  In this paper, we compute the second variation of the first Dirichlet
eigenvalue on extremal domains in general Riemannian manifolds and establish a
criterion for stability. We classify the stable extremal domains in the
2-sphere and higher-dimensional spheres when the boundary is minimal.
Additionally, we establish topological bounds for stable domains in a general
compact Riemannian surface, assuming either nonnegative total Gaussian
curvature or small volume.
