Willmore two-spheres in the four-sphere

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Willmore Submanifolds in a sphere (2002, background): “The conjecture was approached by Willmore [30], Li-Yau [18], Montiel-Ros [20], Ros [23], Langer-Singer [10] and many others(see [29], [31] and references there).”
Energy quantization for Willmore surfaces and applications (2011, methodology): “They were based on the energy quantization for Willmore spheres in codimension 1 and 2 respectively due to R. Bryant [Bry84] and S. Montiel [Mon00].”
The Willmore Conjecture (2014, background): “Ejiri, Montiel, and Musso [14, 42, 45] extended this work and classified Willmore spheres in S4.”