Hypersurfaces in a sphere with constant mean curvature. (English) Zbl 0873.53040
Let \(M^n\) be a closed hypersurface of constant mean curvature immersed in the unit sphere \(S^{n+1}\). Denoting by \(S\) the square of the length of the second fundamental form, the author proves: If \(S<2(n-1)^{1/2}\), then \(M^n\) is a small hypersphere in \(S^{n+1}\). In the case of a not necessarily closed hypersurface the theorem, holds, too, if \(S\) is assumed to be constant (fulfilling the above condition). Furthermore, all \(M^n\) are characterized in the case of equality.
Reviewer: F.Manhart (Wien)
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
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