Isoperimetric inequalities for parametric variational problems

Abstract

We prove isoperimetric inequalities for general parametric variational double integrals F, whose Lagrangians F depend on the position vector X and on the surface normal N. As an essential tool we introduce Sauvigny's F-conformal parameters adapted to the parametric integrand and use the notion of generalized mean and Gaussian curvature. The special cases of minimal surfaces, surfaces of bounded mean curvature and ▪-minimizing surfaces are also discussed.