  We consider the problem of geometric optimization of the lowest eigenvalue
for the Laplacian on a compact, simply-connected two-dimensional manifold with
boundary subject to an attractive Robin boundary condition. We prove that in
the sub-class of manifolds with the Gauss curvature bounded from above by a
constant $K_\circ \ge 0$ and under the constraint of fixed perimeter, the
geodesic disk of constant curvature $K_\circ$ maximizes the lowest Robin
eigenvalue. In the same geometric setting, it is proved that the spectral
isoperimetric inequality holds for the lowest eigenvalue of the
Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin
Laplacians acting on unbounded three-dimensional cones to show that, under a
constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue
is maximized by the circular cone.
