On the stability of solutions of semilinear elliptic equations with Robin boundary conditions on Riemannian manifolds

We investigate existence and nonexistence of stationary stable nonconstant solutions, i.e., patterns, of semilinear parabolic problems in bounded domains on Riemannian manifolds, satisfying Robin boundary conditions. These problems arise in several models in applications, in particular in mathematical biology. We point out the significance both of the nonlinearity and of geometric objects such as the Ricci curvature of the manifold, the second fundamental form of the boundary of the domain, and its mean curvature. Special attention is given to surfaces of revolution and to spherically symmetric manifolds, where we prove refined results.