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 $\mathfrak{g}\geq1$ having the curve $Γ$ as boundary, without any prescription on the conormal. By general lower bound estimates, in case $Γ$ is a circle we prove that such problem is equivalent if restricted to embedded surfaces, we prove that do not exist minimizers, and the infimum equals $β_\mathfrak{g}-4π$, where $β_\mathfrak{g}$ is the energy of the closed minimizing surface of genus $\mathfrak{g}$. 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, $\mathfrak{g}=1$ and the competitors are restricted to a suitable class $\mathcal{C}$ of varifolds including embedded surfaces. We prove that under suitable assumptions minimizers exists in this class of generalized surfaces.