A characterization for geodesic spheres in space forms
Let \(f:M^n \to Q^{n+1}_c\) be an isometric immersion of an oriented Riemannian manifold \(M^n\) into the \((n+1)\)-dimensional complete simply connected Riemannian manifold \(Q^{n+1}_c\) of constant curvature \(c\). The \(m\)-th mean curvature \(H_m\) of \(M^n\) is given by \((^n_m)H_m=S_m\), where \(S_m\) is the \(m\)-th symmetric function of the principal curvatures. Fix a point \(P_{0}\in Q^{n+1}_c\) and consider the function \(r(.)=d(.,P_0)\) where \(d\) is the distance to \(P_0\) in \(Q^{n+1}_c\). Let \(\varphi_{c}(t)\) be the solution of \(y''(t)+cy(t)=0\) with \(y(0)=0\), \(y'(0)=1\) and put \(\mu_{c}(t)=t\varphi'_{c}(t)/\varphi_{c}(t)\). In the paper under review, the author gives the following characterization of geodesic spheres in space forms: Let \(M^n\) be a compact connected oriented Riemannian manifold and let \(f:M^n \to Q^{n+1}_c\) be an isometric immersion, such that \(f\) has its image in a geodesic ball of radius less than \(\pi\over{2 \sqrt c}\) if \(c>0\). Assume that either \(|H_{s}|\leq(\mu_{c}(r)/r)^s\) if \(s\) is odd or \(H_s\leq(\mu_{c}(r)/r)^s\) if s is even, where \(1 \leq s \leq n\). Then \(f(M^n)\) is a geodesic sphere. One of the immediate consequences of the theorem is the following Corollary: Let \(M^n\) be a compact connected oriented immersed hypersurface of the Euclidean space \(E^{n+1}\). If the scalar curvature \({\text{Sc}}\) satisfies \({\text{Sc}}\leq {{n(n-1)}\over r^2}\), then \(M^n\) is a hypersphere. The proof relies on a calculation of \(L_m (r^2)\), where \(L_m\) is the linearized operator of \(S_{m+1}\) arising from normal variations [\textit{R. C. Reilly}, J. Differ. Geom. 8, 465-477 (1973; Zbl 0277.53030)] and the maximum principle for second order elliptic operators.
scientific article; zbMATH DE number 1098491