Willmore spheres in the $3$-sphere revisited

$\def\one{\href{https://dx.doi.org/10.4310/jdg/1214438991}{[1]}}$$\def\two{\href{https://books.google.com/books?id=e0MECAAAQBAJ&lpg=PA227&ots=v8NuefUb1Z&lr&pg=PA227#v=onepage&q&f=false}{[2]}}$$\def\six{\href{https://doi.org/10.1142/9789812792051_0021}{[6]}}$

Bryant $\one$ classified all Willmore spheres in $3$-space to be given by minimal surfaces in $\mathbb{R^3}$ with embedded planar ends. This note provides new explicit formulas for genus $0$ minimal surfaces in R3 with $2k + 1$ embedded planar ends for all $k \geq 4$. Peng and Xiao claimed these examples to exist in $\six$, but in the same paper they also claimed the existence of a minimal surface with $7$ embedded planar ends, which was falsified by Bryant $\two$.

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Received 28 April 2020

Accepted 5 January 2021

Published 16 July 2024