When scanning real shapes, occlusion issues may lead to holes in the reconstructed surface which must be solved using an inpainting technique. When dealing with fabrics with folds, reconstruction gets even more challenging because these occlusion problems become almost inevitable and strong assumptions are implied on the physical model of the inpainted surface. We propose a framework to fill holes in triangle mesh surfaces representing fabrics. The method leverages the developable nature of fabrics to recover the intrinsic geometry of the missing patch in 2D. Our inpainting strategy is then based on a variational method to smoothly incorporate the patch into the surface by minimizing an isometric energy. The proposed approach allows us to produce folds and creases which are difficult to obtain with general purpose hole filling techniques. Moreover, our approach remains relevant in the case where the model is not provided by the digitization of a real fabric as for the acquisition from ancient statues with draperies.