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 \({n_1}+{n_2}=n,\,{R_1^2}+{R_2^2}=1\,{\rm and}\,{R_2^2}={n_2}{R_1^2}\) KeywordsMinimal SurfaceFundamental FormRicci CurvatureJacobi OperatorMinimal HypersurfaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.