The Transverse Entropy Functional and the Sasaki-Ricci Flow

TL;DR

This paper introduces new entropy functionals on Sasaki manifolds that are monotonic along the Sasaki-Ricci flow, leading to bounds on curvature when the basic first Chern class is positive.

Contribution

The paper defines two new functionals inspired by Perelman's work, relating their gradient flow to the transverse Ricci flow on Sasaki manifolds.

Findings

Monotonicity of the new functionals along the flow

Establishment of uniform bounds on transverse scalar curvature

Establishment of uniform bounds on the transverse Ricci potential

Abstract

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.