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

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.