Estimativas para o índice de Morse: resultados de cotação inferior, classificação, lacuna e rigidez

Abstract

In this Doctoral Thesis, we obtain several estimates for the Morse index of minimal hypersurfaces in five different settings. More precisely, (1): We prove the nonexistence of a closed two-sided minimal hypersurface immersed in the real projective space RPn+1 with index two. (2): We prove a gap of the Morse index of an orientable, closed minimal hypersurface immersed in a finite product of spheres. Such estimates are given as a function of the radii and dimensions of the spheres. (3): We study orientable complete f-minimal free boundary hypersurfaces in a domain Ω of the weighted Euclidean space (Rn+1, gcan, e−fdµ). In this case, we get lower bounds for the index by an affine function involving a topological quantity. If the hypersurface is compact, this quantity is its first Betti number. (4): We consider operators of the type ∆f + W − aK on surfaces in weighted Riemannian manifolds (M3, g, e−fdµ), where W is a locally integrable function, K is the Gaussian curvature of the surface, and a is a positive integer. We obtain some results about the topology and volume growth of geodesic ball on f-stable constant weighted mean curvature surfaces. In addition, we also get a lower bound for the first eigenvalue of the stability operator. (5): We obtain monotonicity and density formulas for hypersurfaces with a non-empty boundary, properly embedded in a warped product of type I ×h S2 . Finally, we present a method to calculate a region of stability for totally geodesic cones in a space conformal to warped products of the form I ×h S2 or I ×h R2 with curvature constant Ricci.