  In this paper, we obtain geometric upper bounds for the first eigenvalue
$\lambda_1^J$ of the Jacobi operator for both closed and compact with boundary
hypersurfaces having constant mean curvature (CMC). As an application, we
derive new rigidity results for the area of CMC hypersurfaces under suitable
conditions on $\lambda_1^J$ and the curvature of the ambient space. We also
address the Jacobi-Steklov problem, proving geometric upper bounds for its
first eigenvalue $\sigma_1^J$ and deriving rigidity results related to the
length of the boundary. Additionally, we present some results in higher
dimensions related to the Yamabe invariants.
