The Transverse Entropy Functional and the Sasaki-Ricci Flow

We introduce two new functionals on Sasaki manifolds, inspired by the work of Perelman, which are monotonic along the Sasaki-Ricci flow. We relate their gradient flow, via diffeomorphisms preserving the foliated structure of the manifold, to the transverse Ricci flow. Finally, when the basic first Chern class is positive, we employ these new functionals to prove a uniform $C^{0}$ bound for the transverse scalar curvature, and a uniform $C^{1}$ bound for the transverse Ricci potential along the Sasaki-Ricci flow.