A method is provided including introducing into a heart atrium, an annuloplasty structure having a sleeve with an elongated tubular side wall, anchoring a first section of the sleeve by deploying a first tissue anchor through the first section and into a first portion of annulus tissue, and anchoring a second section of the sleeve by deploying a second tissue anchor through the second section and into a second portion of annulus tissue. A longitudinal portion of the tubular side wall is disposed between the first and second tissue anchors and has first and second lateral parts. The first lateral part is closer to the annulus than the second lateral part is to the annulus, and the second lateral part is opposite the first lateral part and has a larger degree of tension than a degree of tension of the first lateral part.