Two classes of Willmore Surfaces in $\mathbb{S}^2\times \mathbb{S}^2$

Xiaoling Chai; Shimpei Kobayashi; Changping Wang; Zhenxiao Xie

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arXiv · 2602.05303

Two classes of Willmore Surfaces in $\mathbb{S}^2\times \mathbb{S}^2$

Xiaoling Chai·Shimpei Kobayashi·Changping Wang·Zhenxiao Xie

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Abstract

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.

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Xiaoling Chai, Shimpei Kobayashi, Changping Wang, Zhenxiao Xie. 2026-02-05. Two classes of Willmore Surfaces in $\mathbb{S}^2\times \mathbb{S}^2$. https://arxiv.org/abs/2602.05303

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