A gap theorem for hypersurfaces of the sphere with constant scalar curvature one
Let \( x \) be an immersion of a \( n \)-dimensional (connected and oriented) manifold \( M^{n} \) into a space form of dimension \( n+1 \) and curvature \( c \), denoting by \( S_{r} \) the \( r \)-th elementary symmetric function of the principal curvatures \( k_{i} \). An identity of H. Weyl for the Euclidean surface case \( c = 0, n = 2 \) has been extended to the general situation, stating \( \triangle S_{r} = L_{r-1}(S_{1})+\Omega_{r}+\Pi_{r} \), where \( \triangle \) is the Laplace/Beltrami operator of the first fundamental form, \( L_{r-1} \) the second order differential operator of the corresponding Newtonian \( P_{r-1} \), \( \Omega_{r} \) a certain symmetric polynomial in the \( k_{i} \), and \( \Pi_{r} \) a mixed invariant of the shape operator \( A \) and its covariant derivative \( \nabla A \). The \( \Omega_{r} \) can be brought to the invariant form \( \Omega_{r} = \Omega_{r}^{(0)}+c\Omega_{r}^{(1)} \) with \( \Omega_{r}^{(0)}:= 2rS_{2}S_{r}-(r+1)S_{1}S_{r+1}-(r-1)S_{1}^{2}S_{r} \) and \( \Omega_{r}^{(1)}:=nrS_{r}-(n-r+1)S_{1}S_{r-1} \). See for this the reviewer's paper [Math. Ann. 270, 125-145 (1985; Zbl 0536.53054)] which also contains applications to compact hypersurfaces with one constant \( S_{r} \). Various other consequences have been drawn from special cases of the Weyl identities; see the references in this paper. For instance, \textit{S. Y. Cheng} and \textit{S. T. Yau} showed that a compact hypersurface in the unit sphere \( S^{n+1} \) of nonnegative sectional curvature with \( S_{2} = \text{ const.} \geq 0 \) is either a small or great subsphere or a Clifford torus [Math. Ann. 225, 195--204 (1977; Zbl 0349.53041)]. The Newtonians were considered by \textit{K. Voß} in connection with the variation of the \( S_{r} \) [Math. Ann. 131, 180--218 (1956; Zbl 0073.38401)]. Recently, related results and extensions have been established by \textit{P. Kohlmann}, electronically available under www.mathematik.uni-dortmund.de/lsvii/Preprints/. The present authors now deal with the case \( c = 1, S_{2} = 0 \) and gain the following result: If \( M^{n} \) is a compact hypersurface of \( S^{n+1} \) with \( S_{1} \geq 0 \), \( S_{2} = 0 \), \( (n-1)S_{1}+3S_{3} \geq 0 \) then \( (n-1)S_{1}-3S_{3} = 0 \), and \( M^{n} \) is either totally geodesic or a Clifford torus \( S^{n_{1}}(r_{1}) \times S^{n_{2}}(r_{2}) \) such that \( n_{1},n_{2},r_{1},r_{2} \) are compatible with these relations for \( S_{1}, S_{2}, S_{3} \). Their proof amounts to the following (somewhat simpler) arguments: Under these assumptions the Weyl identity for \( r = 2 \) boils down to \( 0 = L_{1}(S_{1})+\Omega_{2}+\Pi_{2} \) with \( \Omega_{2} = -(n-1)S_{1}^2-3S_{1}S_{3} \) and \( \Pi_{2} = |\text{ grad} S_{1}|^{2}-|\nabla A|^{2} \). By a previous lemma of \textit{H. Alencar, M. do Carmo} and \textit{A. G. Colares} [Math. Z. 213, 117--131 (1993; Zbl 0792.53057)] \( S_{2} = \text{ const.} \geq 0 \) implies \( |\text{ grad} S_{1}|\leq |\nabla A|\), thus \( L_{1}(S_{1}) \geq 0 \). Applying Green's formulas for the operator \( L_{1},\) one successively deduces \( L_{1}(S_{1}) = 0 \), \( \text{ grad} S_{1} = 0 \) and \( \nabla A = 0 \), hence the assertion, since the hypersurfaces with parallel shape operator are well known. By the Gauss integrability condition, \( S_{2} = 0 \) is equivalent to \( R = 1 \), \( R \) the scalar curvature, and the inequality between \( S_{1}, S_{3} \) may be rephrased as \( |\sqrt{P_{1}}A|^{2} \leq \text{ trace} P_{1} \). As a consequence the authors claim a characterization of Clifford quadrics in the projective real elliptic space \( P^{n+1} \) by index assumptions. But the proof contains an error. Meanwhile, the authors announced a correction for this part resting on an analogy with \textit{J. Simons}' paper [Ann. Math. (2) 88, 62--105 (1968; Zbl 0181.49702)].
scientific article; zbMATH DE number 1837406