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

TL;DR

This paper investigates the minimization of Willmore energy for surfaces with planar boundary curves, establishing non-existence of minimizers in certain cases and existence in others, with implications for geometric analysis.

Contribution

It provides new results on the existence and non-existence of minimizers for the Willmore energy with boundary conditions, extending understanding of the Plateau-Douglas problem.

Findings

No minimizers exist for the problem when the boundary is a circle, and the infimum is explicitly calculated.

The same non-existence result applies when the boundary is a straight line for asymptotically flat surfaces.

Existence of minimizers is proven for genus-one surfaces with compact boundary within a suitable class of varifolds.

Abstract

For a smooth closed embedded planar curve $\Gamma$, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus $\mathfrak{g}\geq1$ having the curve $\Gamma$ as boundary, without any prescription on the conormal. By general lower bound estimates, in case $\Gamma$ 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 $\beta_\mathfrak{g}-4\pi$, where $\beta_\mathfrak{g}$ is the energy of the closed minimizing surface of genus $\mathfrak{g}$. We also prove that the same result also holds if $\Gamma$ is a straight line for the suitable analogously defined minimization problem on asymptotically flat surfaces.\\ Then we study the case in which $\Gamma$ 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.