have the complete identity set of positive counting function, then $f_1=f_2$ or $f_2=f_3$ or $f_3=f_1$. This result deduces that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicity truncated to level $2$ and...

We prove some finiteness theorems for differential nondegenerate meromorphic mappings of ${ℂ}^m$ into ℙⁿ(ℂ) which share n+3 hyperplanes.