In this research, we introduce a dimension reduction level set method (DR-LSM) for optimizing the shape and topology of structures in heat conduction and other applications on free-form surfaces. By utilizing conformal geometry theory, we map these surfaces onto a 2D rectangle domain, simplifying the problem from 3D to 2D. This transformation is achieved using a level set function defined in the 2D domain, leveraging the intrinsic relation between covariant derivatives on the manifold and Euclidean gradient operators. Our method, which includes the upwind finite difference or fast marching method, significantly reduces computational costs and overcomes the challenges of dynamic boundary evolution on free-form surfaces. It allows the Riemannian Hamilton-Jacobi equation and the heat equation, governing boundary evolution and thermal conduction respectively, to be solved in the Euclidean space. This approach not only streamlines finite element analysis but also preserves the advantages of conventional level set methods. We demonstrate the effectiveness of this unified computational framework through numerical examples, showcasing its potential in designing conformal structures with increasing applications across various fields.