Two classes of Willmore Surfaces in $\mathbb{S}^2\times \mathbb{S}^2$
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915775968706560 |
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| author | Chai, Xiaoling Kobayashi, Shimpei Wang, Changping Xie, Zhenxiao |
| author_facet | Chai, Xiaoling Kobayashi, Shimpei Wang, Changping Xie, Zhenxiao |
| contents | We establish two classification theorems for Willmore surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$. Firstly, we prove that a Willmore surface which is also minimal must be either a special complex curve given by a slice or a diagonal; or, a minimal surface in a totally geodesic submanifold $\mathbb{S}^2 \times \mathbb{S}^1$ described by a solution of the sinh-Gordon equation in one variable. Secondly, we demonstrate that a Willmore surface is of product type if and only if it is the product of an elastic curve in $\mathbb{S}^2$ and a great circle. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_05303 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Two classes of Willmore Surfaces in $\mathbb{S}^2\times \mathbb{S}^2$ Chai, Xiaoling Kobayashi, Shimpei Wang, Changping Xie, Zhenxiao Differential Geometry 53A30, 53A10, 53C42 We establish two classification theorems for Willmore surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$. Firstly, we prove that a Willmore surface which is also minimal must be either a special complex curve given by a slice or a diagonal; or, a minimal surface in a totally geodesic submanifold $\mathbb{S}^2 \times \mathbb{S}^1$ described by a solution of the sinh-Gordon equation in one variable. Secondly, we demonstrate that a Willmore surface is of product type if and only if it is the product of an elastic curve in $\mathbb{S}^2$ and a great circle. |
| title | Two classes of Willmore Surfaces in $\mathbb{S}^2\times \mathbb{S}^2$ |
| topic | Differential Geometry 53A30, 53A10, 53C42 |
| url | https://arxiv.org/abs/2602.05303 |