Minimal hypersurfaces with low index in the real projective space (Q2078592)

A classical result of [\textit{M. do Carmo} et al., Comment. Math. Helv. 75, No. 2, 247--254 (2000; Zbl 0977.53059)] states that the only compact two-sided minimal hypersurfaces with index one in the real projective space \(\mathbb{RP}^{n+1}\) are the totally geodesic spheres and the minimal Clifford hypersurfaces. In this paper, by using some ideas in the min-max theory in finite dimension, the authors show that there exists no connected compact two-sided minimal hypersurface of \(\mathbb{RP}^{n+1}\) with constant scalar curvature and Morse index two. Therefore, combining this with the result of do Carmo-Ritoré-Ros implies that the only connected compact two-sided minimal hypersurface of \(\mathbb{RP}^{n+1}\) with constant scalar curvature and Morse index less or equal than two are the totally geodesic spheres and the Clifford hypersurfaces. As a corollary, the authors also show a gap estimate for Morse index of minimal hypersurface in the round sphere.