The Sasaki–ricci Flow

Abstract

In this paper, we introduce the Sasaki-Ricci flow to study the existence of η-Einstein metrics. In the positive case any η-Einstein metric can be homothetically transformed to a Sasaki-Einstein metric. Hence it is an odd-dimensional counterpart of the Kähler-Ricci flow. We prove its well-posedness and long-time existence. In the negative or null case the flow converges to the unique η-Einstein metric. In the positive case the convergence remains in general open. The paper can be viewed as an odd-dimensional counterpart of Cao's results on the Kähler-Ricci flow.

Second-Order Geometric Flows on Foliated Manifolds

“…The case of Sasaki manifolds. In the case of Sasaki metrics, theorem 6.2 and theorem 6.3 provide an alternative proof of the main results of [23] on Sasaki-Ricci flow.…”

On Sasaki–Ricci solitons and their deformations

“…g be the projection onto W g . We will consider the function (14) S(t, α, ϕ) := Π ⊥ g Π ⊥ t,α,ϕ G t,α,ϕ (s T t,α,ϕ − s 0 t,α,ϕ ) where G t,α,ϕ is the Green operator of d B with respect to the metric g t,α,ϕ . For the metric g t,α,ϕ to be a generalized SRS we need G t,α,ϕ (s T t,α,ϕ − s 0 t,α,ϕ ) to lie in H z t,α,ϕ := H z gt,α,ϕ , so S(t, α, ϕ) = 0.…”

“…They have been extensively studied, one possible motivation is that they are special solutions of the Kähler-Ricci flow (see e.g. [5] and the references therein, see also [14] for an introduction to the transverse Kähler-Ricci flow, called Sasaki-Ricci flow ).…”

Bochner-Type Formulas for Transversally Harmonic Maps

“…One can easily check the following relationship between the Ricci curvature tensor Ric and the transverse Ricci curvature Ric T as follows (cf. [17]),…”