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

TL;DR

This paper classifies Willmore surfaces in the product space , identifying conditions under which they are minimal or of product type, and describes their geometric and differential properties.

Contribution

It provides two new classification theorems for Willmore surfaces in , linking minimality and product structures to specific geometric and differential equations.

Findings

Minimal Willmore surfaces are either special complex curves or solutions to sinh-Gordon equation.

Willmore surfaces of product type are characterized as products of elastic curves and great circles.

Classification theorems connect geometric properties with differential equations.

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.