Compact minimal hypersurfaces with index one in the real projective space

Abstract

Let \({M}^{n} \) be a compact (two-sided) minimal hypersurface in a Riemannian manifold \( \overline{M}^{n+1} .\) It is a simple fact that if \( \overline{M}\) has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If \( \overline{M}=S{n+1} \) is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator. We prove that if \( {\overline{M}}\) is the real projective space \({P^{n+1}}={S^{n+1}}/\{\pm\},\) obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface \({S^n1}({R_1})\times{S^n2}({R_2})\subset{S^{n+1}}\) obtained as the product of two spheres of dimensions \({n_1}, {n_2}\) and radius \({R_1}, {R_2}\), with \({n_1}+{n_2}=n,\,{R_1^2}+{R_2^2}=1\,{\rm and}\,{R_2^2}={n_2}{R_1^2}\)