A Willmore surface is the image of an immersion that is a critical point of the Willmore energy under variations with compact support. Equivalently, it is an immersion that satisfies the Willmore equation.
Every minimal surface is a Willmore surface, as is every round sphere. The Willmore equation is conformally invariant. This relates Willmore surfaces in the 3-sphere to minimal surfaces in 3-dimensional Euclidean space and 3-dimensional hyperbolic space. In particular, suitable hyperbolic minimal surfaces extend through the ideal boundary to Willmore surfaces in the 3-sphere (Bobenko et al. 2019).
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3-Space." Proc. Roy. Soc. A 475, 20190173, 2019.