Reconstructing dynamic 3D garment surfaces with open boundaries from monocular videos is an important problem as it provides a practical and low-cost solution for clothes digitization. Recent neural rendering methods achieve high-quality dynamic clothed human reconstruction results from monocular video, but these methods cannot separate the garment surface from the body. Moreover, despite existing garment reconstruction methods based on feature curve representation demonstrating impressive results for garment reconstruction from a single image, they struggle to generate temporally consistent surfaces for the video input. To address the above limitations, in this paper, we formulate this task as an optimization problem of 3D garment feature curves and surface reconstruction from monocular video. We introduce a novel approach, called REC-MV to jointly optimize the explicit feature curves and the implicit signed distance field (SDF) of the garments. Then the open garment meshes can be extracted via garment template registration in the canonical space. Experiments on multiple casually captured datasets show that our approach outperforms existing methods and can produce high-quality dynamic garment surfaces.