Compact minimal hypersurfaces with index one in the real projective space
Noting that any compact closed minimal hypersurface in a real projective space (of general dimension) cannot be stable, i.e., cannot have index zero, the authors consider the next natural question about hypersurfaces of low index: Which hypersurfaces have index one? They briefly give a bit of history about the subject that demonstrates how natural the question is. For example, analogous questions have been answered in Euclidean three-space and in spherical space and in the three-dimensional projective space and in flat three tori, with some of the answers coming from the authors themselves. The index is defined as the number of negative eigenvalues of the Jacobi operator, and for this operator to be defined on functions over the hypersurface, the authors require that the hypersurface be two-sided (this means that the hypersurface has a globally defined unit normal vector field). The main result is that the only compact two-sided minimal hypersurfaces with index one in the real projective space are the totally geodesic spheres and the minimal Clifford hypersurfaces. Clifford hypersurfaces are products of two spheres whose dimensions add up to one less than the dimension of the ambient projective space, and they are briefly described in this paper. The proof is based on a clever choice of test functions which reduce the problem to a result in a paper of \textit{S. S. Chern, M. do Carmo}, and \textit{S. Kobayashi} [Funct. Anal. Relat. Fields, Conf. Chicago 1968, 59-75 (1970; Zbl 0216.44001)]. The authors then note that totally geodesic spheres and the minimal Clifford hypersurfaces are the only surfaces in real projective space that can be volume preserving stable, which means that the Jacobi operator has no negative eigenvalues in the space of functions on the hypersurfaces which have mean zero (i.e., whose integral over the hypersurface is zero).
scientific article; zbMATH DE number 1523504