The optimal constant in Wente's $L^{\infty }$ estimate

Abstract

We explore some geometric aspects of compensation compactness associated to Jacobian determinants. We provide the optimal constant in Wente's inequality - the original motivation of this work - and go on to give various extensions to geometric situations. In fact we improve Wente's inequality somewhat, making it more appropriate for applications in which optimal results are required. This is demonstrated when we prove an optimal inequality for immersed surfaces of constant mean curvature in ${\mathbb{R}}^3$ , contolling their diameter in terms of their area and curvature.