An implantable structure includes a flexible sleeve, having first and second sleeve ends, and a contracting assembly, which is configured to longitudinally contract the sleeve, and includes a contracting mechanism, which is disposed longitudinally at a first site of the sleeve, and a longitudinal contracting member, having (a) a first member end, (b) a second member end, which is coupled to the sleeve longitudinally at a second site longitudinally between the first site and the second sleeve end, exclusive, and (c) a first member end portion, which is coupled to the contracting mechanism. The sleeve is arranged in a closed loop, such that first and second portions of the sleeve together define a longitudinally overlapping portion of the sleeve. The implantable structure is configured such that the contracting assembly longitudinally applies a longitudinal contracting force only between the first and the second sites, and not along the overlapping portion.