Almost complex surfaces in the nearly K\'ahler $S^3\times S^3$
| Autor: | Bolton, John, Dillen, Franki, Dioos, Bart, Vrancken, Luc |
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| Rok vydání: | 2012 |
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| Druh dokumentu: | Working Paper |
| Popis: | In this paper almost complex surfaces of the nearly K\"ahler $S^3\times S^3$ are studied in a systematic way. We show that on such a surface it is possible to define a global holomorphic differential, which is induced by an almost product structure on the nearly K\"ahler $S^3\times S^3$. We also find a correspondence between almost complex surfaces in the nearly K\"ahler $S^3\times S^3$ and solutions of the general $H$-system equation introduced by Wente, thus obtaining a geometric interpretation of solutions of the general $H$-system equation. From this we deduce a correspondence between constant mean curvature surfaces in $\mathbb R^3$ and almost complex surfaces in the nearly K\"ahler $S^3\times S^3$ with vanishing holomorphic differential. This correspondence allows us to obtain a classification of the totally geodesic almost complex surfaces. Moreover, we will prove that almost complex topological 2-spheres in $S^3\times S^3$ are totally geodesic. Finally, we also show that every almost complex surface with parallel second fundamental form is totally geodesic. Comment: 15 pages |
| Databáze: | arXiv |
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