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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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##### References:
 [1] Aubin T.: Some nonlinear Problems in Riemannian Geometry. Springer, Heidelberg (1998) · Zbl 0896.53003 [2] Berard P., Meyer D.: Inégalités isopérimétriques et applications. Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 513–541 (1982) [3] Druet O.: Sharp local isoperimetric inequalities involving the scalar curvature. Proc. Am. Math. Soc. 130(8), 2351–2361 (2002) · Zbl 1067.53026 · doi:10.1090/S0002-9939-02-06355-4 [4] Gray A.: Tubes. Advanced Book Program. Addison-Wesley, Redwood City (1990) [5] Kapouleas N.: Compact constant mean curvature surfaces in Euclidean three-space. J. Differ. Geom. 33(3), 683–715 (1991) · Zbl 0727.53063 [6] Lee J.M., Parker T.H.: The Yamabe problem. Bull. Am. Math. Soc. (N.S.) 17(1), 37–91 (1987) · Zbl 0633.53062 · doi:10.1090/S0273-0979-1987-15514-5 [7] Li Y.Y.: On a singularly perturbed elliptic equation. Adv. Differ. Equ. 2, 955–980 (1997) · Zbl 1023.35500 [8] Lusternik L., Shnirelman L.: Méthodes topologiques dans les problèmes variationnels. Hermann, Paris (1934) [9] Nardulli, S.: Le profil isopérimétrique d’une variété Riemannienne compacte pour les petits volumes, Thèse de l’Université Paris 11 (2006). arXiv:0710.1849 and arXiv:0710.1396 [10] Ros A.: The isoperimetric problem. In: Hoffman, D. (eds) Global Theory of Minimal Surfaces, Clay Mathematics Proceedings, AMS, Providence (2005) · Zbl 1125.49034 [11] Schoen R., Yau S.T.: Lectures on Differential Geometry. International Press, New York (1994) · Zbl 0830.53001 [12] Takens F.: The minimal number of critical points of a function on a compact manifold and the Lusternic–Schnirelman category. Invent. Math. 6, 197–244 (1968) · Zbl 0198.56603 · doi:10.1007/BF01404825 [13] Ye R.: Foliation by constant mean curvature spheres. Pac. J. Math. 147(2), 381–396 (1991) · Zbl 0722.53022 [14] Willmore T.J.: Riemannian Geometry. Oxford University Press, NY (1993) · Zbl 0797.53002
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