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

TL;DR

This work classifies Willmore surfaces in $\mathbb{S}^2\times\mathbb{S}^2$ by establishing two main results: (i) any Willmore surface that is also minimal falls into one of two rigid families—either a special complex curve (slice or diagonal) or a minimal surface in a totally geodesic $\mathbb{S}^2\times\mathbb{S}^1$ described by a sinh–Gordon equation in one variable; (ii) any Willmore surface of product type must be the product of an elastic curve in $\mathbb{S}^2$ with a great circle. The authors derive the fundamental equations for surfaces in $\mathbb{S}^2\times\mathbb{S}^2$ and the Willmore Euler–Lagrange equation in terms of a comprehensive set of geometric data, then use sinh–Gordon theory and a detailed analysis of the product structure to obtain the classifications. They also connect these results to known examples (e.g., Weierstrass tori) and show that certain candidate minimal–Willmore surfaces are non–Willmore, highlighting rigidity in this non–space-form setting. The findings advance understanding of conformal Willmore geometry in product Kähler–Einstein surfaces and identify explicit rigid subclasses with geometric and PDE descriptions.

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.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Remark 3.1
• Remark 3.2
• Proposition 3.1
• proof
• Remark 3.3
• Example 4.1
• Example 4.2
• Example 4.3