Existence of free boundary disks with constant mean curvature in $\mathbb{R}^3$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910962245697536 |
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| author | Cheng, Da Rong |
| author_facet | Cheng, Da Rong |
| contents | Given a surface $Σ$ in $\mathbb{R}^3$ diffeomorphic to $S^2$, Struwe (Acta Math., 1988) proved that for almost every $H$ below the mean curvature of the smallest sphere enclosing $Σ$, there exists a branched immersed disk which has constant mean curvature $H$ and boundary meeting $Σ$ orthogonally. We reproduce this result using a different approach and improve it under additional convexity assumptions on $Σ$. Specifically, when $Σ$ itself is convex and has mean curvature bounded below by $H_0$, we obtain existence for all $H \in (0, H_0)$. Instead of the heat flow used by Struwe, we use a Sacks-Uhlenbeck type perturbation. As in previous joint work with Zhou (arXiv:2012.13379), a key ingredient for extending existence across the measure zero set of $H$'s is a Morse index upper bound. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_16323 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Existence of free boundary disks with constant mean curvature in $\mathbb{R}^3$ Cheng, Da Rong Differential Geometry Analysis of PDEs Given a surface $Σ$ in $\mathbb{R}^3$ diffeomorphic to $S^2$, Struwe (Acta Math., 1988) proved that for almost every $H$ below the mean curvature of the smallest sphere enclosing $Σ$, there exists a branched immersed disk which has constant mean curvature $H$ and boundary meeting $Σ$ orthogonally. We reproduce this result using a different approach and improve it under additional convexity assumptions on $Σ$. Specifically, when $Σ$ itself is convex and has mean curvature bounded below by $H_0$, we obtain existence for all $H \in (0, H_0)$. Instead of the heat flow used by Struwe, we use a Sacks-Uhlenbeck type perturbation. As in previous joint work with Zhou (arXiv:2012.13379), a key ingredient for extending existence across the measure zero set of $H$'s is a Morse index upper bound. |
| title | Existence of free boundary disks with constant mean curvature in $\mathbb{R}^3$ |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2203.16323 |