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Constant mean curvature spheres in Riemannian manifolds. (English) Zbl 1165.53038
The authors show that there exist embedded spheres with large constant mean curvature in any compact Riemannian manifold \(M\). Specifically, they prove that there is a constant \(\rho_0 > 0\) and a smooth real valued function \[ \phi : M \times (0,\rho_0) \rightarrow {\mathbb R} \] such that, for all \(\rho \in (0, \rho_0)\), if \(p\) is a critical point of \(\phi(\cdot, \rho)\), then there exists an embedded hyper-surface \(S_{p,\rho}^\flat\) with mean curvature equal to \({m \over \rho}\) where \(m = \dim(M)\). \(S_{p,\rho}^\flat\) is obtained by deforming the geodesic sphere of radius \(\rho\) centered at \(p\). This result generalizes one due to R. Ye [Pac. J. Math. 147, No. 2, 381–396 (1991; Zbl 0722.53022)] who was able to construct such families of constant mean curvature spheres by deforming geodesic spheres centered at non-degenerate critical points of the scalar curvature function of \(M\). Unlike Ye’s result, the result of this paper applies even when all the critical points of the scalar curvature are degenerate, in particular, when \(M\) has constant scalar curvature.

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C20 Global Riemannian geometry, including pinching
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