Connecting disclinations by ridges

We consider a thin elastic sheet with a finite number of disclinations in a variational framework in the F\"oppl-von K\'arm\'an approximation. Under the non-physical assumption that the out-of-plane displacement is a convex function, we prove that minimizers display ridges between the disclinations. We prove the associated energy scaling law with upper and lower bounds that match up to logarithmic factors in the thickness of the sheet. One of the key estimates in the proof that we consider of independent interest is a generalization of the monotonicity property of the Monge-Amp\`ere measure.