We introduce WarpCurves, a technique for interactively manipulating an implicit surface using curve-based spatial deformations. Although implicit surfaces have several advantages in 3D modeling, current workflows are limited by the compositional nature of implicit modeling. Wide classes of surface features that are easy to create with the direct manipulation tools available for explicit surface representations are difficult to reproduce using volumetric implicit operations. We describe a novel spatial deformation that can be used to approximate direct surface manipulation. With our method an artist first draws a curve on the current surface to indicate the feature region-of-interest. Deformations applied to this handle curve are transferred to the implicit surface via an automatically constructed C2 continuous space mapping. Additional curves can be added in a hierarchical manner to create complex shapes. Our technique is implemented as a node in the BlobTree hierarchical implicit volume representation, and hence can be used along with other volumetric nodes (operators) such as blending and CSG. Our results show that surface deformations which would be difficult to reproduce using existing volumetric operations can be quickly constructed using warp curves, making them a valuable addition to the implicit modeling toolbox.