In this paper, we introduce a novel approach for computing conformal deformations in ℝ 3 while minimizing curvature-based energies. Curvature-based energies serve as fundamental tools in geometry processing, essential for tasks such as surface fairing, deformation, and approximation using developable or cone metric surfaces. However, accurately computing the geometric embedding, especially for the latter, has been a challenging endeavor. The complexity arises from inherent numerical instabilities in curvature estimation and the intricate nature of differentiating these energies. To address these challenges, we concentrate on conformal deformations, leveraging the curvature tensor as the primary variable in our model. This strategic choice renders curvature-based energies easily applicable, mitigating previous manipulation difficulties. Our key contribution lies in identifying a previously unknown integrability condition that establishes a connection between conformal deformations and changes in curvature. We use this insight to deform surfaces of arbitrary genus, aiming to minimize bending energies or prescribe Gaussian curvature while sticking to positional constraints.