On the Plateau–Douglas problem for the Willmore energy of surfaces with planar boundary curves

Abstract

For a smooth closed embedded planar curve Γ, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus 𝔤 ≥ 1 having the curve Γ as boundary, without any prescription on the conormal. In case Γ is a circle we prove that do not exist minimizers and that the infimum of the problem equals β𝔤 − 4π, where β𝔤 is the energy of the closed minimizing surface of genus 𝔤. We also prove that the same result also holds if Γ is a straight line for the suitable analogously defined minimization problem on asymptotically flat surfaces. Then we study the case in which Γ is compact, 𝔤 = 1 and the competitors are restricted to a suitable class 𝒞 of varifolds that includes embedded surfaces. We prove that under suitable assumptions minimizers exists in this class of generalized surfaces.