Characterizations of $φ$-inner uniform domains and $φ$-uniform domains in metric spaces

In this paper, we study $φ$-uniform/-inner uniform domains in metric spaces. In our first main result, we establish quantitative characterizations of three different definitions of $φ$-uniformity/-inner uniformity in length spaces. In our second main result, we obtain a geometric sufficient condition of $φ$-uniform/-inner uniform domains in doubling length spaces, which extends the corresponding result of Guo-Huang-Wang [arXiv 2025] and Gao-Guo-Huang-Wang [arXiv 2025] for linear case.