We extend the classification of Robert Bryant of Willmore spheres in S3 to
variational branched Willmore spheres S3 and show that they are inverse
stereographic projections of complete minimal surfaces with finite total
curvature in R3 and vanishing flux. We also obtain a classification
of variational branched Willmore spheres in S4, generalising a theorem of
Seb\'{a}stian Montiel. As a result of our asymptotic analysis at branch points,
we obtain an improved C1,1 regularity of the unit normal of variational
branched Willmore surfaces in arbitrary codimension. We also prove that the
width of Willmore sphere min-max procedures in dimension 3 and 4, such as
the sphere eversion, is an integer multiple of 4π.Comment: 74 pages, 1 figur