*Preprint*

**Inserted:** 17 oct 2018

**Last Updated:** 18 oct 2018

**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 $\mathcal{g}\geq 1$ having the curve $\Gamma$ as boundary, without any prescription on the conormal. By a general lower bound estimate, in case $\Gamma$ is a circle or a straight line we prove that the problem is equivalent if restricted to embedded surfaces, we prove that do not exist minimizers, and the infimum equals $\beta_\mathcal{g}-4\pi$, where $\beta_\mathcal{g}$ is the energy of the closed minimizing surface of genus $\mathcal{g}$. Then we study the case in which $\Gamma$ is compact, $\mathcal{g}=1$ and the competitors are restricted to a weaker class $\mathcal{C}$ which includes embedded surfaces, and we prove that the nonexistence of minimizers implies that the infimum equals $\beta_1-4\pi$; therefore we 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 $C^k$ norm for any $k$. Finally we prove that there are curves $\Gamma$ arbitrarily close to the circle for which the classical problem on immersed surfaces coincides with the problem on the class $\mathcal{C}$, there exist minimizers, and such minimizers are smooth embedded surfaces.

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