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Raphael
Raphael
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Considering a Jordan curve $\Gamma\subset \mathbb{R}^3$, we know from Douglas and Rado that there exists a unique minimal disk which bounds $\Gamma$. We also know that there exists exterior solutions, for example graph over $\mathbb{R}^2\setminus \Omega$ where $\Omega$ is the interior of the projection of $\Gamma$ over $\mathbb{R}^2\times\{0\}$ when the projection is one to one. Moreover we know that these surfaces are asymptotic to a plane or a catenoid. My question is about condition over Γ for existence of such minimal surface asymptotic to a plane? and do we know some curve for which there exits several solutions?

Considering a Jordan curve $\Gamma\subset \mathbb{R}^3$, we know from Douglas and Rado that there exists a unique minimal disk which bounds $\Gamma$. We also know that there exists exterior solutions, for example graph over $\mathbb{R}^2\setminus \Omega$ where $\Omega$ is the interior of the projection of $\Gamma$ over $\mathbb{R}^2\times\{0\}$ when the projection is one to one. Moreover we know that these surfaces are asymptotic to a plane or catenoid. My question is about condition over Γ for existence of such minimal surface asymptotic to a plane? and do we know some curve for which there exits several solutions?

Considering a Jordan curve $\Gamma\subset \mathbb{R}^3$, we know from Douglas and Rado that there exists a unique minimal disk which bounds $\Gamma$. We also know that there exists exterior solutions, for example graph over $\mathbb{R}^2\setminus \Omega$ where $\Omega$ is the interior of the projection of $\Gamma$ over $\mathbb{R}^2\times\{0\}$ when the projection is one to one. Moreover we know that these surfaces are asymptotic to a plane or a catenoid. My question is about condition over Γ for existence of such minimal surface asymptotic to a plane? and do we know some curve for which there exits several solutions?

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Raphael
Raphael
  • 198
  • 5

Exterior plateau problem with planar ends

Considering a Jordan curve $\Gamma\subset \mathbb{R}^3$, we know from Douglas and Rado that there exists a unique minimal disk which bounds $\Gamma$. We also know that there exists exterior solutions, for example graph over $\mathbb{R}^2\setminus \Omega$ where $\Omega$ is the interior of the projection of $\Gamma$ over $\mathbb{R}^2\times\{0\}$ when the projection is one to one. Moreover we know that these surfaces are asymptotic to a plane or catenoid. My question is about condition over Γ for existence of such minimal surface asymptotic to a plane? and do we know some curve for which there exits several solutions?