On how to use drift to push the spectral gap of a diffusion on $S^{2}$ to infinity

We prove that on the sphere $S^{2}$, one can find a sequence of divergence free vector fields $\textrm {b}_{n}$ with the property that the spectral gap of the operators $A_{\textrm {b}_{n}}= \Delta +\textrm {b}_{n}\cdot \nabla$ goes to infinity. The proof uses some suitable adapted Faber-Krahn type inequality for functions which are in the kernel of the operator $\textrm {b}_{n}\cdot \nabla$. Questions of this type arise when trying to accelerate Markov Monte Carlo methods by adding convergence enhancing motion.