Apparatus for use at a native atrioventricular valve of a heart of a subject includes a prosthetic valve structure. The prosthetic valve structure comprises a valve ring, a leaflet frame, a plurality of prosthetic leaflets, and an upper skirt. The valve ring circumscribes a longitudinal axis of the prosthetic valve structure. The leaflet frame is disposed within the valve ring. The prosthetic leaflets are coupled to the leaflet frame, and are configured to facilitate upstream-to-downstream bloodflow longitudinally through the prosthetic valve structure. The upper skirt includes an annular skirt fabric, and an annular skirt frame coupled to the valve ring by the skirt fabric. The upper skirt extends radially outward and in an upstream direction from the valve ring. The upper skirt is configured to rest against an atrial portion of the native atrioventricular valve. Other embodiments are also described.