Compact graphs over a sphere of constant second order mean curvature. (English) Zbl 1189.53058

If \(x: M^n\rightarrow \mathbb{R}^{n+1}(c)\) is an oriented hypersurface in the Euclidean space and \(k_1,\dots, k_n\) the principal curvatures of \((M^n,x)\), one may consider \(S_r\), the \(r^{th}\) elementary symmetric functions of the \(k_i\), as a generalization of the mean curvature: \(S_r = k_{i1} \dots k_{ir},\, r = 0, 1,\dots, n\). Many well known theorems on constant mean curvature hypersurfaces can be extended by means of the \(S_r\).
For instance, for a given \(r\), A. Ros has proved that a round sphere is the unique compact embedded hypersurface in Euclidean space for which \(S_r\) is constant [J. Differ. Geom. 27, No. 2, 215–220 (1988; Zbl 0638.53051); Rev. Mat. Iberoam. 3, No. 3–4, 447–453 (1987; Zbl 0673.53003)]. Later on, S. Montiel and A. Ros extended this result to any compact embedded hypersurface contained in an open hemisphere of the sphere \(\mathbb{S}^{n+1}\) [in: Differential geometry. A symposium in honour of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 279–296 (1991; Zbl 0723.53032)].
In the paper under review, the authors obtain a similar result for compact star-shaped hypersurfaces of the sphere, whose \(S_2\) function is a positive constant, and without assuming that it is contained in an open hemisphere.
Let \(x: M^n\rightarrow M^{n+1}(c)\) be an oriented hypersurface with unit normal vector field \(N\) in a real space form \(M^{n+1}(c)\). For a conformal vector field \(V\) on \(M^{n+1}(c)\), the authors define a support function on \(M^{n}\) by \(g(p) = <V,N>(x(p))\). They consider also the \(L_r\) operator (an extension of the Laplacian) given by: \(L_r : D(M) \rightarrow D(M)\), \(L_r(f) = div (P_r\nabla f)\), where \(P_r\) is the \(r^{th}\) Newton tensor. Then, they obtain a formula for \(L_r(g)\) acting on the support function \(g\) in terms of the \(S_r\) (whose Laplacian version has been previously obtained by several authors), and which is the main ingredient to prove: Let \(x: M^n\rightarrow \mathbb{S}^{n+1}\) be a compact star-shaped hypersurface with positive constant \(S_2\). Then \((M^n,x)\) is totally umbilic. As an easy consequence, they also prove that a compact star-shaped minimal hypersurface \(x: M^n\rightarrow \mathbb{S}^{n+1}\) has to be totally geodesic.


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
53C65 Integral geometry
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