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 a 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 (R^{n+1}, g_{can}, e^{−f}dµ). 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 (M^3 , g, e^{−f}dµ), 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 S^2 . 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 S^2 or I ×_h R^2 with curvature constant Ricci
