In this article, we first investigate weighted Minkowski-type inequalities for nearly spherical sets in space forms, focusing on the sets that are $C^1$-close to geodesic spheres. Our results generalize the work of Glaudo (2022, Adv. Math. 408, 108595) by incorporating broader geometric settings and convex weight functions. Additionally, we establish quantitative stability estimates for weighted Alexandrov–Fenchel-type inequalities in $\mathbb {R}^{n+1}$ and $\mathbb {H}^{n+1}$, extending the earlier results of VanBlargan and Wang (2024, Commun. Contemp. Math. 26, 2350026) and Zhou and Zhou (2024, J. Geom. Anal. 34, 376). These inequalities hold for nearly spherical sets that are $W^{2,\infty }$-close to geodesic spheres coupled with general convex weights.

We consider a mixed Steklov–Dirichlet eigenvalue problem on a smooth bounded domain having a spherical hole. In this article, we take Dirichlet condition on the boundary of the spherical hole and Steklov condition on the other boundary component/s. Under certain symmetry assumptions on multiconnected domains in $\mathbb {R}^{n}$ having a spherical hole, we obtain isoperimetric inequalities for the k-th Steklov–Dirichlet eigenvalues for each $k \in \{2, 3, \dots , n+1\}$. We provide examples to emphasise the fact that the symmetry assumptions, on the family of domains considered, are crucial. We also extend Theorem 3.1 of “Gavitone et al. (2023), An isoperimetric inequality for the first Steklov–Dirichlet Laplacian eigenvalue of convex sets with a spherical hole, Pacific Journal of Mathematics, 320(2): 241–259” not only from Euclidean domains to domains in space forms but also from convex domains to star-shaped domains. In particular, we obtain sharp lower and upper bounds for the first Steklov–Dirichlet eigenvalue on the family of all bounded star-shaped domains on the hemisphere as well as on the hyperbolic space.

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a group action. We also prove another topological rigidity result for hypersurfaces of the sphere that involves the spherical image of its usual Gauss map.

In this short note, we prove that an almost umbilical compact hypersurface of a real space form with almost Codazzi umbilicity tensor is embedded, diffeomorphic and quasi-isometric to a round sphere. Then, we derive a new characterisation of geodesic spheres in space forms.

For an isotropic submanifold ${{M}^{n}}(n\underline{\underline{>}}3)$ of a space form ${{\tilde{M}}^{n+p}}(c)$ of constant sectional curvature $c$ , we show that if the mean curvature vector of ${{M}^{n}}$ is parallel and the sectional curvature $K$ of ${{M}^{n}}$ satisfies some inequality, then the second fundamental form of ${{M}^{n}}$ in ${{\tilde{M}}^{n+p}}$ is parallel and our manifold ${{M}^{n}}$ is a space form.