Comparison methods for semilinear elliptic problems on Riemannian manifolds with a Ricci lower bound

In the first part of the article we develop a comparison method for positive solutions of the semilinear Dirichlet problem $\Delta u+f(u)=0$ on domains $\Omega\subset \mathcal M^n$ of a Riemannian manifold $(\mathcal{M}^n,g)$ with a Ricci lower bound $\operatorname{Ric}_g\ge (n-1)k\,g$. Assuming admissibility and structural conditions on $f$, we prove a sharp pointwise gradient comparison, with a rigid characterization of the equality case. As applications, we derive an explicit isoperimetric-type inequality and a quantitative hot-spot localization estimate under natural convexity assumptions. In the second part, on $\mathbb S^n$ we show that isoparametric foliations produce non-rotational $f$-extremal domains, and that these examples descend to smooth quotients under free isometric actions preserving the foliation.