*Preprint*

**Inserted:** 17 oct 2018

**Last Updated:** 26 may 2019

**Year:** 2018

**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, and we prove that the non-existence of minimizers implies that the infimum equals $\beta_1-4\pi$; therefore we use such criterion in order to explicitly find an infinite family of curves $\Gamma$ for which such problem does have minimizers in $\mathcal{C}$. Also such curves $\Gamma$ can be chosen arbitrarily close to a circumference in $C^1$ sense.

**Download:**