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 M n+1. It is a simple fact that if M has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If M = S n+1 (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 M is the real projective space RP n+1 = S n+1 (1)/{±}, 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) × S n2 (R 2) ⊂ S n+1 (1) obtained as the product of two spheres of dimensions n

Isoperimetry and volume preserving stability in real projective spaces

“…As usual, Σ and Σ are smooth away for a closed set of Hausdorff dimension n − 7. By the result of do Carmo, Ritoré, and Ros[10], the only possibility for the former is a Clifford hypersurface; the proof still holds in the singular case by the cut-off argument of Morgan and Ritoré[21] discussed in the previous section. In the latter case we have | Σ| ≥ |RP n |.…”

“…Although there are many topologies for the index one minimal hypersurfaces, the norm of their second fundamental forms can only take two possible values. This striking fact manifested sharply in the proof given in [10]. In contrast, the norm of the second fundamental form of volume preserving stable hypersurfaces ranges over all non-negative numbers.…”

“…The proof of Theorem 1.2 is inspired by the work of do Carmo, Ritoré and Ros in [10] on the classification of two-sided index one minimal hypersurfaces in the real projective space RP n . The question of isoperimetry in RP n is also raised in [10]. As in the volume preserving case, index one minimal hypersurfaces also minimize area for a certain class of variations.…”

Second variation of one-sided complete minimal surfaces

“…Then the Jacobi operator, which acts on the sections of the normal bundle, becomes a Schrödinger operator acting on functions. This case, perhaps the easiest one, has been studied by many people (see [2], [1], [4], [5], [6], [11], [16] and the references therein). For one-sided minimal hypersurfaces and for minimal submanifolds with high codimension, only a few particular situations have been considered (see [7], [8], [9], [10], [12], [13], [14], [15] and the references therein).…”

“…Later, Urbano [16] classified the orientable compact minimal surfaces of S 3 with index less than six, proving that the surface must be either an equator or the Clifford torus which has index five. When M is the real projective space RP 3 , Onhita [9] proved that its only stable compact minimal surface is the totally geodesic real projective plane, and later, Do Carmo, Ritoré and Ros [2] characterized the totally geodesic two-sphere and the Clifford torus as the only orientable two-sided compact minimal surfaces of RP 3 with index one.…”