On the J-flow in Sasakian manifolds

Abstract

We study the space of Sasaki metrics on a compact manifold $M$ by introducing an odd-dimensional analogue of the $J$-flow. That leads to the notion of critical metric in the Sasakian context. In analogy to the Kähler case, on a polarised Sasakian manifold, there exists at most one normalised critical metric. The flow is a tool for texting the existence of such a metric. We show that some results proved by Chen (Commun. Anal. Geom. 12: 837–852, 2004) can be generalised to the Sasakian case. In particular, the Sasaki $J$ -flow is a gradient flow which has always a long-time solution minimising the distance on the space of Sasakian potentials of a polarised Sasakian manifold. The flow minimises an energy functional whose definition depends on the choice of a background transverse Kähler form $\chi$. When $\chi$ has nonnegative transverse holomorphic bisectional curvature, the flow converges to a critical Sasakian structure.

1 Introduction

Sasakian manifolds are the odd-dimensional counterpart of Kähler manifolds and are defined as odd-dimensional Riemannian manifolds $(M,g)$ whose Riemannian cone $(M\times {\mathbb {R}}^+,t^2g+\hbox {d}t^2)$ admits a Kähler structure. These manifolds are important for both geometric and physical reasons. In geometry, they can be used to produce new examples of complete Kähler manifolds, manifolds with special holonomy and Einstein metrics. Moreover, Sasakian manifolds play a role in the study of orbifolds since many Kähler orbifolds can be desingolarised by using Sasakian spaces. In theoretical physics, these manifolds play a central role in the AdS/CFT correspondence (see e.g. [12, 16, 17, 30–33]). We refer to [3, 39] for general theory and recent advancement in the study of these manifolds.

Given a Sasakian manifold, the choice of a Kähler structure on the Riemannian cone determines a unitary Killing vector field $\xi$ of the metric $g$ and an endomorphism $\Phi$ of the tangent bundle to $M$ such that

$$\begin{aligned} \Phi ^2=-\mathrm{Id}+\eta \otimes \xi ,\quad g(\Phi \cdot ,\Phi \cdot )=g(\cdot ,\cdot )-\eta \otimes \eta , \quad g=\frac{1}{2} \mathrm{d}\eta \circ (\mathrm{Id}\otimes \Phi )+\eta \otimes \eta , \end{aligned}$$

$\eta$ being the $1$-form dual to $\xi$ via $g$. It turns out that $\eta$ is a contact form and that $\Phi$ induces a CR-structure $(\mathcal D,J)$ on $M$. Moreover, $\Phi (X)=\mathrm{D}_X\xi$ for every vector field $X$ on $M$, where $\mathrm{D}$ is the Levi-Civita connection of $g$. The quadruple $(\xi ,\Phi ,\eta ,g)$ is usually called a Sasakian structure and the pair $(\xi ,J)$ can be seen as a polarisation of $M$.

The research of this paper is mainly motivated by [4, 21–23] where the study of Riemannian and symplectic aspects of the space of Sasakian potentials $\mathcal {H}$ on a polarised Sasakian manifold is approached. Our approach consists in using an analogue of the $J$-flow in the context of Sasakian Geometry obtaining some results similar to the ones proved in the Kähler case by Chen in [7]. The $J$-flow is a gradient geometric flow of Kähler structures introduced and firstly studied by Donaldson in [13] from the point of view of moment maps and by Chen in [7] in relation to the Mabuchi energy. It is defined as the gradient flow of a functional $J_{\chi }$ defined on the space of normalised Kähler potentials whose definition depends on a fixed background Kähler structure $\chi$. Chen proved in [7] that the flow has always a unique long-time solution which, in the special case when $\chi$ has nonnegative biholomorphic curvature, converges to a critical Kähler metric. Further results about the flow are obtained in [25, 38, 42, 43].

As far as we know, the interest for geometric flows in foliated manifolds comes from [27] where a foliated version of the Ricci flow is introduced. Subsequently, Smoczyk, Wang and Zhang proved in [36] that the transverse Ricci flow preserves the Sasakian condition and study its long-time behaviour generalising the work of Cao in [5] to the Sasakian case. Some deep geometric and analytic aspects of the Sasaki–Ricci flow were further investigated in [8–11].

In analogy to the Kähler case, the Sasaki $J$ -flow introduced in this paper (see Sect. 4 for the precise definition) is the gradient flow of a functional $J_{\chi }:\mathcal {H}\rightarrow {\mathbb {R}}$ whose definition depends on the choice of a transverse Kähler structure $\chi$. Sasakian metrics arising from critical points of the restriction of $J_{\chi }$ to the space of normalised Sasakian potentials $\mathcal {H}_0$, are natural candidates to be canonical Sasakian metrics.

The main result of the paper is the following

Theorem 1.1

Let $(M,\xi ,\Phi ,\eta ,g)$ be a $(2n+1)$-dimensional Sasakian manifold, and let $\chi$ be a transverse Kähler form on $M$. Then, the functional $J_{\chi }:{\mathcal {H}}_0\rightarrow {\mathbb {R}}$ has at most one critical point and the Sasaki $J$-flow has a long-time solution $f$ for every initial datum $f_0$. Furthermore, the length of any smooth curve in $\mathcal {H}_0$ and the distance between any two points decrease under the flow and when the transverse holomorphic bisectional curvature of $\chi$ is nonnegative, $f$ converges to a critical point of $J_{\chi }$ in ${\mathcal {H}}_0$.

The last sentence in the statement of Theorem 1.1 implies that if the transverse Kähler structure $\chi$ has nonnegative transverse holomorphic bisectional curvature, then $J_{\chi }$ has a critical point in ${\mathcal {H}}_0$. We remark that Sasakian manifolds having nonnegative transverse holomorphic bisectional curvature are classified in [24], but in the definition of the Sasaki $J$-flow, $\chi$ is just a transverse Kähler structure not necessarily induced by a Sasaki metric.

From the local point of view, a solution to the Sasaki $J$ -flow can be seen as a collection of solutions to the Kähler $J$-flow on open sets in ${\mathbb {C}}^n$. This fact allows us to use all the local estimates about the Kähler $J$-flow provided in [7]. What is necessary modifying from the Kähler case is the proof of the existence of a short-time solution to the flow (since the flow is parabolic only along transverse directions) and the global estimates. The short-time existence is obtained in Sect. 4 by using a trick introduced in [36], while the global estimates are obtained by using a transverse version of the maximal principle for transversally elliptic operators (see Sect. 5).

2 Preliminaries

In this section, we recall some basic facts about Sasakian Geometry declaring the notation which will be adopted in the rest of the paper.

Let $(M,\xi ,\Phi ,\eta ,g)$ be a $(2n+1)$-dimensional Sasaki manifold. Then, the Reeb vector field $\xi$ specifies a Riemannian foliation on $M$, which is usually denoted by $\mathcal F_{\xi }$, and the tangent bundle to $M$ splits in $TM=\mathcal D\oplus L_{\xi }$, where $L_{\xi }$ is the line bundle generated by $\xi$ and $\mathcal D$ has as fibre over a point $x$ the vector space $\ker \eta _x$. The metric $g$ splits accordingly in $g=g^\mathrm{T}+\eta ^2$, where the degenerate tensor $g^\mathrm{T}$ is called the transverse metric of the Sasakian structure. In the following, we denote by $\nabla ^\mathrm{T}$ the transverse Levi-Civita connection defined on the bundle $\mathcal D$ in terms of the Levi-Civita connection $\mathrm{D}$ of $g$ as

$$\begin{aligned} \nabla ^\mathrm{T}_XY= \left\{ \begin{array}{lll} \mathrm{D}_XY &{}\quad \hbox {if}&{}\quad X\in \Gamma ({\mathcal {D}})\\ {[}\xi ,Y]^{{\mathcal {D}}} &{}\quad \hbox {if}&{}\quad X=\xi , \end{array}\right. \end{aligned}$$

(1)

where the upperscript $\mathcal D$ denotes the orthogonal projection onto $\mathcal D$. This connection induces the transverse curvature

$$\begin{aligned} R^\mathrm{T}(X,Y)Z=\nabla _X^\mathrm{T}\nabla ^\mathrm{T}_YZ-\nabla _Y^\mathrm{T}\nabla ^\mathrm{T}_XZ-\nabla ^\mathrm{T}_{[X,Y]}Z, \end{aligned}$$

(2)

and the transverse Ricci curvature $\mathrm{Ric}^\mathrm{T}$ obtained as the trace of the map $X\mapsto R^\mathrm{T}(X,\cdot )\cdot$ on $\mathcal D$ with respect to $g^\mathrm{T}$. We further recall that a real $p$-form $\alpha$ on $(M,\xi ,\Phi ,\eta ,g)$ is called basic if

$$\begin{aligned} \iota _{\xi }\alpha =0,\quad \iota _{\xi }\hbox {d}\alpha =0, \end{aligned}$$

where $\iota _{\xi }$ denotes the contraction along $\xi$. The set of basic $p$-forms is usually denoted by $\Omega _{B}^p(M)$ and $\Omega _B^0(M)=C^{\infty }_B(M)$. Since the exterior differential operator takes basic forms into basic forms, its restriction $\mathrm{d}_B$ to $\Omega _B(M)=\oplus \Omega _B^p(M)$ defines a cohomological complex. Moreover, $\Phi$ induces a transverse complex structure $J$ on $(M,\xi )$ and a splitting of the space of complex basic forms in forms of type $(p,q)$ in the usual way. Furthermore, the complex extension of $\mathrm{d}_B$ to $\Omega _B(M,{\mathbb {C}})$ splits as $\mathrm{d}_B=\partial _B+\bar{\partial }_B$ and $\bar{\partial }_B^2=0$ (see e.g.  [2] for details). A basic $(1,1)$-form $\chi$ on $(M,\xi ,\Phi ,\eta ,g)$ is said to be positive if

$$\begin{aligned} \chi (Z,\bar{Z})>0, \end{aligned}$$

(3)

for every nonzero section $Z$ of $\Gamma (\mathcal D^{1,0})$. If further $\chi$ is closed, we refer to $\chi$ as to a transverse Kähler form. Note that condition (3) depends only on the transverse complex structure $J$ and on $\xi$, since $\chi$ is basic. Every such a $\chi$ induces the global metric

$$\begin{aligned} g_{\chi }(\cdot ,\cdot )=\chi (\cdot ,\Phi \cdot )+\eta (\cdot )\,\eta (\cdot ), \end{aligned}$$

on $M$. The metric $g_{\chi }$ induces a transverse Levi-Civita connection $\nabla ^{\chi }$ and a transverse curvature $R^{\chi }$ as in (1) and (2) (here it is important that $\chi$ is basic in order to define $\nabla ^{\chi }$).

2.1 Adapted coordinates

Let $(M,\xi ,\Phi ,\eta ,g)$ be a Sasakian manifold. We can always find local coordinates $\{z^1,\dots ,z^{n},z\}$ taking values in ${\mathbb {C}}^n\times {\mathbb {R}}$ such that

$$\begin{aligned} \xi =\partial _z,\quad \Phi (\hbox {d}{z^j})=i\,\hbox {d}{z^j},\quad \Phi (\hbox {d}{\bar{z}^j})=-i\,\hbox {d}{\bar{z}^j}. \end{aligned}$$

(4)

A function $h$ is basic if and only if it does not depend on the variable $z$, and we usually denote by $h_{,i_1\dots i_r\bar{j}_1\dots \bar{j}_l}$ the space derivatives of $h$ along $\partial _{z^{i_1}},\dots ,\partial _{z^{i_r}}, \partial _{\bar{z}^{ j_1}},\dots ,\partial _{\bar{z}^{ j_l}}$. We denote by $A_{i_1\dots i_r\bar{j}_1\dots \bar{j}_l}$ (without “,”) the components of the basic tensor $A$. Furthermore, when a function $f$ depends also on a time variable $t$, we use notation $\dot{f}$ to denote its time derivative. In the case when $f$ depends on two time variables $(t,s)$, we write $\partial _t f$ and $\partial _s f$, to distinguish the two derivatives.

For instance, the metric $g$ and the transverse symplectic form $\hbox {d}\eta$ locally write as

$$\begin{aligned} g=g_{i\bar{j}}\hbox {d}z^i\hbox {d}\bar{z}^{j}+\eta ^2,\quad \hbox {d}\eta =2ig_{i\bar{j}}\hbox {d}z^i\wedge \hbox {d}\bar{z}^{j}, \end{aligned}$$

where the $g_{i\bar{j}}$ are all basic functions. In particular, the transverse metric $g^\mathrm{T}$ writes as $g^\mathrm{T}=g_{i\bar{j}}\mathrm{d}z^i\mathrm{d}\bar{z}^{ j}$ and a Sasakian structure can be regarded as a collection of Kähler structures each one defined on an open set of ${\mathbb {C}}^n$. Observe that conditions (4) depend only on $(\xi ,J)$, and therefore, they hold for every Sasakian structure compatible with $(\xi ,J)$. This fact is crucial in the Proof of Theorem 1.1.

In this paper, we make sometimes use of special foliated coordinates with respect to a transverse Kähler form $\chi$. Indeed, once a transverse Kähler form $\chi$ on the Sasakian manifold $(M,\xi ,\Phi ,\eta ,g)$ is fixed, we can always find foliated coordinates $\{z^1,\dots ,z^{n},z\}$ around any fixed point $x$ such that if $\chi =\chi _{i\bar{j}}\,\hbox {d}z^i\wedge \hbox {d}\bar{z}^j$, then

$$\begin{aligned} \chi _{i\bar{j}}=\delta _{ij},\quad \partial _{z^r}\chi _{i\bar{j}}=0, \text{ at }\quad x. \end{aligned}$$

Moreover, we can further require that the transverse metric $g^\mathrm{T}$ takes a diagonal expression at $x$.

2.2 The space of the Sasakian potentials and the definition of $J$-flow

Following [4, 21–23], given a Sasakian manifold $(M,\xi ,\Phi ,\eta ,g)$, we consider

$$\begin{aligned} {\mathcal {H}}=\{ h \in C_B^{\infty }(M,{\mathbb {R}})\,\,:\,\, \eta _h=\eta + \mathrm{d}^c h\ \mathrm{is \ a \ contact \ form}\}, \end{aligned}$$

where $\mathrm{d}^ch$ is the $1$-form on $M$ defined by $(\mathrm{d}^ch)(X)=-\frac{1}{2} \hbox {d}h(\Phi (X))$. Every $h\in {\mathcal {H}}$ induces the Sasakian structure $(\xi ,\Phi _h,\eta _h,g_h)$ where

$$\begin{aligned} \begin{aligned} \Phi _h=\Phi -(\xi \otimes (\eta _h-\eta ))\circ \Phi ,\quad g_h=\frac{1}{2}\,\hbox {d}\eta _h\circ (\mathrm{Id}\otimes \Phi _h)+\eta _h\otimes \eta _h. \end{aligned} \end{aligned}$$

Notice that

$$\begin{aligned} \eta _h\wedge (\hbox {d}\eta _h)^n=\eta \wedge (\hbox {d}\eta _h)^n. \end{aligned}$$

All the Sasakian structures induced by the functions in ${\mathcal {H}}$ have the same Reeb vector field and the same transverse complex structure. It is rather natural to restrict our attention to the space $\mathcal {H}_0$ of normalised Sasakian potentials. ${\mathcal {H}}_0$ is defined as the zero set of the functional $I:{\mathcal {H}}\rightarrow {\mathbb {R}}$ defined through its first variation by

$$\begin{aligned} \frac{\partial }{\partial t}I(f)=\frac{1}{2^nn!}\int _M \dot{f}\,\eta \wedge \hbox {d}\eta _{f}^n,\quad I(0)=0, \end{aligned}$$

where $f$ is a smooth curve in ${\mathcal {H}}$ (see [21, formula (14)] for an explicit formulation of $I$). The pair $(\xi ,J)$ can be seen as a polarisation of the Sasakian manifold (see [4]). Notice that $\mathcal {H}$ is open in $C^{\infty }_B(M,{\mathbb {R}})$ and has the natural Riemannian metric

$$\begin{aligned} (\varphi ,\psi )_h:=\frac{1}{2^nn!}\int _M\, \varphi \psi \,\eta \wedge (\hbox {d}\eta _h)^n. \end{aligned}$$

(5)

The covariant derivative of (5) along a smooth curve $f=f(t)$ in $C^{\infty }_B(M,{\mathbb {R}})$ takes the following expression

$$\begin{aligned} D_t\psi =\dot{\psi }-\frac{1}{4}\,\langle \mathrm{d}_B\psi ,\mathrm{d}_B\dot{f}\rangle _f, \end{aligned}$$

where $\psi$ is an arbitrary smooth curve in $C^{\infty }_B(M,{\mathbb {R}})$ and $\langle \cdot ,\cdot \rangle _f$ is the pointwise scalar product induced by $g_f$ on basic forms (see [21, 23]). Note that $D_t$ can be alternatively written as

$$\begin{aligned} D_t\psi =\dot{\psi }-\frac{1}{2}\mathrm{Re} \langle \partial _B\psi ,\partial _B\dot{f}\rangle _f \end{aligned}$$

which has the following local expression

$$\begin{aligned} D_t\psi =\dot{\psi }-\frac{1}{4}g_f^{\bar{j}k}(\psi _{,k}\dot{f}_{,\bar{j}}+\psi _{,\bar{j}}\dot{f}_{,k}). \end{aligned}$$

Moreover, a curve $f=f(t)$ in ${\mathcal {H}}$ is a geodesic if and only if it solves

$$\begin{aligned} \ddot{f}-\frac{1}{4} |\mathrm{d}_B \dot{f}|^2_f=0. \end{aligned}$$

(6)

Furthermore, W. He proved in [23] that ${\mathcal {H}}$ is an infinite dimensional symmetric space whose curvature can be written as

$$\begin{aligned} R_h(\psi _1,\psi _2)\psi _3=-\frac{1}{16}\{\{\psi _1,\psi _2\}_f,\psi _3\}_h, \end{aligned}$$

where $\{,\,\}_h$ is the Poisson bracket on $C^{\infty }_B(M,{\mathbb {R}})$ induced by the contact form $\eta _h$.

As in the Kähler case, it is still an open problem to establish when two points in ${\mathcal {H}}$ can be connected by a geodesic path. Fortunately, Guan and Zhang proved in [22] that this can be always done in a weak sense. More precisely, the role of ${\mathcal {H}}$ is replaced with its completion $\bar{\mathcal {H}}$ with respect to the $C_{w}^2$-norm (see [22] for details) and the geodesic Eq. (6) with

$$\begin{aligned} \left( \ddot{f}-\frac{1}{4} |\mathrm{d}_B \dot{f}|^2_f\right) \,\eta \wedge \hbox {d}\eta _f^n=\epsilon \,\eta \wedge \hbox {d}\eta ^n. \end{aligned}$$

(7)

Then, by definition, a $C^{1,1}$-geodesic is a curve in $\bar{{\mathcal {H}}}$ obtained as weak limit of solutions to (7), and from [22], it follows that for every two points in ${\mathcal {H}}$, there exists a $C^{1,1}$-geodesic connecting them.

Now, we can introduce the Sasakian version of the $J$-flow. The definition depends on the choice of a transverse Kähler form $\chi$. Note that

$$\begin{aligned} \eta _h\wedge \chi ^n=\eta \wedge \chi ^n\ne 0, \end{aligned}$$

for every $h\in \ \mathcal {H}$, since $\chi$ and $\mathrm{d}^c_Bh$ are both basic forms.

Proposition 2.1

Let $f_0,f_1\in \mathcal {H}$ and $f:[0,1]\rightarrow \mathcal {H}$ be a smooth path satisfying $f(0)=f_0, \, f(1)=f_1$. Then

$$\begin{aligned} A_{\chi }(f):=\int _0^1\int _M \dot{f}\,\chi \wedge \eta \wedge (\mathrm {d}\eta _f)^{n-1}\,\mathrm {d}t, \end{aligned}$$

depends only on $f_0$ and $f_1$.

Proof

Following the approach of Mabuchi in [28], let $\psi (s,t):=sf(t)$ and let $\Psi$ be the $2$-form on the square $Q=[0,1]\times [0,1]$ defined as

$$\begin{aligned} \Psi (s,t):=\left( \int _M \partial _t\psi \, \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-1}\right) \,\hbox {d}t+ \left( \int _M \partial _s\psi \, \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-1}\right) \,\hbox {d}s. \end{aligned}$$

We show that $\Psi$ is closed as $2$-form on $Q$:

$$\begin{aligned} \hbox {d}\Psi (s,t)= & {} \,\frac{\hbox {d}}{\hbox {d}s}\left( \int _M \partial _t\psi \, \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-1}\right) \,\hbox {d}t\wedge \hbox {d}s\\&-\frac{\hbox {d}}{\hbox {d}t}\left( \int _M \partial _s\psi \, \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-1}\right) \,\hbox {d}t\wedge \hbox {d}s\\= & {} \,(n-1)\,s\,i\left( \int _M \dot{f}\, \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-2}\wedge \partial _B \bar{\partial }_B f\right. \\&\qquad \qquad \quad \qquad \left. +\int _M f\, \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-2}\wedge \bar{\partial }_B \partial _B \dot{f}\right) \hbox {d}t\wedge \hbox {d}s\\= & {} \,(n-1)\,s\,i\left[ \int _M \hbox {d}\left( \dot{f}\, \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-2}\wedge \bar{\partial }_B f\right) \right. \\&\qquad \qquad \quad \qquad \left. -\int _M\partial _B\dot{f}\, \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-2}\wedge \bar{\partial }_B f+\right. \\&\qquad \qquad \quad \qquad \left. +\int _M \hbox {d}\left( f\, \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-2}\wedge \partial _B \dot{f}\right) \right. \\&\qquad \qquad \quad \qquad \left. -\int _M \bar{\partial }_B f\wedge \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-2}\wedge \partial _B \dot{f}\right] \hbox {d}t\wedge \hbox {d}s\\= & {} \,(n-1)\,s\,i\left[ \int _M\partial _B\dot{f}\wedge \bar{\partial }_B f \wedge \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-2}\right. \\&\qquad \qquad \quad \qquad \left. -\int _M \partial _B \dot{f}\wedge \bar{\partial }_B f\wedge \eta \wedge \chi \wedge (\hbox {d}\eta _\psi )^{n-2} \right] \hbox {d}t\wedge \hbox {d}s\\= & {} \,0. \end{aligned}$$

Therefore the Gauss-Green Theorem implies that

$$\begin{aligned} \int _{\partial Q}\Psi =0, \end{aligned}$$

and the claim follows. $\square$

In view of the last proposition, we can write $A_{\chi }(f_0,f_1)$ instead of $A_{\chi }(f)$.

Definition 2.2

The Sasaki $J$ -functional is the map $J_{\chi }:\mathcal {H}\rightarrow {\mathbb {R}}$ defined as

$$\begin{aligned} J_{\chi }(h)=\frac{1}{2^{n-1}(n-1)!} A_{\chi }(0,h). \end{aligned}$$

Alternatively, we can define $J_{\chi }$ through its first variation by

$$\begin{aligned} \partial _t J_{\chi }(f)=\int _M \frac{1}{2^{n-1}(n-1)!}\dot{f}\,\chi \wedge \eta \wedge (\hbox {d}\eta _f)^{n-1},\quad J_{\chi }(0)=0, \end{aligned}$$

(8)

and then apply Proposition 2.1 to show that the definition is well posed. Note that

$$\begin{aligned} \partial _t J_{\chi }(f)=\frac{1}{2} (\dot{f}\chi ,\hbox {d}\eta )_f, \end{aligned}$$

and therefore

$$\begin{aligned} \partial _t J_{\chi }(f)=\frac{1}{2^nn!} \int _M \dot{f}\sigma _f\,\eta \wedge \hbox {d}\eta _{f}^n, \end{aligned}$$

where for $h\in \mathcal {H}$

$$\begin{aligned} \sigma _h=g^{\bar{b}a}_h\chi _{a\bar{b}} , \end{aligned}$$

the components and the derivatives are computed with respect to transverse holomorphic coordinates, and with the upper indices in $g_h$, we denote the components of the inverse matrix.

If we restrict $J_{\chi }$ to $\mathcal {H}_0,$ then $h\in \mathcal H_0$ is a critical point of $J_{\chi }:\mathcal {H}_0\rightarrow {\mathbb {R}}$ if and only if

$$\begin{aligned} \int _M k\,\eta \wedge \chi \wedge \hbox {d}\eta _h^{n-1}=0, \end{aligned}$$

for every $k$ in the tangent space to ${\mathcal {H}}_0$ at $h$, i.e., if and only if $2n \,\eta \wedge \chi \wedge \hbox {d}\eta _h^{n-1}=c\, \eta \wedge \hbox {d}\eta _h^{n}$, where

$$\begin{aligned} c=\frac{2n\int _M \chi \wedge \eta \wedge \hbox {d}\eta ^{n-1}}{\int _M \eta \wedge \hbox {d}\eta ^n}. \end{aligned}$$

(9)

Given $h\in \mathcal {H}_0$, we can rewrite the condition of being a critical point of $J_{\chi }$ as

$$\begin{aligned} \sigma _h=c. \end{aligned}$$

(10)

Therefore, if $f_0\in {\mathcal {H}}_0$ is fixed, the evolution equation

$$\begin{aligned} \dot{f}=c-\sigma _f,\quad f(0)=f_0, \end{aligned}$$

(11)

can be seen as the gradient flow of $J_{\chi }:\mathcal {H}_0\rightarrow {\mathbb {R}}$.

Definition 2.3

A Sasakian structure $(\xi ,\Phi _h,\eta _h,g_h)$ is called critical if $h$ satisfies (10). We will refer to (11) as to the Sasaki $J$ -flow.

3 Technical results and critical Sasaki metrics

Let $(M,\xi ,\Phi ,\eta ,g)$ be a $(2n+1)$-dimensional compact Sasakian manifold and let $f=f(t)$ be a smooth curve in the space of normalised Sasakian potentials ${\mathcal {H}}_0$. Then

$$\begin{aligned} \frac{\partial }{\partial t} \eta \wedge (\hbox {d}\eta _f)^n= \Delta _f\dot{f}\, \eta \wedge (\hbox {d}\eta _f)^n, \end{aligned}$$

(12)

where for $h\in {\mathcal {H}}_0, \, \Delta _h$ denotes the basic Laplacian

$$\begin{aligned} \Delta _h\psi =-\partial _B^*\partial _B\psi =\,g_h^{\bar{j}r}\psi _{,r\bar{j}},\quad \text{ for }\quad \psi \in C^{\infty }_B(M, {\mathbb {R}}). \end{aligned}$$

A direct computation yields

$$\begin{aligned} \dot{\sigma }_f=-g_f^{\bar{p} m}\, \dot{f}_{,m\bar{l}}\,g_f^{\bar{l} q}\,\chi _{q\bar{p}}=-\langle i\partial _B\bar{\partial }_B \dot{f},\chi \rangle _f, \end{aligned}$$

(13)

where, given $\alpha$ and $\beta$ in $\Omega _B^{(p,q)}(M,{\mathbb {C}})$, we set

$$\begin{aligned} \langle \alpha ,\beta \rangle _h=\alpha _{i_1\dots i_p\bar{j}_1\dots \bar{j}_q}\cdot \bar{\beta }_{r_1\dots r_p\bar{s}_1\dots \bar{s}_q} g^{\bar{r}_1i_1}_{h}\cdots g^{\bar{r}_pi_p}_{h}\cdot g_h^{\bar{j}_1 s_1}\cdots g_{h}^{\bar{j}_q s_q}, \end{aligned}$$

and

$$\begin{aligned} (\alpha ,\beta )_h=\frac{1}{2^nn!}\int _M \langle \alpha ,\beta \rangle _h\,\eta \wedge \hbox {d}\eta _{h}^n. \end{aligned}$$

In particular, if $\alpha =\alpha _i\,\hbox {d}z^i$ and $\beta =\beta _j\,\hbox {d}z^j$ are transverse forms of type $(1,0)$, by writing $\chi =i\chi _{a\bar{b}}\hbox {d}z^a\wedge \bar{z}^b$, we have

$$\begin{aligned} \langle \chi ,\alpha \wedge \bar{\beta }\rangle _h=i\chi _{a\bar{b}}\bar{\alpha }_{r}\beta _j g^{\bar{r} a}_hg^{\bar{b} j}_h. \end{aligned}$$

The following technical lemma will be useful in the sequel.

Lemma 3.1

Let $u\in C_B^\infty (M,\mathbb {R})$ and $f$ be a smooth path in $C^{\infty }_B(M,{\mathbb {R}})$. Then

• (i) $(\Delta _f\dot{f},u\sigma )_f=-(\partial _B \dot{f},\sigma \partial _B u)_f-(u\partial _B\dot{f},\partial _B \sigma )_f;$

• (ii) $(\bar{\partial }_B\partial _B \dot{f},u\chi )_f=-i(u\,\partial _B \dot{f},\partial _B\sigma )_f-(\chi , \partial _B u \wedge \bar{\partial }_B \dot{f})_f;$

• (iii) $(\dot{f},\dot{\sigma })_f=\frac{1}{2} (\partial _B (\dot{f})^2,\partial _B\sigma )_f-i(\chi , \partial _B \dot{f} \wedge \bar{\partial }_B \dot{f})_f.$

where $\sigma =g_f^{\bar{k}r}\chi _{r\bar{k}}$.

Proof

• (i) $(\Delta _f\dot{f},\dot{u}\sigma )_f=-(\partial _B^*\partial _B\dot{f},u \sigma )_f=-(\partial _B \dot{f},\sigma \partial _B u)_f-(u\partial _B\dot{f},\partial _B\sigma )_f.$

• (ii) Since the Laplacian is self-adjoint we have:

  $$\begin{aligned} 2^nn!i(\bar{\partial }_B\partial _B \dot{f},u\chi )_f= & {} -\int _M ug_f^{\bar{c} j}g_f^{\bar{b} a}\dot{f}_{,j\bar{b}}\,\chi _{a\bar{c}}\,\eta \wedge (\hbox {d}\eta _f)^n\\= & {} \int _M u_{,\bar{b}}g_f^{\bar{c} j}g_f^{\bar{b} a}\chi _{a\bar{c}}\,\dot{f}_{,j}\,\eta \wedge (\hbox {d}\eta _f)^n+\int _M ug_f^{\bar{c} j}g_f^{\bar{b} a}\chi _{a{\bar{b}},\bar{c}}\,\dot{f}_{,j}\,\eta \wedge (\mathrm{d}\eta _f)^n\\= & {} \int _M u_{,\bar{b}}g_f^{\bar{c} j}g_f^{\bar{b} a}\chi _{a\bar{c}}\,\dot{f}_{,j}\,\eta \wedge (\hbox {d}\eta _f)^n+\int _M ug_f^{\bar{c} j}\sigma _{,\bar{c}}\,\dot{f}_{,j}\,\eta \wedge (\hbox {d}\eta _f)^n\\= & {} 2^nn!(u\partial _B \dot{f},\partial _B\sigma )_f-2^nn!i(\chi , \partial _B u\wedge \bar{\partial }_B \dot{f})_f. \end{aligned}$$

• (iii) By using (13) and (ii), we have

  $$\begin{aligned} (\dot{f},\dot{\sigma })_f= & {} -i(\partial _B\bar{\partial }_B\dot{f},\dot{f}\chi )_f=(\dot{f}\,\partial _B \dot{f},\partial _B\sigma )_f-i(\chi , \partial _B \dot{f} \wedge \bar{\partial }_B \dot{f})_f\\= & {} \frac{1}{2} (\partial _B (\dot{f})^2,\partial _B\sigma )_f-i(\chi , \partial _B \dot{f} \wedge \bar{\partial }_B \dot{f})_f \end{aligned}$$

  as required.$\square$

The following proposition is about the uniqueness of critical Sasaki metrics in ${\mathcal {H}}_0$ and it is analogue to the Kähler case.

Proposition 3.2

$J_{\chi }:\mathcal {H}_0\rightarrow {\mathbb {R}}$ has at most one critical point.

Proof

Let $f$ be a curve in the space $\bar{{\mathcal {H}}}$ obtained as completion of ${\mathcal {H}}$ with respect to the $C^2_w$-norm. Then, taking into account the definition of $J_{\chi }$, Lemma 3.1 and Eqs. (12), (13), we have

$$\begin{aligned} \partial _t^2\,J_{\chi }(f)= & {} (\ddot{f},\sigma _f)_f+\frac{1}{2}(\Delta _f\dot{f},\dot{f}\sigma _f)_f+i(\dot{f}\bar{\partial }_B\partial _B\dot{f},\chi )_f\\= & {} \frac{1}{2^nn!}\int _M\left( \ddot{f}-\frac{1}{2}|\partial _B \dot{f}|_g^2 \right) \sigma _f\,\eta \wedge (\hbox {d}\eta _f)^{n}-i(\chi ,\partial _B \dot{f}\wedge \bar{\partial }_B\dot{f})_f. \end{aligned}$$

Therefore, if $f$ solves the modified geodesic equation (7), then

$$\begin{aligned} \partial _t^2\,J_{\chi }(f)\ge -i(\chi ,\partial _B \dot{f}\wedge \bar{\partial }_B\dot{f})_f\ge 0. \end{aligned}$$

Let us assume now to have two critical points $f_0$ and $f_1$ of $J_{\chi }$ in ${\mathcal {H}}_0$ and denote by $\bar{{\mathcal {H}}}_0$ the completion of ${\mathcal {H}}_0$ with respect to the $C^2_w$-norm. Then, in view of [22], there exists a $C^{1,1}$-gedesic $f$ in $\bar{{\mathcal {H}}}_0$ such that $f(0)=f_0$ and $f(1)=f_1$. Let $h(t)=J_{\chi }(f(t))$. Then, since $f_0$ and $f_1$ are critical points of $J_{\chi }$, we have $\dot{h}(0)=\dot{h}(1)=0$. Since $\ddot{h}\ge 0$, it as to be $\ddot{h}\equiv 0$ which implies $\partial _B \dot{f}=0$ and $\dot{f}(t)$ is constant for every $t\in [0,1]$. Finally, since $f$ is a curve in $\bar{{\mathcal {H}}}_0$, then $I(f)=0$ and therefore $\dot{f}=0$, which implies $f_0=f_1$, as required. $\square$

On a compact $3$-dimensional Sasaki manifold, the existence of a critical metric is always guaranteed. Indeed, if $(M,\xi ,\Phi ,\eta ,g)$ is a compact $3$-dimensional Sasaki manifold with a fixed background transverse Kähler form $\chi$, then we can write:

$$\begin{aligned} \chi =\frac{1}{4}\langle \chi ,\mathrm{d}\eta \rangle \,\hbox {d}\eta ,\quad \hbox {d}\eta _h= \left( 1-\frac{1}{2} \Delta _Bh\right) \, \hbox {d}\eta , \end{aligned}$$

where the scalar product and the basic Laplacian are computed with respect to the metric induced by $\eta$. Hence, $\eta _h=\eta +\mathrm{d}^ch$ induces a critical metric if and only if $h$ solves:

$$\begin{aligned} \Delta _Bh=2-\frac{1}{c}\langle \chi ,\hbox {d}\eta \rangle ,\quad c=\frac{2\int _M\eta \wedge \chi }{\int _M\eta \wedge \hbox {d}\eta }, \end{aligned}$$

which has always a solution since:

$$\begin{aligned} \int _M \left( 2-\frac{1}{c}\langle \chi ,\hbox {d}\eta \rangle \right) \eta \wedge \hbox {d}\eta =0. \end{aligned}$$

In higher dimensions, there is a cohomological obstruction to the existence of a critical metric similar to the one in the Kähler case.

Recall that if $(M,\omega )$ is a Kähler $2n$-dimensional manifold (with $n>1$) with a fixed background Kähler metric $\chi$, then the existence of a $J_{\chi }$-critical normalised Kähler potential on $(M,\omega )$ implies that $[c\omega -\chi ]$ is a Kähler class in $H^2(M,{\mathbb {R}})$ (see [14]). In [6], Chen proved that such a condition is sufficient for the existence of a critical metric on complex surfaces, while in the recent paper [25], Lejmi and Székelhyidi provide an example where the condition is satisfied, but the $J$-flow does not converge. In [38], Song and Weinkove find a necessary and sufficient condition for the convergence of the flow in terms of a ($n-1, n-1$)-form. Some further results about the convergence have been obtained in [42, 43]. The Sasakian context is quite similar. Indeed, given a Sasakian manifold $(M,\xi ,\Phi ,\eta ,g)$ with a fixed background transverse Kähler form $\chi$, then if $h\in {\mathcal {H}}_0$ is a critical normalised Sasakian potential, then $\frac{c}{2}\hbox {d}\eta _h-\chi$ is a transverse Kähler form. Hence, it is rather natural to conjecture that the existence of a Sasakian potential $h$ satisfying $\frac{c}{2}\hbox {d}\eta _h-\chi >0$, implies the existence of a critical Sasaki metric, and we expect that the results in [38, 42, 43] could be generalised to the Sasakian case.

The following proposition is about the existence of a critical Sasaki metric in dimension $5$:

Proposition 3.3

Let $(M,\xi ,\Phi ,\eta ,g)$ be a compact $5$-dimensional Sasaki manifold. Assume that there exists a map $h\in \mathcal {H}_0$ such that $\frac{c}{2}\,(\mathrm{d}\eta + \mathrm{dd}^ch)-\chi$ is a transverse Kähler form. Then, there exists a critical Sasaki metric on $M$.

Proof

Up to rescaling $\eta$, we may assume $c=1$. A function $h\in \mathcal {H}_0$ is critical if and only if

$$\begin{aligned} 2\,\eta \wedge \chi \wedge \left( \frac{1}{2} \hbox {d}\eta + \hbox {d} \hbox {d}^c h\right) = \eta \wedge \left( \frac{1}{2} \hbox {d}\eta +\hbox {d}\hbox {d}^ch\right) ^2. \end{aligned}$$

Let $\Omega =\frac{1}{2} \hbox {d}\eta -\chi$. Then, our hypothesis implies that $\Omega$ is a transverse Kähler form and moreover by substituting we get

$$\begin{aligned} (\Omega +\hbox {d}\hbox {d}^c h)^2=\chi ^2. \end{aligned}$$

Finally, the Calabi-Yau theorem in Kähler foliations [15] implies the statement. $\square$

4 Well posedness of the Sasaki $J$-flow

Theorem 4.1

The Sasaki $J$-flow is well posed, i.e., for every initial datum $f_0$, system (11) has a unique maximal solution $f$ defined in $[0,\epsilon _{\max })$, for some positive $\epsilon _{\max }$.

Proof

Since $\mathcal {H}$ is not open in $C^{\infty }(M,{\mathbb {R}})$, to apply the standard parabolic theory, we have to use a trick adopted by Smoczyk, Wang and Zhang for showing the short-time existence of the Sasaki–Ricci flow in [36]. Since the functional $F:\mathcal {H}\rightarrow {\mathbb {R}}$ defined as

$$\begin{aligned} F(f)=\xi ^2(f)+\sigma _f, \end{aligned}$$

is elliptic, the standard parabolic theory implies that the geometric flow

$$\begin{aligned} \dot{f}=c-\xi ^2(f)-\sigma _f,\quad f(0)=f_0, \end{aligned}$$

(14)

has a unique maximal solution $f\in C^{\infty }(M\times [0,\epsilon _\mathrm{max}),{\mathbb {R}})$, for some $\epsilon _\mathrm{max}>0$. Of course if $f(\cdot ,t)$ is a solution to (14) which is basic for every $t$ and $I(f)=0$, then $f$ solves (11). We first show that if $f_0$ is basic, then the solution $f$ to (14) holds basic for every $t\in [0,\epsilon _\mathrm{max})$. We have

$$\begin{aligned} \partial _t \xi (f)=\xi (\dot{f})=\xi (-\xi ^2(f)-g^{\bar{k} r}\chi _{r\bar{k}}). \end{aligned}$$

Moreover, since the components of $\chi$ are basic, we have

$$\begin{aligned} \xi (g^{\bar{k} r}\chi _{r\bar{k}})=-g^{\bar{k} l}_f(\xi (f_{,l\bar{m}}))g^{\bar{m} r}_f\chi _{r\bar{k}}= g^{\bar{k} l}_f\xi (f)_{,l\bar{m}}\,g^{\bar{m} r}_f\chi _{r\bar{k}}=-\langle \hbox {d}\hbox {d}^c_B\xi (f),\chi \rangle _f, \end{aligned}$$

i.e.

$$\begin{aligned} \partial _t \xi (f)=-\xi ^3(f)+\langle \hbox {d}\hbox {d}^c_B\xi (f),\chi \rangle _f. \end{aligned}$$

(15)

Equation (15) is parabolic in $\xi (f)$, and then, since the solution to a parabolic problem is unique, if $\xi (f_0)=0, \, \xi (f(t))=0$ for every $t\in [0,\epsilon _\mathrm{max})$, as required. Finally, we show that if $f_0$ is normalised, then $I(f)=0$ for every $t\in [0,\epsilon _\mathrm{max})$. We have

$$\begin{aligned} \partial _tI(f)=\frac{1}{2^nn!}\int _M \dot{f}\eta \wedge \hbox {d}\eta _{f}^n=\frac{1}{2^nn!}\int _M (c-\sigma _f)\,\eta \wedge \hbox {d}\eta _{f}^n, \end{aligned}$$

and since $c\int _M\,\eta \wedge \hbox {d}\eta _{f}^n=\int _M\sigma _f\,\eta \wedge \hbox {d}\eta _{f}^n$ we have $\partial _tI(f)=0$. Therefore, since $I(f_0)=0, \, I(f)=0$ for every $t\in [0,\epsilon _{\max })$ and the claim follows. $\square$

Remark 4.2

Alternatively, the short-time existence of the Sasaki $J$-flow can be obtained by invoking the short-time existence of any second-order transversally parabolic equation on compact manifolds foliated by Riemannian foliations. A proof of the latter result can be found in [1].

In analogy to the Kähler case, let $\mathrm{En} :\mathcal H_0\rightarrow {\mathbb {R}}$ be the energy functional

$$\begin{aligned} \mathrm{En}(h)=\frac{1}{2^nn!}\int _M\sigma _h^2\,\eta \wedge (\hbox {d}\eta _h)^{n}=(\sigma _h,\sigma _h)_{h}^2. \end{aligned}$$

Proposition 4.3

The following items hold:

• 1. $\mathrm{En}$ has the same critical points of $J_\chi$ and it is strictly decreasing along the Sasaki $J$-flow;

• 2. any critical point of $\mathrm{En}$ is a local minimiser;

• 3. the length of any curve in ${\mathcal {H}}_0$ and the distance of any two points in ${\mathcal {H}}_0$ decrease under the $J$-flow.

Proof

• 1. Let $f\!:[0,1]\rightarrow \mathcal {H}_0$ be a smooth curve. Then, by using (13) and Lemma 3.1, the first variation of En reads:

  $$\begin{aligned} \partial _t \mathrm{En}(f)= & {} \frac{1}{2^nn!}\partial _t\int _M\sigma _f^2\,\eta \wedge (\hbox {d}\eta _f)^{n}\\= & {} 2(\sigma _f,\dot{\sigma }_f)_f+(\sigma _f^2,\Delta _f\dot{f})_f\\= & {} 2(\sigma _f\partial _B \dot{f},\partial _B\sigma _f)-2i(\chi ,\partial _B\dot{f}\wedge \bar{\partial }_B \sigma _f)_f-2(\partial _B\sigma _f,\sigma _f\partial _B\dot{f})_f\\= & {} -2i(\chi ,\partial _B\dot{f}\wedge \bar{\partial }_B \sigma _f)_f. \end{aligned}$$

  Along the Sasaki $J$-flow one has $\dot{f}=c-\sigma _f$, thus:

  $$\begin{aligned} \partial _t \mathrm{En}(f)=-2i(\chi ,\partial _B\sigma _f\wedge \bar{\partial }_B \sigma _f)_f\le 0, \end{aligned}$$

  and $\mathrm{En}$ is strictly decreasing along the $J$-flow. Moreover, if $h\in {\mathcal {H}}_0$ is a critical point of $\mathrm{En}$, then $\partial _B \sigma _h=0$ which implies that $h$ is critical if and only if $\sigma _h=c$.

• 2. Now, we compute the second variation of $\mathrm{En}$. Let $f:(-\delta ,\delta ) \times (-\delta ,\delta ) \rightarrow {\mathcal {H}}_0$ be a smooth map in the variables $(t,s)$. Assume that $f(0,0)=h$ is a critical point of $\mathrm{En}$ and let $u=\partial _{t}f_{|(0,0)}, \, v=\partial _s f_{|(0,0)}$. Then, we have

  $$\begin{aligned} \partial _s\partial _t \mathrm{En}(\alpha )=\frac{1}{2^{n-1}n!}i\partial _s\left( \int _M\langle \chi ,\partial _B\partial _t\alpha \wedge \bar{\partial }_B \sigma _\alpha \rangle _\alpha \,\eta \wedge (\hbox {d}\eta _\alpha )^n\right) , \end{aligned}$$

  and

  $$\begin{aligned} \partial _s\partial _t \mathrm{En}(\alpha )_{|(0,0)}= & {} \frac{1}{2^{n-1}n!}\int _M\langle \chi ,\partial _Bu\wedge \bar{\partial }_B \partial _s\sigma _{\alpha |(0,0)}\rangle _h\,\eta \wedge (\hbox {d}\eta _h)^n\\= & {} 2(\chi ,\partial _Bu\wedge \bar{\partial }_B \partial _s\sigma _{\alpha |(0,0)})_h, \end{aligned}$$

  since $\sigma _h$ is constant. Now

  $$\begin{aligned} 2(\chi ,\partial _Bu\wedge \bar{\partial }_B \partial _s\sigma _{\alpha |(0,0)})_h=2 (\chi ,\partial _s\sigma _{\alpha |(0,0)}\, \partial _B\bar{\partial }_Bu)_h, \end{aligned}$$

  and formula (13) implies

  $$\begin{aligned} \partial _s\partial _t \mathrm{En}(f)_{|(0,0)}=\frac{1}{2^{n-1}n!}\int _M\langle i\partial _B\bar{\partial }_B u,\chi \rangle _h\langle i\partial _B\bar{\partial }_B v,\chi \rangle _h\,\eta \wedge (\hbox {d}\eta _h)^n, \end{aligned}$$

  which implies that $\partial _s\partial _t \mathrm{En}(f)_{|(0,0)}$ is positive definite as symmetric form.

• 3. Given smooth curve $u:[0,1]\rightarrow {\mathcal {H}}_0$ in ${\mathcal {H}}_0$ and $h\in \mathcal H_0$ we denote by

  $$\begin{aligned} \mathcal {L}(h,u) =\frac{1}{2^{n}n!}\int _0^1\int _M \dot{u}^2 \ \eta \wedge (\hbox {d}\eta _h)^n\wedge \hbox {d}s=(\dot{u},\dot{u})_{h}, \end{aligned}$$

  the square of the length of $u$ with respect to the Sasaki metric induced by $h$. Let $f:[0,\epsilon )\times [0,1]\rightarrow \mathcal H_0$ and assume that $t\mapsto f(t,s)$ is a solution to the $J$-flow for every $s\in [0,1]$. Then, by using Lemma 3.1, we have

  $$\begin{aligned} 2^{n}n!\partial _t\mathcal {L}(f,f)= & {} \partial _t\left[ \int _0^1\int _M (\partial _s f)^2\, \eta \wedge (\hbox {d}\eta _f)^n\wedge \hbox {d}s\right] \\= & {} \int _0^1\int _M 2\partial _s\partial _tf\partial _s f+ (\partial _sf)^2\Delta _f(\partial _tf)\, \eta \wedge (\hbox {d}\eta _f)^n\wedge \hbox {d}s\\= & {} \int _0^1\int _M -2\partial _s\sigma _f\partial _s f- (\partial _sf)^2\Delta _f(\sigma _f)\, \eta \wedge (\hbox {d}\eta _f)^n\wedge \hbox {d}s\\= & {} -\int _0^1\left[ 2(\partial _s\sigma _f,\partial _sf)_f+((\partial _sf)^2,\Delta _f \sigma _f)_f \right] \hbox {d}s\\= & {} -\int _0^1\left[ 2(\partial _s\sigma _f,\partial _sf)_f+(\partial _B(\partial _sf)^2,\partial _B \sigma _f)_f \right] \hbox {d}s\\= & {} 2i\int _0^1(\chi ,\partial _B \partial _sf\wedge \bar{\partial }_B \partial _sf)_f\,\hbox {d}s\le 0 \end{aligned}$$

  and the equality holds if and only if $\partial _sf(t,s)$ is constant in $s$.$\square$

5 A maximum principle for basic maps and tensors

In this section, we introduce a basic principle for transversally elliptic operators on Sasakian manifolds. The principle will be applied in the next section to compute the $C^2$-estimate about the solutions to (11).

Let $(M,\xi ,\Phi ,\eta )$ be a Sasakian manifold. By a smooth family of basic linear partial differential operators $\{E\}_{t\in [0,\epsilon )}$, we mean a smooth family of operators $E(\cdot ,t):C_B^{\infty }(M,{\mathbb {R}})\rightarrow C^{\infty }_B(M,{\mathbb {R}})$ which can be locally written as

$$\begin{aligned} E(h(y),t)=\sum _{1\le |k|\le m} a_k(y,t) \frac{\partial ^{|k|}}{\partial y^{k_1}\cdots \partial y^{k_r}}h(y) \end{aligned}$$

for every $h\in C_B^{\infty }(M,{\mathbb {R}})$, where $\{y^1,\dots ,y^{2n},z\}$ are real coordinates on $M$ such that $\xi =\partial _z$. The maps $a_k$ are assumed to be smooth and basic in the space coordinates (see [15] for a detailed descriptions of these operators on compact manifolds foliated by Riemannian foliations). Observe that $E$ can be regarded as a functional $E:C_B^{\infty }(M\times [0,\epsilon ),{\mathbb {R}})\rightarrow C_B^{\infty }(M\times [0,\epsilon ),{\mathbb {R}})$ in a natural way. We further make the strong assumption on $E$ to satisfy

$$\begin{aligned} E(h(x,t),t)\le 0, \end{aligned}$$

(16)

whenever the complex Hessian $\hbox {d}d^ch$ of $h$ is nonpositive at the point $(x,t)\in M\times [0,\epsilon ).$

Proposition 5.1

(Maximum principle for basic maps). Assume that $h\in C_B^{\infty }(M\times [0,\epsilon ),\mathbb {R})$ satisfies

$$\begin{aligned} \partial _th(x,t)-E(h(x,t),t)\le 0. \end{aligned}$$

Then

$$\begin{aligned} \sup _{(x,t)\in M\times [0,\epsilon )} h(x,t)\le \sup _{x\in M} h(x,0). \end{aligned}$$

Proof

Fix $\epsilon _0\in (0,\epsilon )$ and let $h_{\uplambda }:M\times [0,\epsilon _0]\rightarrow {\mathbb {R}}$ be the map $h_{\uplambda }(x,t)=h(x,t)-{\uplambda }t$. Assume that $h_{\uplambda }$ achieves its global maximum at $(x_0,t_0)$ and assume by contradiction that $t_0>0$. Then $\partial _th_{\uplambda }(x_0,t_0)\ge 0$ and $\hbox {d}\hbox {d}^ch_{\uplambda }(x_0,t_0)$ is nonpositive. Therefore, condition (16) implies $E(h_{\uplambda }(x_0,t_0),t_0)\le 0$ and consequently

$$\begin{aligned} \partial _th_{\uplambda }(x_0,t_0)-E(h_{\uplambda }(x_0,t_0),t_0)\ge 0. \end{aligned}$$

Since $\partial _th_{\uplambda }=\partial _th-{\uplambda }$ and $E(h_{\uplambda }(x,t),t)=E(h(x,t),t)$, we have

$$\begin{aligned} 0\le \partial _th(x_0,t_0)-E(h(x_0,t_0),t_0)-{\uplambda }\le -{\uplambda }, \end{aligned}$$

which is a contradiction. Therefore, $h_{\uplambda }$ achieves its global maximum at a point $(x_0,0)$ and

$$\begin{aligned} \sup _{M\times [0,\epsilon _0]} h\le \sup _{M\times [0,\epsilon _0]} h_{\uplambda }+{\uplambda }\epsilon _0\le \sup _{x\in M} h(x,0)+{\uplambda }\epsilon _0. \end{aligned}$$

Since the above inequality holds for every $\epsilon _0\in (0,\epsilon )$ and ${\uplambda }>0$, the claim follows. $\square$

A similar result can be stated for tensors:

Proposition 5.2

(Maximum principle for basic tensors). Let $\kappa$ be a smooth curve of basic $(1,1)$-forms on $M$ for $t\in [0,\epsilon )$. Assume $\kappa$ nonpositive and such that

$$\begin{aligned} \partial _t\kappa _{i\bar{j}}(x,t)-E(\kappa _{i\bar{j}}(x,t),t)=N_{i\bar{j}}(x,t), \end{aligned}$$

where $N$ is a nonpositive basic form and the components are with respect to foliated coordinates. Then, $\kappa$ is nonpositive for every $t\in [0,\epsilon )$.$\square$

Proof

The proof is very similar to the case of functions. We show that for every positive ${\uplambda }, \, \kappa _{{\uplambda }}=\kappa -t{\uplambda }\hbox {d}\eta$ is nonpositive. Assume by contradiction that this is not true. Then, there exists a ${\uplambda }$, a first point $(x_0,t_0)\in M\times [0,\epsilon )$ and $g$-unitary $(1,0)$-vector $Z\in \mathcal {D}_{x_0}^{1,0}$ such that $\kappa _{{\uplambda }}(Z,\bar{Z})=0$. We extend $Z$ to a basic and unitary vector field in a small enough neighbourhood $U$ of $x$ and consider the map $f_{{\uplambda }}:U\times [0,t_0]\rightarrow {\mathbb {R}}$ given by $f_{{\uplambda }}=\kappa _{{\uplambda }}(Z,\bar{Z})$. Then, $f_{\uplambda }$ has a maximum at $(x_0,t_0)$ and so

$$\begin{aligned} \partial _tf_{{\uplambda }}\ge 0,\quad E(f_{{\uplambda }}(x_0,t_0),t_0)\le 0, \end{aligned}$$

at $(x_0,t_0)$. Now since

$$\begin{aligned} E(f_{{\uplambda }}(x,t),t)=E(f_0(x,t),t), \end{aligned}$$

we have

$$\begin{aligned} 0\le \partial _t(f_{\uplambda })=E(f_{\uplambda },\cdot )+N(Z,\bar{Z})-\frac{{\uplambda }}{2}\le 0, \end{aligned}$$

at $(x_0,t_0)$, which implies

$$\begin{aligned} N(Z,\bar{Z})\ge \frac{{\uplambda }}{2} , \end{aligned}$$

at $(x_0,t_0)$, which is a contradiction.

In the following, we will apply the two propositions when $E$ is the operator $\tilde{\Delta }_f$ depending on a smooth curve $f$ in $\mathcal {H}$ defined by:

$$\begin{aligned} \tilde{\Delta }_f(h,t)=g_f^{\bar{k} p}g_f^{\bar{q} j}\chi _{j\bar{k}} h_{,a\bar{b}}. \end{aligned}$$

6 Second-order estimates

The following two lemmas provide the a priori estimates we need to prove the main theorem.

Lemma 6.1

Let $f:M\times [0,\epsilon )\rightarrow {\mathbb {R}}$ be a solution to (11), with $\epsilon <\infty$. Then

$$\begin{aligned} \sigma _f\le \min _{x\in M}\sigma _f(x,0) \end{aligned}$$

and there exists a uniform constants $C$, depending only on $f_0$, such that

$$\begin{aligned} \gamma _f(x,t)\le \sup _{x\in M} \gamma _f(x,0)\,\mathrm{e}^{C\epsilon } \end{aligned}$$

where $\gamma _f=\chi ^{\bar{j} k}(g_f)_{k\bar{j}}$.

Proof

The upper bound of $\sigma _f$ easily follows from the definition of $J_{\chi }$ and Proposition 5.1. Indeed, differentiating (11) in $t$ we have $\ddot{f}=-\partial _t\sigma _f=g_f^{\bar{a} b}g_f^{\bar{k} j} \chi _{b\bar{k}}\dot{f}_{,j\bar{a}}=-\tilde{\Delta }_f\sigma _f,$ i.e.,

$$\begin{aligned} \partial _t\sigma _f=\tilde{\Delta }_f\sigma _f \end{aligned}$$

and Proposition 5.1 implies the first inequality. About the upper bound of $\gamma _f$, we have

$$\begin{aligned} \partial _t\gamma _f=\chi ^{\bar{j} k}\partial _t\left[ (g_f)_{k\bar{j}}\right] =\chi ^{\bar{j} k}\dot{f}_{,k\bar{j}}. \end{aligned}$$

Since $f$ solves (11), we have

$$\begin{aligned} \dot{f}_{,a}=g_f^{\bar{k} p}(g_f)_{p\bar{q},a}g_f^{\bar{q} j}\chi _{j\bar{k}}-g_f^{\bar{k} j}\chi _{j\bar{k},a} \end{aligned}$$

and

$$\begin{aligned} \dot{f}_{,a\bar{b}}= & {} -2g_f^{\bar{k} s}(g_f)_{\bar{r}s,\bar{b}}g_f^{\bar{r} p}(g_f)_{p\bar{q},a}g_f^{\bar{q} j}\chi _{j\bar{k}}+\tilde{\Delta }_f[(g_f)_{a\bar{b}}]\nonumber \\&+g_f^{\bar{k} p}(g_f)_{p\bar{q},a}g_f^{\bar{q} j}\chi _{j\bar{k},\bar{b}}+g_f^{\bar{k} s}(g_f)_{\bar{r}s,\bar{b}}g_f^{\bar{r} j}\chi _{j\bar{k},a}-g_f^{\bar{k} j}\chi _{j\bar{k},a\bar{b}}. \end{aligned}$$

(17)

Let $R^\mathrm{T}=R^\mathrm{T}(\chi )$ be the transverse curvature of $\chi$ and $\mathrm{Ric}^\mathrm{T}(\chi )$ its transverse Ricci tensor (see Sect. 2). The components of $R^\mathrm{T}$ with respect to foliated coordinates read as $R_{j\bar{k} a\bar{b}}^\mathrm{T}=-\chi _{j\bar{k}, a\bar{b}}+\chi ^{\bar{p} q}\chi _{j\bar{p}, a}\chi _{q\bar{k}, \bar{b}}$.

Fix a point $(x_0,t_0)\in M\times [0,\epsilon )$ and special foliated coordinates for $\chi$ around it (see Subsection 2.1). We may further assume without loss of generality that $(g_f)_{j\bar{k}}={\uplambda }_j\delta _{j k}$ at $(x_0,t_0)$. Then

$$\begin{aligned} \dot{f}_{,a\bar{b}}=\,\sum _{k,r=1}^n \frac{-2}{{\uplambda }_k^2{\uplambda }_r}(g_f)_{\bar{r}k,\bar{b}}(g_f)_{r\bar{k},a} +\tilde{\Delta }_f[(g_f)_{a\bar{b}}] -\sum _{k=1}^n \frac{1}{{\uplambda }_k}\chi _{k\bar{k},a\bar{b}}\quad \text{ at } (x_0,t_0) \end{aligned}$$

(18)

and

$$\begin{aligned} \partial _t\gamma _f\!=\!\sum _{a=1}^n\dot{f}_{,a\bar{a}}\!=\! \sum _{a=1}^n\left[ \sum _{k,r=1}^n \frac{-2}{{\uplambda }_k^2{\uplambda }_r}|(g_f)_{k\bar{r},\bar{a}} |^2 \!+\!\tilde{\Delta }_f[(g_f)_{a\bar{a}}] -\sum _{k=1}^n \frac{1}{{\uplambda }_k}\chi _{k\bar{k},a\bar{a}}\right] \quad \text{ at } (x_0,t_0), \end{aligned}$$

i.e.

$$\begin{aligned} \partial _t\gamma _f= \sum _{a=1}^n\left( \sum _{k,r=1}^n \frac{-2}{{\uplambda }_k^2{\uplambda }_r}|(g_f)_{k\bar{r},\bar{a}} |^2 +\tilde{\Delta }_f[(g_f)_{a\bar{a}}]\right) -\sum _{k=1}^n\frac{1}{{\uplambda }_k}\mathrm{Ric}^\mathrm{T}_{k\bar{k}} \quad \text{ at } (x_0,t_0). \end{aligned}$$

Now a direct computation yields

$$\begin{aligned} \tilde{\Delta }_f\gamma _f=\sum _{a=1}^n \tilde{\Delta }_f[(g_f)_{a\bar{a}}]-\sum _{a,k=1}^n\frac{{\uplambda }_a}{{\uplambda }_k^2} R^\mathrm{T}_{a\bar{a}k\bar{k}} \quad \text{ at } (x_0,t_0) \end{aligned}$$

and therefore

$$\begin{aligned} \partial _t\gamma _f-\tilde{\Delta }_f\gamma _f=\sum _{a,k=1}^n\left( \sum _{r=1}^n \frac{-2}{{\uplambda }_k^2{\uplambda }_r}|(g_f)_{k\bar{r},\bar{a}} |^2 +\frac{{\uplambda }_a}{{\uplambda }_k^2} R^\mathrm{T}_{a\bar{a}k\bar{k}}\right) -\sum _{k=1}^n \frac{1}{{\uplambda }_k}\mathrm{Ric}^\mathrm{T}_{k\bar{k}} \quad \text{ at } (x_0,t_0). \end{aligned}$$

Observe that

$$\begin{aligned} \sum _{k=1}^n\frac{1}{{\uplambda }_k}=\sigma _f(x_0,t_0)\le C_1,\qquad \sum _{k=1}^n{\uplambda }_k=\gamma _f(x_0,t_0), \end{aligned}$$

where $C_1=\min _{x\in M}\sigma _{f}(x,0)$. Thus for all $k=1,\dots , n$ we have

$$\begin{aligned} \frac{1}{{\uplambda }_k}\le C_1, \qquad {\uplambda }_k\le \gamma _f(x_0,t_0). \end{aligned}$$

Since $M$ is compact, there exists a constant $C_2$ such that $\mathrm{Ric}^\mathrm{T}-C_2\chi$ is nonnegative and therefore at $(x_0,t_0)$ we have

$$\begin{aligned} \left| \frac{1}{{\uplambda }_k}\mathrm{Ric}^\mathrm{T}_{k\bar{k}}\right| \le nC_1C_2, \quad \left| \sum _{a,k=1}^n\frac{{\uplambda }_a}{{\uplambda }_k^2} R^\mathrm{T}_{a\bar{a}k\bar{k}}\right| \le C_1^2 \left| \sum _{a=1}^n{\uplambda }_a \mathrm{Ric}^\mathrm{T}_{a\bar{a}}\right| \le nC_1^2C_2\gamma _f. \end{aligned}$$

Thus there exists a constant $C$ such that

$$\begin{aligned} \partial _t\gamma _f-\tilde{\Delta }_f\gamma _f\le C\gamma _f+C. \end{aligned}$$

Let $F:=\mathrm{e}^{-Ct}\gamma _f -Ct$. Then

$$\begin{aligned} \partial _t F-\tilde{\Delta }_f F=e^{-Ct}\left( -c\gamma _f+\partial _t\gamma _f-\tilde{\Delta }\gamma _f\right) -C, \end{aligned}$$

and by Proposition 5.1 we have

$$\begin{aligned} \sup _{(x,t)\in M\times [0,\epsilon )}F\le \sup _{x\in M}F(x,0)=\sup _{x\in M} \gamma _f(x,0), \end{aligned}$$

which implies

$$\begin{aligned} \sup _{(x,t)\in M\times [0,\epsilon )}\gamma _t=\sup _{x\in M} \gamma _f(x,0)e^{C\epsilon } \end{aligned}$$

as required. $\square$

In order to get a uniform lower bound for $\hbox {d}\eta _f$, we need to add an hypothesis on the bisectional curvature of $\chi$ (see Theorem 6.2 below). Observe that the existence of a uniform lower bound without further assumption would imply the existence of a critical metric in $\mathcal {H}_0$ for each choice of $\eta$ and $\chi$, in contrast with the necessary condition $\frac{c}{2}\hbox {d}\eta _f-\chi >0$.

Theorem 6.2

Assume that the transverse bisectional curvature of $\chi$ is nonnegative and let $f:M\times [0,\epsilon )\rightarrow {\mathbb {R}}$ be a solution to (11). Then, there exists a constant $C$ depending only on the initial datum $f_0$ such that $C\chi -\mathrm {d}\eta _f$ is a transverse Kähler form for every $t\in [0,\epsilon )$.

Proof

Let $\kappa =\frac{1}{2} \hbox {d}\eta _f-C\chi$ where $C$ is a constant chosen big enough to have $\kappa$ nonpositive at $t=0$. Then, $\kappa$ is a time-dependent basic $(1,1)$-form which is nonpositive at $t=0$. We apply Proposition 5.2 to show that $\kappa$ is nonpositive for every $t\in [0,\epsilon )$. Once a system of foliated coordinates $\{z^k,z\}$ is fixed, we have $\partial _t \kappa _{a\bar{b}}=\dot{f}_{,a\bar{b}}$ and formula (17) implies

$$\begin{aligned} \partial _t \kappa _{a\bar{b}}= & {} -2g_f^{\bar{k} s}(g_f)_{\bar{r}s,\bar{b}}g_f^{\bar{r} p}(g_f)_{p\bar{q},a}g_f^{\bar{q} j}\chi _{j\bar{k}}+g_f^{\bar{k} p}(g_f)_{p\bar{q},a\bar{b}}g_f^{\bar{q} j}\chi _{j\bar{k}}\\&+g_f^{\bar{k} p}(g_f)_{p\bar{q},a}g_f^{\bar{q} j}\chi _{j\bar{k},\bar{b}}+g_f^{\bar{k} s}(g_f)_{\bar{r}s,\bar{b}}g_f^{\bar{r} j}\chi _{j\bar{k},a}-g_f^{\bar{k} j}\chi _{j\bar{k},a\bar{b}}\\= & {} -2g_f^{\bar{k} s}(g_f)_{\bar{r}s,\bar{b}}g_f^{\bar{r} p}(g_f)_{p\bar{q},a}g_f^{\bar{q} j}\chi _{j\bar{k}}+\tilde{\Delta }\left[ (g_f)_{a\bar{b}}\right] \\&+g_f^{\bar{k} p}(g_f)_{p\bar{q},a}g_f^{\bar{q} j}\chi _{j\bar{k},\bar{b}}+g_f^{\bar{k} s}(g_f)_{\bar{r}s,\bar{b}}g_f^{\bar{r} j}\chi _{j\bar{k},a}-g_f^{\bar{k} j}\chi _{j\bar{k},a\bar{b}}, \end{aligned}$$

i.e.

$$\begin{aligned} \partial _t \kappa _{a\bar{b}}-\tilde{\Delta }\left[ (g_f)_{a\bar{b}}\right]= & {} -2g_f^{\bar{k} s}(g_f)_{\bar{r}s,\bar{b}}g_f^{\bar{r} p}(g_f)_{p\bar{q},a}g_f^{\bar{q} j}\chi _{j\bar{k}}\nonumber \\&+g_f^{\bar{k} p}(g_f)_{p\bar{q},a}g_f^{\bar{q} j}\chi _{j\bar{k},\bar{b}}+g_f^{\bar{k} s}(g_f)_{\bar{r}s,\bar{b}}g_f^{\bar{r} j}\chi _{j\bar{k},a}-g_f^{\bar{k} j}\chi _{j\bar{k},a\bar{b}}.\qquad \end{aligned}$$

(19)

We apply Proposition 5.2 using as $N$ the basic form defined by the right-hand part of formula (19). To this end, we have to show that $N$ is nonpositive. That can be easily done as follows: fix a point $(x,t)\in M\times [0,\epsilon )$ and an arbitrary unitary vector field $Z\in \mathcal D_{x}^{1,0}$. Then, we can find foliated coordinates $\{z^k,z\}$ around $x$ which are special for $\chi$ and such that: $Z=\partial _{z^1|x}$ and $g_f$ takes a diagonal expression with eigenvalues ${\uplambda }_k$ at $(x,t)$. Then we have

$$\begin{aligned} N(Z,\bar{Z})=-2\sum _{k,r=1}^n\frac{1}{{\uplambda }_k^2{\uplambda }_r}|(g_f)_{k\bar{r},\bar{1}}|^2-\sum _{k=1}^n\frac{1}{{\uplambda }_k^2}R^\mathrm{T}(\chi )_{k\bar{k}1\bar{1}} \end{aligned}$$

at $(x,t)$ and the claim follows. $\square$

7 Proof of the main theorem

The Proof of Theorem 1.1 is based on the second-order estimates provided in Sect. 6 and on the following result in Kähler geometry.

Theorem 7.1

Let $B$ be an open ball about $0$ in ${\mathbb {C}}^n$ and let $\omega , \chi$ be two Kähler forms on $B$. Let $f:M\times [0,\epsilon )\rightarrow {\mathbb {R}}$ be solution to the Kähler $J$-flow

$$\begin{aligned} \dot{f}=c-g_f^{\bar{k} r}\chi _{r\bar{k}}, \end{aligned}$$

where $g_f$ is the metric associated to $\omega _f=\omega +\hbox {d}d^cf$. Assume that $\omega _f$ is uniformly bounded in $B\times [0,\epsilon )$. Then $f$ is $C^{\infty }$-bounded in a small ball about $0$.

As explained in [7], the theorem can be proved by using the well-known Evans and Krylov’s interior estimate (see [19] for a proof of the estimates in the complex case).

Proof of Theorem 1.1

The proof of the long-time existence consists in showing that every solution $f$ to (11) has a $C^{\infty }$-bound. Let $f:M\times [0,\epsilon _{\max })\rightarrow \mathbb {R}$ be the solution to (11) with initial condition $f_0\in {\mathcal {H}}_0$ and assume by contradiction $\epsilon _{\max }< \infty$. Lemma 6.1 implies that the second derivatives of $f$ are uniformly bounded in $M$. Since $f$ can be regarded as a collection of solutions to the Kähler $J$-flow on small open balls in ${\mathbb {C}}^n$, Theorem 7.1 implies that $f$ is $C^\infty$-uniformly bounded in $M$. Therefore, $f$ converges in $C^{\infty }$-norm to a smooth function $\tilde{f}$ as $t$ tends to $\epsilon _{\max }^{-}$. Since $\partial _t f$ is basic for every $t\in [0,\epsilon _{\max }), \, \tilde{f}$ is basic and by the well posedness of the Sasaki $J$-flow, the solution $f$ can be extended after $\epsilon _{\max }$ contradicting its maximality.

The proof of the long-time existence in the case when $\chi$ has nonnegative transverse holomorphic bisectional curvature, is obtained exactly as in the Kähler case. Let $f:M\times [0,\infty )\rightarrow {\mathbb {R}}$ be a solution to the Sasaki $J$-flow. Since $\chi$ has nonnegative holomorphic bisectional curvature, Theorem 6.2 implies that $f$ has a uniform $C^{\infty }$-bound and Ascoli-Arzelà Theorem implies that given a sequence $t_j\in [0,\infty ), \, t_j\rightarrow \infty , \, f_{t_j}$ has a subsequence converging in $C^\infty$-norm to function $f_{\infty }$ as $t_j\rightarrow \infty$. Therefore, $f$ converges to a critical map $f_{\infty }\in \mathcal {H}_0$. $\square$