Compactness theorems and an isoperimetric inequality for critical points of elliptic parametric functionals

Abstract.

We consider parametric variational double integrals ${\cal F}$ with elliptic Lagrangians F depending on the surface normal and prove a compactness theorem for ${\cal F}$-critical immersions. As a key ingredient for the relevant a priori estimates we use F. Sauvigny's F-conformal parameters adapted to the parametric integrand F. As a by-product of our analysis we obtain an isoperimetric inequality for ${\cal F}$-critical immersions generalizing the classical isoperimetric inequality for minimal surfaces.