  We examine the renormalized free energy of the free Dirac fermion and the
free scalar on a (2+1)-dimensional geometry $\mathbb{R} \times \Sigma$, with
$\Sigma$ having spherical topology and prescribed area. Using heat kernel
methods, we perturbatively compute this energy when $\Sigma$ is a small
deformation of the round sphere, finding that at any temperature the round
sphere is a local maximum. At low temperature the free energy difference is due
to the Casimir effect. We then numerically compute this free energy for a class
of large axisymmetric deformations, providing evidence that the round sphere
globally maximizes it, and we show that the free energy difference relative to
the round sphere is unbounded below as the geometry on $\Sigma$ becomes
singular. Both our perturbative and numerical results in fact stem from the
stronger finding that the difference between the heat kernels of the round
sphere and a deformed sphere always appears to have definite sign. We
investigate the relevance of our results to physical systems like monolayer
graphene consisting of a membrane supporting relativistic QFT degrees of
freedom.
