Willmore two-spheres in the four-sphere

Montiel, Sebastián

doi:10.1090/S0002-9947-00-02571-X

Willmore two-spheres in the four-sphere
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Trans. Amer. Math. Soc. 352 (2000), 4469-4486 Request permission

Abstract:

Genus zero Willmore surfaces immersed in the three-sphere $\mathbb {S}^3$ correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are $4\pi k$, where $k\in \mathbb {N}^*$, with $k\ne 2,3,5,7$. When the ambient space is the four-sphere $\mathbb {S}^4$, the regular homotopy class of immersions of the two-sphere $\mathbb {S}^2$ is determined by the self-intersection number $q\in \mathbb {Z}$; here we shall prove that the possible critical values are $4\pi (|q|+k+1)$, where $k\in \mathbb {N}$. Moreover, if $k=0$, the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibration $\mathbb {P}^3\rightarrow \mathbb {S}^4$, from a rational curve in $\mathbb {P}^3$ and, if $k\ne 0$, via stereographic projection, from a minimal surface in $\mathbb {R}^4$ with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in some $\mathbb {P}^2\subset \mathbb {P}^3$ or (equivalently) when the minimal surface of $\mathbb {R}^4$ is complex with respect to a suitable complex structure of $\mathbb {R}^4$.

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