Multiply connected freeform surface features are widely encountered in industrial components, where toolpath generation often suffers from discontinuities, sharp turns, non-uniform scallop heights, and incomplete boundary coverage. This paper proposes a scalar-field variational optimization method for milling that produces continuous, boundary-conforming, and non-self-intersecting toolpaths with smoother transitions, more uniform spacing, and reduced redundant path length. A feasible singularity-free initial scalar field with boundary-conforming iso-level sets is first constructed via conformal slit mapping. The optimization is then reformulated as a topology-preserving mesh deformation process governed by boundary-synchronous updates, whereby the continuity, boundary-conformity, and non-self-intersection requirements of the toolpath are converted into mesh-shape constraints maintained throughout the iterative optimization. As a result, the proposed method achieves globally optimized path spacing and improved scallop-height uniformity while preserving trajectory smoothness. Milling experiments show that, compared with a state-of-the-art conformal slit mapping-based method, the proposed approach improves machining efficiency by 14.24%, enhances scallop-height uniformity by 5.70%, and reduces milling impact-induced vibrations by over 10%. The proposed strategy provides an effective solution for high-performance machining of complex multiply connected freeform components.

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