Eigenvalues and energy functional with monotonicity formulae under the CR Yamabe flow on a closed pseudohermitian 3-manifold

Abstract

In this paper, we show that the first positive eigenvalue of $-\Delta_b+cW$ is nondecreasing under the CR Yamabe flow on a closed pseudohermitian 3-manifold. We construct families of functionals of the type of $\mathcal{F}$-functional of Perelman and show that these energy functionals are nondecreasing under the coupled Yamabe flow on a closed pseudohermitian 3-manifold. We also show the monotonicity property under the normalized CR Yamabe flow.