Sharp bounds for higher Steklov-Dirichlet eigenvalues on domains with spherical holes

We consider mixed Steklov-Dirichlet eigenvalue problem on smooth bounded domains in Riemannian manifolds. Under certain symmetry assumptions on multiconnected domains in ${\mathbb{R}}^n$ with a spherical hole, we obtain isoperimetric inequalities for $k$-th Steklov-Dirichlet eigenvalues for $2\le k\le n+1$. We extend Theorem 3.1 of \cite{gavitone2023isoperimetric} from Euclidean domains to domains in space forms, that is, we obtain sharp lower and upper bounds of the first Steklov-Dirichlet eigenvalue on bounded star-shaped domains in the unit $n$-sphere and in the hyperbolic space.