Flowing to free boundary minimal surfaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918511393112064 |
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| author | Rupflin, Melanie Struwe, Michael Wright, Christopher |
| author_facet | Rupflin, Melanie Struwe, Michael Wright, Christopher |
| contents | We introduce a flow that is designed to flow maps $u:Σ\to \mathbb{R}^n$ which map the boundary of a general domain surface $Σ$ into a given (not necessarily connected) submanifold $N\hookrightarrow \mathbb{R}^n$ towards a free boundary (branched) minimal immersion supported by $N$. In the case when $Σ$ is the unit disc $D$, this task can be achieved by means of the Plateau-flow introduced in the work [15] of the second author. When $Σ\neq D$, however, also the conformal type of the domain metric plays a role and it no longer suffices to deform the trace of the given map into a half-harmonic map as in [15]. In order to overcome this issue, here we combine ideas of the Plateau-flow from [15] with ideas of the Teichmüller harmonic flow from [12], in order to flow both an initial map $u_0$ with trace $u_0\colon\partial Σ\to N$ and an initial domain metric $g_0$ in a way that produces, as time tends to infinity, a half-harmonic map from $\partial Σ$ into $N$ whose harmonic extension is conformal and hence is a (branched) minimal immersion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19574 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Flowing to free boundary minimal surfaces Rupflin, Melanie Struwe, Michael Wright, Christopher Analysis of PDEs Differential Geometry We introduce a flow that is designed to flow maps $u:Σ\to \mathbb{R}^n$ which map the boundary of a general domain surface $Σ$ into a given (not necessarily connected) submanifold $N\hookrightarrow \mathbb{R}^n$ towards a free boundary (branched) minimal immersion supported by $N$. In the case when $Σ$ is the unit disc $D$, this task can be achieved by means of the Plateau-flow introduced in the work [15] of the second author. When $Σ\neq D$, however, also the conformal type of the domain metric plays a role and it no longer suffices to deform the trace of the given map into a half-harmonic map as in [15]. In order to overcome this issue, here we combine ideas of the Plateau-flow from [15] with ideas of the Teichmüller harmonic flow from [12], in order to flow both an initial map $u_0$ with trace $u_0\colon\partial Σ\to N$ and an initial domain metric $g_0$ in a way that produces, as time tends to infinity, a half-harmonic map from $\partial Σ$ into $N$ whose harmonic extension is conformal and hence is a (branched) minimal immersion. |
| title | Flowing to free boundary minimal surfaces |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2605.19574 |