Open book structures and unicity of minimal submanifolds

Hardt, R.; Rosenberg, Harold

Annales de l'Institut Fourier, Tome 40 (1990) no. 3, pp. 701-708.

Nous démontrons l’unicité de certaines sous-variétés minimales, par exemple, les anneaux de Clifford dans $S^3$. L’idée est de regarder le placement de la sous-variété minimale par rapport au feuilletage singulier de $S^3$ par des anneaux de Clifford de bord deux géodésiques fixées à une distance $\pi /2$.

We prove unicity of certain minimal submanifolds, for example Clifford annuli in $S^3$. The idea is to consider the placement of the submanifold with respect to the (singular) foliation of $S^3$ by the Clifford annuli whose boundary are two fixed great circles a distance $\pi /2$ apart.

[GT] D. Gilbarg & N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1977. | MR | Zbl

[H] J. Hass, Minimal surfaces in foliated manifolds, Comm. Math. Helv., 61 n°1 (1986), 1-32. | MR | Zbl

[HS] R. Hardt & L. Simon, Boundary regularity and embedded solutions for the oriented Plateau problem, Ann. of Math., 110 (1979), 439-486. | MR | Zbl

[L] H.B. Lawson, Complete minimal surfaces in Sn, Ann. of Math., 92 (1970), 335-374. | MR | Zbl

[M] F. Morgan, On finiteness of the number of stable minimal hypersurfaces with a fixed boundary, Indiana U. Math. J., 35 (1986), 779-833. | MR | Zbl

[Sh] M. Shiffman, On surfaces of stationary bounded by two circles, or convex curves, in parallel planes, Ann. of Math., 3 (1956), 77-90. | MR | Zbl

[So] B. Solomon, On the Gauss map of an area-minimizing hypersurface, J. Diff. Geom., 24 (1984), 221-232. | MR | Zbl