Isoperimetric inequalities for parametric variational problems (Q697664)

Isoperimectric inequalities for general parametric variational double integrals \({\mathcal F}\), whose Lagrangians \(F\) depend on the position vector \(X\) and the surface normal \(N\), are established. As an essential tool Sauvigny's \(F\)-conformal parameters adapted to the parametric integrand are introduced and use is made of the notion of generalised mean and Gaussian curvatures. The special cases of minimal surfaces, surfaces of bounded mean curvature and \({F}\)-minimising surfaces are also discussed.