Journal of Differential Geometry

Classification of Willmore two-spheres in the 5-dimensional sphere

Xiang Ma, Changping Wang, and Peng Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The classification of Willmore two-spheres in the $n$-dimensional sphere $S^n$ is a long-standing problem, solved only when $n = 3, 4$ by Bryant, Ejiri, Musso and Montiel independently. In this paper we give a classification when $n = 5$. There are three types of such surfaces up to Möbius transformations: (1) superconformal surfaces in $S^4$; (2) minimal surfaces in $R^5$; (3) adjoint transforms of superconformal minimal surfaces in $R^5$. In particular, Willmore surfaces in the third class are not $S$-Willmore (i.e., without a dual Willmore surface).

To show the existence of Willmore two-spheres in $S^5$ of type (3), we describe all adjoint transforms of a superconformal minimal surface in $R^n$ and provide some explicit criterions on the immersion property. As an application, we obtain new immersed Willmore two-spheres in $S^5$ and $S^6$, which are not $S$-Willmore.

Article information

J. Differential Geom., Volume 106, Number 2 (2017), 245-281.

Received: 26 February 2015
First available in Project Euclid: 14 June 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Ma, Xiang; Wang, Changping; Wang, Peng. Classification of Willmore two-spheres in the 5-dimensional sphere. J. Differential Geom. 106 (2017), no. 2, 245--281. doi:10.4310/jdg/1497405626.

Export citation