Geometric Pluripotential Theory on Sasaki Manifolds

Abstract

We extend profound results in pluripotential theory on Kähler manifolds (Darvas in arXiv:1902.01982, 2019) to Sasaki setting via its transverse Kähler structure. As in Kähler case, these results form a very important piece to solve the existence of Sasaki metrics with constant scalar curvature in terms of properness of $\mathcal {K}$-energy, considered by the first named author in He (arXiv:1802.03841, 2019). One main result is to generalize Darvas’ theory on the geometric structure of the space of Kähler potentials in Sasaki setting. Along the way we extend most of corresponding results in pluripotential theory to Sasaki setting via its transverse Kähler structure.

Appendix

1.1 Approximation Through Type-I Deformation and Regularity of Rooftop Envelop

Using Type-I deformation, we can obtain the following approximation of irregular Sasaki structure $(M, \xi , \eta , g)$, which would be important for us; see [55] and in particular [13, Theorem 7.1.10] for the approximation. Suppose $\xi$ is irregular, then the Reeb flow generates an isometry in $\text {Aut}(M, \xi , \eta , g)$. Let $T^k\subset \text {Aut}(M, \xi , \eta , g)$ ($k\ge 2$) be the torus generated by $\xi$ and denote $\mathfrak {t}$ to be its Lie algebra. We can then choose $\rho _i\rightarrow 0, \rho _i\in \mathfrak {t}$ such that $\xi _i=\xi +\rho _i$ is quasiregular. Define

$$\begin{aligned} \eta _i=\frac{\eta }{1+\eta (\rho _i)}, \Phi _i=\Phi -\frac{1}{1+\eta (\rho _i)}\Phi \rho _i\otimes \eta , \omega _i^T=\frac{1}{2}\mathrm{{d}}\eta _i, g_i=\eta _i\otimes \eta _i+\omega _i^T(\mathbb {I}\otimes \Phi _i), \end{aligned}$$

(6.1)

where $\Phi$ is the (1, 1) tensor field defined on the contact bundle $\mathcal {D}=\text {Ker}(\eta )$. We recall the following:

Theorem 6.1

(Approximation of irregular Sasaki structure) Let $(M, \xi , \eta , g)$ be an irregular Sasaki structure on a compact manifold M. Then we can choose $\rho _i\rightarrow 0$ such that $\xi _i$ is quasiregular and (6.1) define a quasiregular Sasaki structure which is invariant under the action of $T^k$, the torus generated by $\xi$ in $\text {Aut}(M, \xi , \eta , g)$.

Lemma 6.1

Let $(M, \xi , \eta , g)$ be a Sasaki structure on a compact manifold M. Consider a torus $T\subset \text {Aut}(M, \xi , \eta , g)$ and $\xi _i\in \mathfrak {t}$. Choose $\xi _i=\xi +\rho _i$ for $\rho _i$ sufficiently small. Consider two Sasaki structures $(\xi , \eta , \Phi , g)\leftrightarrow (\xi _i, \eta _i, \Phi _i, g_j)$ via Type-I deformation. Then we have the following. Suppose u is T invariant and $u\in \text {PSH}(M, \xi , \omega ^T)$ with $|d \Phi \mathrm{{d}}u|\le C_0$. Then for $\rho _i$ sufficiently small, there exists positive constant $\epsilon _i\rightarrow 0$ (as $\rho _i\rightarrow 0$) such that,

$$\begin{aligned} (1-\epsilon _i) u\in \text {PSH}(M, \xi _i, \omega _i^T). \end{aligned}$$

(6.2)

Similarly, suppose $|d \Phi \mathrm{{d}}u|\le C_0$ and $u\in \text {PSH}(M, \xi _i, \omega _i^T)$, then there exists positive constant $\epsilon _i \rightarrow 0$ as $i\rightarrow \infty$, such that

$$\begin{aligned} (1-\epsilon _i) u\in \text {PSH}(M, \xi , \omega ^T). \end{aligned}$$

(6.3)

Proof

Since u is $T^k$-invariant, hence u is a basic function with respect to both $\xi$ and $\xi _i$. We write

$$\begin{aligned} \omega _i^T+\sqrt{-1}\partial ^i_B\bar{\partial }^i_B u =\omega ^T_i+\frac{1}{2} \mathrm{{d}}\Phi _i d u. \end{aligned}$$

Using (6.1), we compute

$$\begin{aligned} \omega ^T_i+\frac{1}{2} \mathrm{{d}}\Phi _i d u=&\frac{\omega ^T}{1+\eta (\rho _i)}+\eta \wedge d\left( \frac{1-\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}\right) +\frac{1}{2}\mathrm{{d}}\Phi \mathrm{{d}}u+2\omega ^T \frac{\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}\nonumber \\ =&\frac{1+2\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}\omega ^T+\frac{1}{2}\mathrm{{d}}\Phi \mathrm{{d}}u+\eta \wedge d\left( \frac{1-\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}\right) \nonumber \\ =&\,\omega ^T+\frac{1}{2}\mathrm{{d}}\Phi \mathrm{{d}}u+\left( \frac{1+2\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}-1\right) \omega ^T+\eta \wedge d\left( \frac{1-\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}\right) . \end{aligned}$$

(6.4)

If $|\mathrm{{d}}\Phi d u|\le C_0$, then (6.4) implies that $|\mathrm{{d}}\Phi _i d u|\le C_1$ (vice versa). Moreover, when $\rho _i\rightarrow 0$,

$$\begin{aligned} \frac{1+2\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}\rightarrow 1,\;\;\; d\left( \frac{1-\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}\right) \rightarrow 0. \end{aligned}$$

We can then choose $\epsilon _i\rightarrow 0$ as $\rho _i\rightarrow 0$, such that

$$\begin{aligned} \omega ^T_i+\frac{1}{2} \mathrm{{d}}\Phi _i \mathrm{{d}} (u(1-\epsilon _i))\ge 0. \end{aligned}$$

This proves (6.2). Note that given the relation of $\Phi$ and $\Phi _i$, then $|d \Phi \mathrm{{d}}u|\le C_0$ implies that $|\mathrm{{d}}\Phi _i d u|$ is uniformly bounded (we suppose $\rho _i$ is uniformly small in smooth topology). Interchanging $\xi$ and $\xi _i$, this proves (6.3). $\square$

Remark 6.1

Note that the complex structure on the cone remains unchanged under Type-I deformation [50, Lemma 2.2]. The transverse holomorphic structure is changed since the foliation is changed, due to the change of Reeb vector foliation; on the other hand, the contact bundle $\mathcal {D}$ remains unchanged. Note that $(\mathcal {D}, \Phi )$ and $(\mathcal {D}, \Phi _i)$ can be identified to transverse holomorphic tangent bundle $T^{1, 0}(\mathcal {F}_\xi )$ and $T^{1, 0}(\mathcal {F}_{\xi _i})$ (the foliations are different). Since the term $\eta \wedge d\left( \frac{1-\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}\right)$ vanishes on $\mathcal {D}$ and $\left( \frac{1+2\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}-1\right) \omega ^T$ involves with only $\mathrm{{d}}u$, hence the above statement holds if we only assume that $|\mathrm{{d}}u|$ is uniformly bounded. Since we shall not need this, we skip the argument. However, it seems that assumption like $|\mathrm{{d}}u|\le C$ is necessary and we are not able to extend this to $\text {PSH}(M, \xi , \omega ^T)$.

As mentioned above, we fix a torus $T\subset \text {Aut}(N, \xi , \eta , g)$ and consider $\rho _i\in \mathfrak {t}$ sufficiently small. Let $\xi _i=\xi +\rho _i$ and let $(\xi _i, \eta _i, g_i, \Phi _i)$ be the Type-I deformation of $(\xi , \eta , g, \Phi )$.

Lemma 6.2

Let $\rho _i\rightarrow 0$. Suppose a sequence of T-invariant functions $u_i\in \text {PSH}(M, \xi _i, \omega _i^T)$ with $|\mathrm{{d}}\Phi d u_i|_{\omega ^T}\le C_0$ converges to $u\in \text {PSH}(M, \xi , \omega ^T)$. Then $|\mathrm{{d}}\Phi \mathrm{{d}}u|_{\omega ^T}\le C_0$ and we have the following weak convergence of the measure

$$\begin{aligned} \left( \omega _i^T+\frac{1}{2}\mathrm{{d}}\Phi _i d u_i\right) ^n\wedge \eta _i\rightarrow \left( \omega ^T+\frac{1}{2}\mathrm{{d}}\Phi d u\right) ^n\wedge \eta . \end{aligned}$$

Proof

By (6.4) and $|\mathrm{{d}}\Phi d u_i|_{\omega ^T}\le C_0$, $\omega _i^T+\frac{1}{2}\mathrm{{d}}\Phi _i d u_i$ and $\omega ^T+\frac{1}{2}\mathrm{{d}}\Phi d u_i$ differ by a term with small $L^\infty$ norm, hence we only need to prove that

$$\begin{aligned} \left( \omega ^T+\frac{1}{2}\mathrm{{d}}\Phi d u_i\right) ^n\wedge \eta _i\rightarrow \left( \omega ^T+\frac{1}{2}\mathrm{{d}}\Phi d u\right) ^n\wedge \eta . \end{aligned}$$

Note that $\eta _i=\eta /(1+\eta (\rho _i))$ converges smoothly to $\eta$, then the above follows from the weak convergence of $(\omega ^T+\frac{1}{2}\mathrm{{d}}\Phi d u_i)^n\wedge \eta$. $\square$

Next we give a proof of Theorem 3.1 in Sasaki setting, regarding the regularity of envelop construction.

Lemma 6.3

Assume $\beta >0$ and $u,v \in \text {PSH}(M,\xi ,\omega ^T) \cap L^{\infty }$. If

$$\begin{aligned} (\omega _v^T)^n \wedge \eta \ge e^{\beta (v-f)}(\omega ^T)^n \wedge \eta ,\; (\omega _u^T)^n \wedge \eta \le e^{\beta (u-f)}(\omega ^T)^n \wedge \eta \end{aligned}$$

then $v \le u$.

Proof

By the comparison principle (3.6)

$$\begin{aligned} \int _{\{u<v\}}(\omega ^T_v)^n \wedge \eta \le \int _{\{u<v\}}(\omega ^T_u)^n \wedge \eta . \end{aligned}$$

Then we have

$$\begin{aligned} \int _{\{u<v\}} e^{\beta (v-f)}(\omega ^T)^n \wedge \eta \le \int _{\{u<v\}} e^{\beta (u-f)}(\omega ^T)^n \wedge \eta . \end{aligned}$$

It follows that $\{u<v \}$ has zero Lebesgue measure and $v \le u$ almost everywhere with respect to Lebesgue measure. Moreover, we have $v \le u$ everywhere on M since they are $\omega ^T$-plurisubharmonic. $\square$

Using the same computation to the transverse complex Monge–Ampere equation (as in the complex Monge–Ampere equation [57, p. 99]), we can obtain the following Laplacian estimate.

Lemma 6.4

Suppose $u \in \mathcal {H}$ is a solution for the equation

$$\begin{aligned} (\omega _u^T)^n \wedge \eta =e^g (\omega ^T)^n \wedge \eta . \end{aligned}$$

Then

$$\begin{aligned} \triangle ^{\omega _u^T}\log \text {Tr}_{\omega ^T}\omega _u^T \ge \frac{\triangle ^{\omega ^T}g}{\text {Tr}_{\omega ^T}\omega _u^T}-2B\text {Tr}_{\omega _u^T}\omega ^T \end{aligned}$$

where $B>0$ is a constant which depends on $\omega$.

Theorem 6.2

Given $f\in C^\infty _B(M)$, then we have the following estimate

$$\begin{aligned} \Vert P(f)\Vert _{C^{1, \bar{1}}}\le C(M, \omega ^T, g, \Vert f\Vert _{C^{1, \bar{1}}}). \end{aligned}$$

Moreover, if $u_1, \ldots , u_k\in \mathcal {H}_\Delta$, where we use the notation

$$\begin{aligned} \mathcal {H}_\Delta =\{u\in \text {PSH}(M, \xi , \omega ^T): \Vert u\Vert _{C^{1, \bar{1}}}<\infty \} \end{aligned}$$

then $P(u_1, \ldots , u_k)\in \mathcal {H}_\Delta$.

Proof

The first result was proved by Berman–Demailly [8] in Kähler setting. Since all quantities are basic and only transverse Kähler structure is involved, the argument as in Kähler setting has a direct adaption; see [30, Theorem A.7] for details in Kähler setting.

For each $\beta >0$, consider the equation

$$\begin{aligned} \left( \omega ^T_{u_\beta }\right) ^n\wedge \eta =e^{\beta (u_\beta -f)} (\omega ^T)^n\wedge \eta . \end{aligned}$$

(6.5)

This reads locally

$$\begin{aligned} \frac{\det (g_{i\bar{j}}^T+{u_\beta }_{i\bar{j}})}{\det (g_{i\bar{j}}^T)}=e^{\beta (u_{\beta }-f)}. \end{aligned}$$

The transverse version of Aubin–Yau theorem implies that there exists a unique solution $u_\beta$ for any $\beta >0$ and a smooth function f. The unique solution $u_\beta$ satisfies the following:

$$\begin{aligned} \Vert u_\beta -P(f)\Vert _{C^0}\rightarrow 0, \beta \rightarrow \infty \end{aligned}$$

(6.6)

and there exists $\beta _0>0$ and a uniform constant C such that $\beta \ge \beta _0$,

$$\begin{aligned} -n<\Delta ^{\omega ^T} u_\beta \le C. \end{aligned}$$

(6.7)

To prove (6.6), we choose $x_0 \in M$ such that $u_{\beta }-f$ obtains its maximum at $x_0$. Combining with Eq. (6.5), we have

$$\begin{aligned} \sqrt{-1}\partial _B \overline{\partial }_B (f-u_{\beta }) \ge 0 \end{aligned}$$

and

$$\begin{aligned} u_{\beta }-f = \frac{1}{\beta }\log \frac{(\omega _{u_{\beta }}^T)^n \wedge \eta }{(\omega ^T)^n \wedge \eta } \le \frac{1}{\beta }\log \frac{(\omega _f^T)^n \wedge \eta }{(\omega ^T)^n \wedge \eta } \end{aligned}$$

at $x_0$. It follows that

$$\begin{aligned} u_{\beta }-\frac{C}{\beta } \le f \end{aligned}$$

on M where $C=\sup _{M}\log \frac{(\omega _f^T)^n \wedge \eta }{(\omega ^T)^n \wedge \eta }$. By the definition (3.24) we have

$$\begin{aligned} u_{\beta }-P(f) \le \frac{C}{\beta }. \end{aligned}$$

(6.8)

On the other hand, we choose $v \in \mathcal {H}$ and $L>0$ such that

$$\begin{aligned} \omega _v^T \ge L \omega ^T\;\text {and}\; v \le f. \end{aligned}$$

One can choose $\beta _1>2$ such that $\epsilon =\frac{2n\log \beta }{\beta }<1$ for all $\beta \ge \beta _1$. Take $\beta _2=\max \{\frac{1}{L},\beta _1\}$, then for any $\beta \ge \beta _2$, we have

$$\begin{aligned} 0< \delta , \epsilon <1\;\text {and}\; e^{-\beta \epsilon } \le \delta ^n L^n \end{aligned}$$

where $\delta =\frac{1}{\beta }$. It follows that

$$\begin{aligned} u_{\delta ,\epsilon }:=(1-\delta )P(f)+\delta v-\epsilon \le f-\epsilon \end{aligned}$$

and

$$\begin{aligned} (\omega _{u_{\delta ,\epsilon }}^T)^n \wedge \eta \ge \delta ^n (\omega ^T_v)^n \ge \delta ^nL^n(\omega ^T)^n \wedge \eta \ge e^{-\beta \epsilon } (\omega ^T)^n \wedge \eta \ge e^{\beta (u_{\delta ,\epsilon }-f)} (\omega ^T)^n \wedge \eta . \end{aligned}$$

By Eq. ((6.5)) and Lemma (6.3), we have

$$\begin{aligned} u_{\delta ,\epsilon }=\left( 1-\frac{1}{\beta }\right) P(f)+\frac{v}{\beta }-\frac{2n\log \beta }{\beta } \le u_{\beta } \end{aligned}$$

and

$$\begin{aligned} P(f) \le \frac{\beta }{\beta -1}u_{\beta }+\frac{2n \log \beta }{\beta -1} -\frac{1}{\beta -1} \inf _M v. \end{aligned}$$

Combined with Eq. ((6.8)) we can derive that

$$\begin{aligned} \frac{1}{\beta } \inf _M v-\frac{1}{\beta } \sup _M f-\frac{2n\log \beta }{\beta } \le u_{\beta }-P(f) \le \frac{C}{\beta } \end{aligned}$$

(6.9)

for $\beta \ge \beta _2$. Then (6.6) follows immediately.

It is standard to deduce the lower bound in (6.7) from the fact $\omega ^T+\sqrt{-1}\partial _B\overline{\partial }_B u_{\beta } \ge 0$. By Eq. ((6.5)) and Lemma 6.4, we have

$$\begin{aligned} 2B\text {Tr}_{\omega _{u_{\beta }}^T} \omega ^T +\triangle ^{\omega _{u_{\beta }}^T}\log \text {Tr}_{\omega ^T}\omega _{u_{\beta }}^T \ge \beta \frac{\triangle ^{\omega ^T}(u_{\beta }-f)}{\text {Tr}_{\omega ^T}\omega _{\beta }^T}. \end{aligned}$$

It follows that

$$\begin{aligned} 2nB +\triangle ^{\omega _{u_{\beta }}^T}(\log \text {Tr}_{\omega ^T}\omega _{u_{\beta }}^T -Bu_{\beta })\ge \beta \frac{\text {Tr}_{\omega ^T}\omega ^T_{u_{\beta }}-2n-\triangle ^{\omega ^T}f}{\text {Tr}_{\omega ^T}\omega _{\beta }^T} \end{aligned}$$

and

$$\begin{aligned} \beta \text {Tr}_{\omega ^T}\omega _{u_{\beta }}^Te^{-Bu_{\beta }}\le & {} \beta (2n+\triangle ^{\omega ^T}f)e^{-Bu_{\beta }}\\&+\,[2nB+\triangle ^{\omega _{u_{\beta }}^T}\log ( \text {Tr}_{\omega ^T}\omega _{u_{\beta }}^Te^{-Bu_{\beta }})]\text {Tr}_{\omega ^T}\omega _{u_{\beta }}^Te^{-Bu_{\beta }}. \end{aligned}$$

Assume that $\text {Tr}_{\omega ^T}\omega _{u_{\beta }}^Te^{-Bu_{\beta }}$ obtains its maximum s at $x_1 \in M$ and $C_1=\sup _M (2n+\triangle ^{\omega ^T}f)$, then we have

$$\begin{aligned} \beta s \le \beta C_1e^{-Bu_{\beta }(x_1)}+2nBs. \end{aligned}$$

By the inequality (6.9) and $P(f) \le f$, $u_{\beta }$ is uniformly bounded. Hence we obtain an upper bound for $\text {Tr}_{\omega ^T}\omega _{u_{\beta }}^Te^{-Bu_{\beta }}$ if $\beta \ge \beta _0=\max \{3nB,\beta _2\}$. We conclude that $\triangle ^{\omega ^T}u_{\beta } \le C$ for $\beta \ge \beta _0$.

The first statement follows from (6.6) and (6.7).

For the second statement, first note that we only need to show that if $u_0, u_1\in \mathcal {H}_\Delta$, then $P(u_0, u_1)\in \mathcal {H}_\Delta$. Let $u_t$ be the geodesic segment connecting $u_0, u_1$, then by Lemma 3.9, we know that $u_t\in \mathcal {H}_\Delta$ (see [8] and [47] for Kähler setting). Now we have already known $P(u_0, u_1)=\inf _{t\in [0, 1]} u_t$, then by [31, Proposition 4.4] (applied to each foliation chart), $\Delta u_t$ is uniformly bounded. This shows that $P(u_0, u_1)\in \mathcal {H}_\Delta$. $\square$

More generally, one can obtain results as in [31] that $P(f_1, \ldots , f_n)\in C^{1, \bar{1}}_B$ given $f_1, \ldots , f_n\in C^{1, \bar{1}}_B$. The point is that given two functions $f_1, f_2$, $h=\min \{f_1, f_2\}$ satisfy $\Delta h\le \max \{\Delta f_1, \Delta f_2\}$ in viscosity sense, writing $h=\frac{f_1+f_2}{2}-\frac{|f_1-f_2|}{2}$. The argument as in [30, Theorem A.7] applies using the maximum principle in viscosity sense. Since we do not need this, we shall skip the details.

1.2 Complex Monge–Ampere Operator and Intrinsic Capacity on Compact Sasaki Manifolds

We discuss briefly the Bedford–Taylor theory on Sasaki manifolds. For details for complex Monge–Ampere operator, see Bedford–Taylor [2]. We also extend intrinsic Monge–Ampere capacity to Sasaki setting, see [43] for Kähler setting.

Given a Sasaki structure, there is a splitting of tangent bundle $TM=L\xi \otimes \mathcal {D}$, where $\mathcal {D}=\text {Ker}(\eta )$, with $\Phi : \mathcal {D}\rightarrow \mathcal {D}$ inducing a splitting $\mathcal {D}\otimes \mathbb {C}=\mathcal {D}^{1, 0}\oplus \mathcal {D}^{0, 1}$. Hence the subbundle $\Lambda ^{2p}(\mathcal {D}^*)$ of $\Lambda ^{2p}M$ is well defined and $\Phi$ induces a splitting to give bidegree of forms in $\Lambda ^{2p}(\mathcal {D}^*)$. Note that we have the following,

$$\begin{aligned} \Lambda ^{2p}(\mathcal {D}^*)=\{\theta : \theta \in \Lambda ^{2p}M, \iota _\xi \theta =0\}. \end{aligned}$$

We do not assume that $\theta \in \Lambda ^{2p}(\mathcal {D}^*)$ is basic. That is, the coefficients of $\theta$ might not be invariant under the Reeb flow. A simple observation shows that if $\theta \in \Lambda ^{2p}(\mathcal {D}^*)$, then $\theta$ is basic if it is closed, $d\theta =0$ (since $\iota _\xi \theta =0$). Hence a closed 2p-form in $\Lambda ^{2p}(\mathcal {D}^*)$ is basic and can be regarded as a transverse closed 2p-form, defined as in [58]. In general, $d\Lambda ^{2p}(\mathcal {D}^*)$ is not in $\Lambda ^{2p+1}(\mathcal {D}^*)$.

Next, we give a very brief discussion of transverse positive closed currents of bidegree of (p, p) on M, $0\le p\le n$; see [58] for similar treatment. We simply treat them as closed differential forms of bidegree (p, p) in $\Lambda ^{2p}(\mathcal {D}^*)$ with measurable coefficients which are invariant under the Reeb flow. Its total variation is controlled by

$$\begin{aligned} \Vert T\Vert :=\int _M T\wedge (\omega ^T)^{n-p}\wedge \eta . \end{aligned}$$

Given $\phi \in \text {PSH}(M, \xi , \omega ^T)$, we write $\phi \in L^1(T)$ if $\phi$ is integrable with respect to the measure $T\wedge (\omega ^T)^{n-p}\wedge \eta$. In this case, the current $\phi T$ is well defined and we write

$$\begin{aligned}&\omega _\phi \wedge T:=\omega ^T\wedge T+\mathrm{{dd}}^c_B(\phi T)\\&\omega _\phi \wedge T\wedge (\omega ^T)^{n-p-1}\wedge \eta = T\wedge (\omega ^T)^{n-p}\wedge \eta +\mathrm{{dd}}^c_B(\phi T)\wedge (\omega ^T)^{n-p-1}\wedge \eta . \end{aligned}$$

The positivity is a local notion and we simply think T as a positive closed (p, p)-form on each foliation chart. Hence $\omega _\phi \wedge T$ is also a transverse closed positive $(p+1, p+1)$ form. Note that we think transverse positive closed currents of bidegree of (p, p)-type as a linear functional on $\Lambda ^{n-p, n-p}(\mathcal {D}^*)$, hence the test forms are of bidegree $(n-p, n-p)$. A main point is that test forms are not restricted to basic forms. In other words, given such a current T and $\gamma \in \Lambda ^{n-p, n-p}(\mathcal {D}^*)$, we have the following pairing:

$$\begin{aligned} \gamma \rightarrow \int _M \gamma \wedge T\wedge \eta . \end{aligned}$$

When $\phi \in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$, it follows that $\phi \in L^1(T)$ for any transverse positive closed current T of bidegree (p, p) and hence one can define inductively $\omega _\phi ^k\wedge (\omega ^T)^{n-k}$; in particular, this leads to the definition of transverse complex Monge–Ampere operator $\omega _\phi ^n$ of bidegree (n, n). Moreover, the cocycle condition on transverse holomorphic structure ensures that $\omega _\phi ^k\wedge (\omega ^T)^{n-k}$ is well defined on M. In particular, $\omega _\phi ^n\wedge \eta$ defines a positive Borel measure on M.

It is more convenient to consider this construction locally in foliations charts $W_\alpha =(-\delta , \delta )\times V_\alpha$. By taking test forms $\gamma \in \Lambda ^{n-p, n-p}(\mathcal {D}^*)$ with compact support, we can consider $T\wedge \eta$ on a foliation chart for a transverse positive closed (p, p) current T. In particular, this give a local description of the complex Monge–Ampere measures $\omega _\phi ^k\wedge (\omega ^T)^{n-k}\wedge \eta$. By taking test functions f supported in a foliation chart, the measure $\omega _\phi ^k\wedge (\omega ^T)^{n-k}\wedge \eta$ for each k is regarded as the product measure $\omega _\phi ^k\wedge (\omega ^T)^{n-k}\wedge \mathrm{{d}}x$ on $W_\alpha$, where $\xi =\partial _x$ is the Reeb direction. Note that $\omega _\phi ^k\wedge (\omega ^T)^{n-k}$ is defined on $V_\alpha$ as the usual way in Kähler setting, and the cocycle condition on transverse holomorphic structure ensures that $\omega _\phi ^k\wedge (\omega ^T)^{n-k}$ is well defined as a transverse positive closed current of bidegree (n, n). On each foliation chart, we have $\omega _\phi ^k\wedge (\omega ^T)^{n-k}\wedge \eta =\omega _\phi ^k\wedge (\omega ^T)^{n-k}\wedge \mathrm{{d}}x$ as a product measure. This coincides with the local description given by van Coevering [58, Section 2].

Moreover, when $u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$, $\mathrm{{d}}u\wedge \mathrm{{d}}^c_B v\wedge T$ can also be defined, where T is a transverse closed positive current of bidegree $(n-1, n-1)$. By the polarization formula we only need to define $\mathrm{{d}}u\wedge \mathrm{{d}}^c_B u\wedge T$. By adding a positive constant if necessary, we assume $u\ge 0$. Then we define

$$\begin{aligned} \mathrm{{d}}u\wedge \mathrm{{d}}^c_B u\wedge T:=\frac{1}{2} \mathrm{{dd}}^c_B (u^2)\wedge T- u\mathrm{{dd}}^c_B u\wedge T. \end{aligned}$$

(6.10)

In particular, $\mathrm{{d}}u\wedge \mathrm{{d}}^c_B u\wedge T$ is positive if T is a transverse closed positive current of bidegree $(n-1, n-1)$. We can then define $\mathrm{{d}}u\wedge \mathrm{{d}}^c_B u\wedge T\wedge \eta$ as a positive Borel measure. Using the polarization formula, we have the following Cauchy–Schwarz inequality, for $u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$,

$$\begin{aligned} \left| \int _M \mathrm{{d}}u\wedge \mathrm{{d}}^c_B v\wedge T\wedge \eta \right| ^2\le \left( \int _M \mathrm{{d}}u\wedge \mathrm{{d}}^c_B u \wedge T\wedge \eta \right) \left( \int _M dv\wedge \mathrm{{d}}^c_B v \wedge T\wedge \eta \right) . \end{aligned}$$

(6.11)

We also record the following Stokes’ theorem in Sasaki setting, and its proof follows the Bedford–Taylor theory as in Kähler setting via approximation (Lemma 3.1); see [58, Theorem 2.3.1, Proposition 2.3.2].

Lemma 6.5

Let $u, v, \phi \in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$, then for each $0\le k\le n-1$, we have

$$\begin{aligned} \int _M u \mathrm{{dd}}^c_B v\wedge \omega _\phi ^{k}\wedge (\omega ^T)^{n-k-1}\wedge \eta =&\int _M v \mathrm{{dd}}^c_B u \wedge \omega _\phi ^{k}\wedge (\omega ^T)^{n-k-1}\wedge \eta \nonumber \\ =&-\int _M d u\wedge \mathrm{{d}}^c_B v \wedge \omega _\phi ^{k}\wedge (\omega ^T)^{n-k-1}\wedge \eta . \end{aligned}$$

(6.12)

We record a basic inequality in Sasaki setting, usually referred to Chern–Levine–Nirenberg inequality.

Proposition 6.1

(Chern–Levine–Nirenberg inequalities) Let T be a positive closed current of bidegree (p, p) on M and $\phi \in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$. Then $\Vert \omega _\phi \wedge T\Vert =\Vert T\Vert$. Moreover, if $\psi \in \text {PSH}(M, \xi , \omega ^T)\cap L^1(T)$, then $\psi \in L^1(\omega _\phi \wedge T)$ and

$$\begin{aligned} \Vert \psi \Vert _{L^1(T\wedge \omega _\phi )}\le \Vert \psi \Vert _{L^1(T)}+(2\max \{\sup \psi , 0\}+\sup \phi -\inf \phi ) \Vert T\Vert . \end{aligned}$$

(6.13)

Proof

By Stokes’ theorem, we have $\int _M \mathrm{{dd}}^c_B (\phi T)\wedge (\omega ^T)^{n-p-1}\wedge \eta =0$, hence

$$\begin{aligned} \Vert \omega _\phi \wedge T\Vert =\int _M \omega ^T\wedge T\wedge (\omega ^{T})^{n-p-1}\wedge \eta =\Vert T\Vert . \end{aligned}$$

To prove (6.13), we first assume $\psi \le 0, \phi \ge 0$. By assumption, $\psi \in L^1(T)$, then

$$\begin{aligned} \Vert \psi \Vert _{L^1(T\wedge \omega _\phi )}:= & {} \int _M -\psi T\wedge \omega _\phi \wedge (\omega ^T)^{n-p-1}\wedge \eta =\Vert \psi \Vert _{L^1(T)}\\&+\int _M -\psi \mathrm{{dd}}^c_B(\phi T)\wedge (\omega ^T)^{n-p-1}\wedge \eta . \end{aligned}$$

By Stokes’ theorem, we compute

$$\begin{aligned} \int _M -\psi \mathrm{{dd}}^c_B(\phi T)\wedge (\omega ^T)^{n-p-1}\wedge \eta =&\int _M \mathrm{{dd}}^c_B(-\psi ) \wedge \phi T\wedge (\omega ^T)^{n-p-1}\wedge \eta \\ \le&\int _M \phi T\wedge (\omega ^T)^{n-p}\wedge \eta \\ \le&\sup _M \phi \int _M T\wedge (\omega ^T)^{n-p}\wedge \eta =(\sup _M\phi ) \Vert T\Vert . \end{aligned}$$

Now suppose $\sup \psi >0$. Replacing $\phi$ by $\phi -\inf \phi$, we compute

$$\begin{aligned} \Vert \psi \Vert _{L^1(T\wedge \omega _\phi )}\le \int _M (2\sup \psi -\psi )T\wedge \omega _\phi \wedge (\omega ^T)^{n-p-1}\wedge \eta . \end{aligned}$$

The same argument as above leads to (6.13) for the general case. $\square$

For a Borel subset E on a Sasaki manifold $(M,\xi ,\omega ^T)$, we define the capacity as

$$\begin{aligned} \text {cap}_{\omega ^T}(E):=\sup \left\{ \int _E \omega _{\varphi }^n \wedge \eta : \varphi \in \text {PSH}(M,\xi ,\omega ^T), 0 \le \varphi \le 1 \right\} . \end{aligned}$$

It is obvious that $\text {cap}_{\omega ^T}(\cup _{k=1}^{\infty }E_k)\le \sum \nolimits _{k=1}^{\infty }\text {cap}_{\omega ^T}(E_k)$ for a sequence of Borel sets $E_k$. We have the following:

Proposition 6.2

Let $\phi \in \text {PSH}(M, \xi , \omega ^T)$ with $0\le \phi \le 1$ and $\psi \in \text {PSH}(M, \xi , \omega ^T)$ such that $\psi \le 0$. Then

$$\begin{aligned} \int _M -\psi \omega _\phi ^n\wedge \eta \le \int _M (-\psi )(\omega ^T)^n\wedge \eta +n \int _M (\omega ^T)^n\wedge \eta . \end{aligned}$$

(6.14)

Proof

We only need to prove (6.14) for canonical cutoffs $\psi _k=\max \{\psi , -k\}$ ($-\psi _k$ increases to $-\psi$ and we can apply monotone convergence theorem). We have the following:

$$\begin{aligned} \int _M -\psi _k\omega _\phi ^n\wedge \eta =&\int _M -\psi _k\omega _\phi ^{n-1}\wedge (\omega ^T+\sqrt{-1}\partial _B\bar{\partial }_B \phi )\wedge \eta \\ =&\int _M -\psi _k\omega _\phi ^{n-1}\wedge \omega ^T\wedge \eta +\int _M -\psi _k \omega _\phi ^{n-1}\wedge \sqrt{-1}\partial _B\bar{\partial }_B \phi \wedge \eta \\ =&\int _M -\psi _k\omega _\phi ^{n-1}\wedge \omega ^T\wedge \eta +\int _M \phi \omega _\phi ^{n-1}\wedge (-\sqrt{-1}\partial _B\bar{\partial }_B \psi _k)\wedge \eta \\ \le&\int _M -\psi _k\omega _\phi ^{n-1}\wedge \omega ^T\wedge \eta +\int _M (\omega _\phi )^{n-1}\wedge \omega ^T\wedge \eta \\ \le&\int _M -\psi _k\omega _\phi ^{n-1}\wedge \omega ^T\wedge \eta +\int _M (\omega ^T)^{n}\wedge \eta . \end{aligned}$$

We can then proceed inductively to obtain (6.14). Note that the argument above is a special case of (6.13). $\square$

Proposition 6.3

Suppose that $u \in \text {PSH}(M,\xi ,\omega ^T)$ and $u\le 0$. Then for $t>0$ we have

$$\begin{aligned} \text {cap}_{\omega ^T}(\{u <-t\}) \le \frac{1}{t}\left( \int _M (-u) (\omega ^T)^n\wedge \eta +n\int _M(\omega ^T)^n\wedge \eta \right) . \end{aligned}$$

Proof

This is a direct consequence of Proposition 6.2. Denote $K_t=\{u<-t\}$, then

$$\begin{aligned} \int _{K_t}\omega _\phi ^n\wedge \eta \le&\frac{1}{t}\int _M -\psi \omega _\phi ^n\wedge \eta \\ \le&\frac{1}{t}\left( \int _M -\psi (\omega ^T)^n\wedge \eta +n \int _M (\omega ^T)^n\wedge \eta \right) . \end{aligned}$$

$\square$

Proposition 6.4

Suppose that $u_k, u \in \text {PSH}(M,\xi ,\omega ^T) \cap L^{\infty }$ and $u_k$ decreases to u. Then for $\delta >0$ we have

$$\begin{aligned} \text {cap}_{\omega ^T}(\{u_k>u+\delta \}) \rightarrow 0, k\rightarrow \infty . \end{aligned}$$

Proof

This proceeds exactly the same as in [43, Proposition 3.7]. We sketch the argument briefly. We assume $\text {Vol}(M)=1$ for simplicity. Fix $\delta >0$ and $\phi \in \text {PSH}(M, \xi , \omega ^T)$ such that $0\le \phi \le 1$. We have

$$\begin{aligned} \int _{\{u_k>u+\delta \}} \omega _\phi ^n\wedge \eta \le \delta ^{-1}\int _M (u_k-u) \omega _\phi ^n\wedge \eta . \end{aligned}$$

By Stokes’ theorem, we write

$$\begin{aligned} \int _M (u_k-u) \omega _\phi ^n\wedge \eta =&\int _M (u_k-u) \wedge \omega ^T\wedge \omega _\phi ^{n-1}\wedge \eta \\&+\int _M (u_k-u) \wedge \mathrm{{dd}}^c_B \phi \wedge \omega _\phi ^{n-1}\wedge \eta \\ =&\int _M (u_k-u) \wedge \omega ^T\wedge \omega _\phi ^{n-1}\wedge \eta \\&-\int _M \mathrm{{d}}(u_k-u) \wedge \mathrm{{d}}^c_B \phi \wedge \omega _\phi ^{n-1}\wedge \eta . \end{aligned}$$

By the Cauchy–Schwartz inequality, setting $f_k=u_k-u\ge 0$,

$$\begin{aligned} \left| \int _M \mathrm{{d}}(u_k-u) \wedge \mathrm{{d}}^c_B \phi \wedge \omega _\phi ^{n-1}\wedge \eta \right| ^2\le & {} \int _M d f_k\wedge \mathrm{{d}}^c_B f_k\wedge \wedge \omega _\phi ^{n-1}\\&\wedge \eta \int _M d \phi \wedge \mathrm{{d}}^c_B \phi \wedge \wedge \omega _\phi ^{n-1}\wedge \eta . \end{aligned}$$

We compute

$$\begin{aligned} \int _M d \phi \wedge \mathrm{{d}}^c_B \phi \wedge \wedge \omega _\phi ^{n-1}\wedge \eta= & {} \int _M \phi (-\mathrm{{dd}}^c_B\phi )\wedge \omega _\phi ^{n-1}\wedge \eta \\\le & {} \int _M \phi \omega ^T\wedge \omega _\phi ^{n-1}\wedge \eta \le 1. \end{aligned}$$

Similarly, we compute

$$\begin{aligned} \int _M d f_k\wedge \mathrm{{d}}^c_B f_k\wedge \wedge \omega _\phi ^{n-1}\wedge \eta= & {} \int _M f_k (\mathrm{{dd}}^c_B u-d\mathrm{{d}}^c_B u_k)\wedge \omega _\phi ^{n-1}\wedge \eta \\\le & {} \int _M f_k \omega _u\wedge \omega _\phi ^{n-1}\wedge \eta . \end{aligned}$$

Combining all these together gives

$$\begin{aligned} \int _M (u_k-u) \omega _\phi ^n\wedge \eta\le & {} \int _M (u_k-u) \wedge \omega ^T\wedge \omega _\phi ^{n-1}\wedge \eta \\&+ \left( \int _M (u_k-u) \omega _u\wedge \omega _\phi ^{n-1}\wedge \eta \right) ^{1/2}. \end{aligned}$$

Suppose $u_k-u\le c_0$ for a fixed positive constant $c_0\ge 1$. Then we have

$$\begin{aligned} \int _M (u_k-u) \omega _\phi ^n\wedge \eta\le & {} \sqrt{c_0} \left( \int _M (u_k-u) \wedge \omega ^T\wedge \omega _\phi ^{n-1}\wedge \eta \right) ^{1/2}\\&+ \left( \int _M (u_k-u) \omega _u\wedge \omega _\phi ^{n-1}\wedge \eta \right) ^{1/2}. \end{aligned}$$

Hence we have

$$\begin{aligned} \int _M (u_k-u) \omega _\phi ^n\wedge \eta \le \sqrt{2c_0} \left( \int _M (u_k-u) \wedge (\omega ^T+\omega _u)\wedge \omega _\phi ^{n-1}\wedge \eta \right) ^{1/2}. \end{aligned}$$

We can proceed inductively by replacing $\omega _\phi$ by $\omega ^T+\omega _u$ to obtain

$$\begin{aligned} \int _M (u_k-u) \omega _\phi ^n\wedge \eta \le (\sqrt{2c_0})^n \left( \int _M (u_k-u) \wedge (\omega ^T+\omega _u)^{n}\wedge \eta \right) ^{1/2^n}. \end{aligned}$$

The dominated convergence theorem implies the right-hand side goes to zero, independent of $\phi$. This completes the proof. $\square$

As a consequence, we have the following:

Theorem 6.3

Let $\varphi \in \text {PSH}(M,\xi ,\omega ^T)$, then for any $\epsilon >0$ there exists an open subset $O_{\epsilon } \subset M$ such that $\text {cap}_{\omega ^T}(O_{\epsilon }) < \epsilon$ and $\varphi$ is continuous on $M-O_{\epsilon }$.

Proof

By Proposition 6.3 there exists $t_0>0$ such that $\text {cap}_{\omega ^t}(O_0) <\frac{\epsilon }{2}$ for the open subset $O_0=\{u<-t_0\}$. Take the cutoff $u_{t_0}=\max \{u,-t_0\} \in \text {PSH}(M,\xi ,\omega ^T)$, then there exists a sequence $u_k \in \mathcal {H}$ decreasing to u. By Proposition 6.4, we can choose a subsequence $u_{k_j}$ such that $\text {cap}_{\omega ^T}(O_j) < \frac{\epsilon }{2^{j+1}}$ for the open subset $O_j=\{u_{k_j}>u+\frac{1}{j}\}$. Then for the open subset $O_{\epsilon }=\cup _{j=0}^{\infty }O_j$ we have $\text {cap}_{\omega ^T}(O)<\epsilon$. Moreover $u_{K_j}$ converges uniformly to u on $M-O_{\epsilon }$, hence u is continuous on $M-O_{\epsilon }$. $\square$

Remark 6.2

The discussions above are taken from Kähler setting [43, Section 3]. Note that in (6.13) it is necessary to replace $\sup \psi$ by $\max \{\sup \psi , 0\}$ (similarly one needs to replace $\sup _X\psi$ by $\max \{\sup _X \psi , 0\}$ in [43, Proposition 3.1]).

We also need the following uniqueness in Sasaki setting, see [44, Theorem 3.3].

Theorem 6.4

Suppose $u, v\in \mathcal {E}_1(M, \xi , \omega ^T)$ such that

$$\begin{aligned} \omega _u^n\wedge \eta =\omega _v^n\wedge \eta , \end{aligned}$$

then $u-v=\text {const}$.

Proof

This follows exactly as in [44, Theorem 3.3] and we sketch the argument. The first step is that for $u\in \mathcal {E}_1(M, \xi , \omega ^T)$ and its canonical cutoffs $u_j=\max \{u, -j\}$, then $\nabla u_j\in L^2(\mathrm{{d}}\mu _g)$ and has uniformly bounded $L^2$ norm (see [44, Proposition 3.2]). We can assume that $u\le 0$ and hence $u_j\le 0$. Then for $\phi \in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$ such that $\phi \le 0$, we know that, for any basic positive closed of $(n-1, n-1)$ type.

$$\begin{aligned} \int _M (-\phi )\omega \wedge T= & {} \int _M (-\phi ) (\omega _\phi -\mathrm{{dd}}^c_B\phi )\wedge T=\int _M (-\phi )\omega _\phi \wedge T\\&+\int _M \mathrm{{d}}\Phi \wedge \mathrm{{d}}^c_B \phi \wedge T\le \int _M (-\phi )\omega _\phi \wedge T. \end{aligned}$$

An inductive argument applies to $T=\omega _\phi ^k\wedge (\omega ^T)^{n-k-1}$, we get that

$$\begin{aligned} 0\le \int _M \mathrm{{d}}\Phi \wedge \mathrm{{d}}^c_B \phi \wedge T\le \int _M (-\phi )\omega _\phi ^n\wedge \eta . \end{aligned}$$

(6.15)

Taking $\phi =u_j$ in (6.15) and noting that the right-hand side is uniformly bounded, we get $\nabla u_j$ is uniformly bounded in $L^2(\mathrm{{d}}\mu _g)$, hence $\nabla u\in L^2(\mathrm{{d}}\mu _g)$.

We assume that $u, v\le -1$ and $\text {Vol}(M)=1$. Set $f=(u-v)/2$ and $h=(u+v)/2$. We need to establish that $\nabla f=0$ by showing that $\int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge (\omega ^T)^{n-1}\wedge \eta =0$. If we assume u, v are bounded, then we have

$$\begin{aligned} \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega ^{n-1}_h\wedge \eta \le \sum \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega _u^k\wedge \omega ^{n-1-k}_v\wedge \eta =-\int _M \frac{f}{2}(\omega _u^n-\omega _v^n)\wedge \eta , \end{aligned}$$

(6.16)

where we use the fact that $\mathrm{{dd}}^c_B f=(\omega _u-\omega _v)/2$. We shall also establish the following a priori bound, when u, v are bounded,

$$\begin{aligned} \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge (\omega ^T)^{n-1}\wedge \eta \le 3^n \left( \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega ^{n-1}_h\wedge \eta \right) ^{1/2^{n-1}}. \end{aligned}$$

(6.17)

We apply (6.16) and (6.17) to the canonical cutoffs $u_j, v_j$ (writing $f_j, h_j$ correspondingly and using Proposition 3.15),

$$\begin{aligned} \lim \int _M \mathrm{{d}}f_j\wedge \mathrm{{d}}^c_B f_j\wedge (\omega ^T)^{n-1}\wedge \eta =0. \end{aligned}$$

We can then conclude that

$$\begin{aligned} \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge (\omega ^T)^{n-1}\wedge \eta =0. \end{aligned}$$

This implies that $u-v$ is a constant. To establish (6.17), we need several observations as follows. First observe that for $l=n-2, \ldots , 0$,

$$\begin{aligned} \int _M (-h)\omega ^{2+l}_h\wedge (\omega ^T)^{n-2-l}\wedge \eta \le \int _M (-h)(\omega ^T)^n\wedge \eta \le 1, \end{aligned}$$

where the last inequality follows from $-h\le 1$ and the normalization of the volume. We can then apply the following inequality inductively for $T=\omega _h^l\wedge (\omega ^T)^{n-l-1}$ such that

$$\begin{aligned} \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega ^T\wedge T\wedge \eta \le 3 \left( \int _M\mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega _h\wedge T\wedge \eta \right) ^{1/2}, \end{aligned}$$

(6.18)

which proves (6.17). Now we establish (6.18). We write

$$\begin{aligned} \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega ^T=\mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega _h-\mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \mathrm{{dd}}^c_B h \end{aligned}$$

hence we obtain, integrating by parts,

$$\begin{aligned} \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega ^T\wedge T\wedge \eta= & {} \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega _h\wedge T\wedge \eta \\&+\int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_Bh\wedge \frac{\omega _u-\omega _v}{2}\wedge T\wedge \eta . \end{aligned}$$

By Cauchy–Schwartz inequality, we have

$$\begin{aligned} \left| \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_Bh\wedge \omega _u\wedge T\wedge \eta \right| ^2\le & {} 4 \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_Bf\wedge \omega _h\wedge T\wedge \eta \int _M \mathrm{{d}}h\\&\,\wedge \mathrm{{d}}^c_Bh\wedge \omega _h\wedge T\wedge \eta . \end{aligned}$$

We can get a similar control

$$\begin{aligned} \left| \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_Bh\wedge \omega _v\wedge T\wedge \eta \right| ^2\le & {} 4 \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_Bf\wedge \omega _h\wedge T\wedge \eta \int _M \mathrm{{d}}h\\&\,\wedge \mathrm{{d}}^c_Bh\wedge \omega _h\wedge T\wedge \eta . \end{aligned}$$

Clearly, we have the following ($h\le 0, S=\omega _h^l\wedge (\omega ^T)^{n-l-2}$)

$$\begin{aligned} \int _M \mathrm{{d}}h\wedge \mathrm{{d}}^c_Bh\wedge \omega _h\wedge S\wedge \eta \le \int _M (-h) \omega _h^2\wedge S\wedge \eta \le 1. \end{aligned}$$

Combining these estimate altogether we conclude that,

$$\begin{aligned} \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega ^T\wedge S\wedge \eta\le & {} \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega _h\wedge T\wedge \eta \\&+\,2\left( \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_Bf\wedge \omega _h\wedge T\wedge \eta \right) ^{1/2}. \end{aligned}$$

The last observation is that

$$\begin{aligned} \int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge \omega _h\wedge S\wedge \eta= & {} \frac{1}{4}\int _M (u-v)(\omega _v-\omega _u)\wedge \omega _h\wedge S\wedge \eta \\\le & {} \int _M (-h) \omega _h^2\wedge S\wedge \eta \le 1. \end{aligned}$$

This completes the proof of (6.18) by combining two inequalities above. $\square$

1.3 Functionals in Finite-Energy Class $\mathcal {E}_1$ and Compactness

We discuss briefly well-known functionals in Kähler geometry and their properties over finite-energy class $\mathcal {E}_1$, see [30, Section 3.8]. The energy functionals include Monge–Ampere energy $\mathbb {I}$ and Aubin’s I-functional on $\mathcal {E}_1$, see [1, 4, 5, 30] for Kähler setting. These results have a direct adaption in Sasaki setting. Recall Aubin’s I-functional in Sasaki setting, for $u, v\in \mathcal {H}$

$$\begin{aligned} I(u, v):=I(\omega _u, \omega _v)=\frac{1}{n!}\int _M (v-u) (\omega _u^n-\omega _v^n)\wedge \eta . \end{aligned}$$

(6.19)

We also recall the J-functional

$$\begin{aligned} J(u, v):=J(\omega _u, \omega _v)=\frac{1}{n!}\int _M (v-u) \omega _u^n\wedge \eta -\mathbb {I}_{\omega _u}(v), \end{aligned}$$

(6.20)

where the $\mathbb {I}_{\omega _u}(v)$-functional is given by

$$\begin{aligned} \mathbb {I}_{\omega _u}(v)=\frac{1}{(n+1)!}\int _M (v-u)\sum _{k=0}^n \omega _u^k\wedge \omega _v^{n-k}\wedge \eta . \end{aligned}$$

(6.21)

We define the $\mathbb {I}$-functional (with the base $\omega ^T$) on $\mathcal {H}$,

$$\begin{aligned} \mathbb {I}_{\omega ^T}(u)=\frac{1}{(n+1)!}\int _M u\sum _{k=0}^n \omega _u^k\wedge \omega ^{n-k}_T\wedge \eta . \end{aligned}$$

(6.22)

The $\mathbb {I}$-functional is also called the Monge–Ampère energy, since if $t\rightarrow v_t\in \mathcal {H}$ is smooth, then we have (as in Kähler setting),

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t}\mathbb {I}(v_t)=\frac{1}{n!}\int _M \dot{v}_t \omega _{v_t}^n \wedge \eta . \end{aligned}$$

(6.23)

We mention that I is symmetric with respect to u, v but J is not. I, J are both defined on the metric level, independent of the choice of normalization of potentials u, v; while $\mathbb {I}_{\omega _u}(v)$ depends on the normalization of u, v. When u, v are bounded, then Bedford–Taylor theory allows to integrate by parts and the I-functional takes the formula

$$\begin{aligned} I(\omega _u, \omega _v)=\frac{1}{(n+1)!}\sum _{j=0}^{n-1} \int _M \mathrm{{d}}(u-v)\wedge \mathrm{{d}}^c_B(u-v)\wedge \omega _u^j\wedge \omega _v^{n-1-j}\wedge \eta . \end{aligned}$$

(6.24)

Hence it is non-negative.

We need more information about $\mathbb {I}$-functional, see [30, Section 3.7] for Kähler setting. These properties in Sasaki setting follow in a rather straightforward way given pluripotential theory extended to Sasaki setting. We include these facts here for completeness.

Proposition 6.5

Given $u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$, the following cocycle condition holds

$$\begin{aligned} \mathbb {I}(u)-\mathbb {I}(v)=\frac{1}{(n+1)!}\sum _{k=0}^n\int _M (u-v)\omega _u^k\wedge \omega _v^{n-k}\wedge \eta =\mathbb {I}_{\omega _u}(v). \end{aligned}$$

(6.25)

Moreover, we have $\mathbb {I}(u)$ is concave in u in the sense that,

$$\begin{aligned} \frac{1}{n!}\int _M (u-v)\omega _u^n\wedge \eta \le \mathbb {I}(u)-\mathbb {I}(v)\le \frac{1}{n!}\int _M (u-v)\omega _v^n\wedge \eta . \end{aligned}$$

(6.26)

As a direct consequence, if $u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$ such that $u\ge v$. Then $\mathbb {I}(u)\ge \mathbb {I}(v)$.

Proof

This follows almost identical as in [30, Proposition 3.8], given the pluripotential theory established in Sasaki setting in the paper. We sketch the argument. When $u, v\in \mathcal {H}$, this follows exactly the same as in Kähler setting, by taking $h_t=(1-t)u+tv$ and then use (6.23) to compute directly. When $u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$, we then use $u_k, v_k\in \mathcal {H}$ decreasing to u, v (Lemma 3.1), respectively. Using Bedford–Taylor’s theorem in Sasaki setting [58, Theorem 2.3.1], we proceed exactly as in Kähler setting to conclude that $\mathbb {I}(u_k)\rightarrow \mathbb {I}(u)$, etc. For the estimate (6.26), we compute

$$\begin{aligned} \int _M (u-v)\omega ^k_u\wedge \omega _v^{n-k}\wedge \eta =&\int _M (u-v)\omega ^{k-1}_u\wedge \omega _v^{n-k+1}\wedge \eta \\ {}&+\int _M (u-v)\sqrt{-1}\partial \bar{\partial } (u-v)\wedge \omega _u^{k-1}\wedge \omega _v^{n-k}\wedge \eta \\ =&\int _M (u-v)\omega ^{k-1}_u\wedge \omega _v^{n-k+1}\wedge \eta \\&-\int _M \sqrt{-1}\partial (u-v)\wedge \bar{\partial } (u-v)\wedge \omega _u^{k-1}\wedge \omega _v^{n-k}\wedge \eta \\ \le&\int _M (u-v)\omega ^{k-1}_u\wedge \omega _v^{n-k+1}\wedge \eta . \end{aligned}$$

Using the estimate inductively for the terms in (6.25) leads to (6.26). Clearly, $\mathbb {I}(u)$ is concave in u given (6.26). $\square$

The monotonicity property allows to define $\mathbb {I}(u)$ for $u\in \text {PSH}(M, \xi , \omega ^T)$ through the limit process, using the canonical cutoffs $u_k=\max \{u, -k\}$

$$\begin{aligned} \mathbb {I}(u)=\lim _{k\rightarrow \infty }\mathbb {I}(\max \{u, -k\}). \end{aligned}$$

Though the above limit is well defined, it may equal $-\infty$. It turns out $\mathbb {I}(u)$ is finite exactly on $\mathcal {E}_1(M, \xi , \omega ^T)$. We record some further properties of $\mathbb {I}(u)$ for $u\in \mathcal {E}_1(M, \xi , \omega ^T)$. The proofs are almost identical and we shall skip the details, see [30, Propositions 3.40, 3.42, 3.43; Lemma 3.41].

Proposition 6.6

Let $u\in \text {PSH}(M, \xi , \omega ^T)$. Then $-\infty <\mathbb {I}(u)$ if and only if $u\in \mathcal {E}_1(M, \xi , \omega ^T)$. Moreover,

$$\begin{aligned} |\mathbb {I}(u_0)-\mathbb {I}(u_1)|\le d_1(u_0, u_1), u_0, u_1\in \mathcal {E}_1(M, \xi , \omega ^T). \end{aligned}$$

(6.27)

Proposition 6.7

Suppose $u_0, u_1\in \mathcal {E}_1(M, \xi , \omega ^T)$ and $t\rightarrow u_t$ is the finite-energy geodesic connecting $u_0, u_1$. Then $t\rightarrow \mathbb {I}(u_t)$ is linear in t. We also have the following distance formula:

$$\begin{aligned} d_1(u_0, u_1)=\mathbb {I}(u_0)+\mathbb {I}(u_1)-2\mathbb {I}(P(u_0, u_1)). \end{aligned}$$

In particular, $d_1(u_0, u_1)=\mathbb {I}(u_0)-\mathbb {I}(u_1)$ if $u_0\ge u_1$.

We have the following (see [30, Lemma 3.47])

Lemma 6.6

Suppose $u, u^j, v, v^j\in \mathcal {E}_1(M, \xi , \omega ^T)$ and $u^j\searrow u$ and $v^j\searrow v$. Then the following hold:

$$\begin{aligned} I(u, v)=I(u, \max {\{u, v\}})+I(\max {\{u, v\}}, v). \end{aligned}$$

(6.28)

Moreover, $\lim _{j\rightarrow \infty } I(u^j, v^j)=I(u, v)$.

Proof

By Proposition 3.8, we have

$$\begin{aligned} \chi _{\{v>u\}}\omega ^n_{\max \{u, v\}}\wedge \eta =\chi _{\{v>u\}}\omega ^n_v\wedge \eta . \end{aligned}$$

Hence it follows that

$$\begin{aligned} I(u, \max {\{u, v\}})=\frac{1}{(n+1)!}\int _{\{v>u\}} (u-v)(\omega _v^n-\omega _u^n)\wedge \eta . \end{aligned}$$

Interchange $u\leftrightarrow v$, we get $I(v, \max {\{u, v\}})=\int _{\{u>v\}} (u-v)(\omega _v^n-\omega _u^n)\wedge \eta$. This proves (6.28). We write

$$\begin{aligned} I(u^j, v^j)=I(u^j, \max {\{u^j, v^j\}})+I(v^j, \max {\{u^j, v^j\}}). \end{aligned}$$

Since $u^j, v^j\le \max \{u^j, v^j\}$, we can apply Proposition 3.15 to conclude $I(u^j, \max {\{u^j, v^j\}})\rightarrow I(u, \max {\{u, v\}})$ and $I(v^j, \max {\{u^j, v^j\}})\rightarrow I(v, \max {\{u, v\}})$, using the formula (6.19). This completes the proof. $\square$

We have the following well-known inequalities:

Proposition 6.8

For $u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$, we have

$$\begin{aligned} \frac{1}{n+1}I(u, v)\le J(u, v)\le \frac{n}{n+1} I(u, v). \end{aligned}$$

Moreover, J(u, v) is convex in v since $\mathbb {I}_{\omega ^T}(v)$ is concave in v.

Proof

This is well known, by direct computation [38, Proposition 4.2.1] for $u, v\in \mathcal {H}$. A direct approximation argument using Lemma 3.1 shows that this can be generalized for $u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$. $\square$

The functionals ($I, J, \mathbb {I}$) are well defined for $u, v\in \mathcal {E}_1(M, \xi , \omega ^T)$ [see Proposition (3.16)]. Note that (6.26) and Proposition 6.8 both hold in $\mathcal {E}_1(M, \xi , \omega ^T)$ (see [4] for Kähler setting). This follows by an approximation argument applying Proposition 3.15. Next we prove the following, as a direct adaption of [5, Theorem 1.8],

Lemma 6.7

There exists a positive $C=C(n)$ such that for $u, v, w\in \mathcal {E}_1(M, \xi , \omega ^T)$, then

$$\begin{aligned} I(u, v)\le C(I(u, w)+I(v, w)). \end{aligned}$$

(6.29)

Proof

With Lemma 6.6, we only need to argue (6.29) holds for bounded potentials, with u, v, w replaced by canonical cutoffs $u_k, v_k, w_k$. The proof follows exactly as in [5, Theorem 1.8, Lemma 1.9]. and we include the proof for completeness. For $u, v, \psi \in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty$, set

$$\begin{aligned} \Vert \mathrm{{d}}(u-v)\Vert _\psi :=\left( \int _M \mathrm{{d}}(u-v)\wedge \mathrm{{d}}^c_B(u-v)\wedge \omega _\psi ^{n-1}\wedge \eta \right) ^{\frac{1}{2}}. \end{aligned}$$

Using (6.24), it is straightforward to see that

$$\begin{aligned} \Vert \mathrm{{d}}(u-v)\Vert _{\frac{u+v}{2}}^2\le I(u, v)\le 2^{n-1}\Vert \mathrm{{d}}(u-v)\Vert _{\frac{u+v}{2}}^2. \end{aligned}$$

(6.30)

We need the following, there exists a constant $C=C(n)$ for $u, v, \psi \in \text {PSH}{M, \xi , \omega ^T}\cap L^\infty$, we have the following (see [5, Lemma 1.9]),

$$\begin{aligned} \Vert \mathrm{{d}}(u-v)\Vert ^2_\psi \le C I(u, v)^{1/2^{n-1}}\left( I(u, \psi )^{1-1/2^{n-1}}+I(v, \psi )^{1-1/2^{n-1}}\right) . \end{aligned}$$

(6.31)

With (6.31) we prove (6.29). Taking $\phi =\frac{u+v}{2}$, the triangle inequality gives,

$$\begin{aligned} \Vert \mathrm{{d}}(u-v)\Vert _\phi \le \Vert \mathrm{{d}}(u-w)\Vert _\phi +\Vert d(v-w)\Vert _\phi . \end{aligned}$$

Using (6.30) and (6.31), we have

$$\begin{aligned} I(u, v)\le 2^{n-1} \Vert \mathrm{{d}}(u-v)\Vert _{\phi }^2\le&\, C (\Vert \mathrm{{d}}(u-w)\Vert _\phi ^2+\Vert d(v-w)\Vert _\phi ^2)\\ \le&\, CI(u, w)^{1/2^{n-1}}\left( I(u, \phi )^{1-1/2^{n-1}}+I(w, \phi )^{1-1/2^{n-1}}\right) \\&+CI(v, w)^{1/2^{n-1}}\left( I(v, \phi )^{1-1/2^{n-1}}+I(w, \phi )^{1-1/2^{n-1}}\right) . \end{aligned}$$

By Proposition 6.8, we have

$$\begin{aligned} I(u, \phi )\le nI(u, v), I(v, \phi )\le nI(v, u), I(w, \phi )\le n (I(w, u)+I(w, v)). \end{aligned}$$

It follows that

$$\begin{aligned} I(u, v)\le & {} C\left( I(u, w)^{\frac{1}{2^{n-1}}}+I(v, w)^{\frac{1}{2^{n-1}}}\right) \left( I(u, v)^{1-1/2^{n-1}}\right. \\&\left. +\,I(u, w)^{1-1/2^{n-1}}+I(v, w)^{1-1/2^{n-1}}\right) . \end{aligned}$$

We assume $I(u, v)\ge \max \{I(u, w), I(v, w)\}$ (otherwise we are done). Hence it follows

$$\begin{aligned} I(u, v)^{1/2^{n-1}}\le C \left( I(u, w)^{\frac{1}{2^{n-1}}}+I(v, w)^{\frac{1}{2^{n-1}}}\right) . \end{aligned}$$

This is sufficient to prove that

$$\begin{aligned} I(u, v)\le C (I(u, w)+I(v, w)). \end{aligned}$$

Now we establish (6.31) (see [5, Lemma 1.9]). First observe that

$$\begin{aligned} \Vert \mathrm{{d}}(u-v)\Vert _\psi \le \Vert \mathrm{{d}}(u-\psi )\Vert _\psi +\Vert d(v-\psi )\Vert _{\psi }\le I(u, \psi )^{1/2}+I(v, \psi )^{1/2}. \end{aligned}$$

Hence we have

$$\begin{aligned} \Vert \mathrm{{d}}(u-v)\Vert _\psi ^2\le 2(I(u, \psi )+I(v, \psi )). \end{aligned}$$

Hence if $I(u, v)\ge I(u, \psi )+I(v, \psi )$, clearly we have

$$\begin{aligned} \Vert \mathrm{{d}}(u-v)\Vert _\psi ^2\le & {} 2(I(u, \psi )+I(v, \psi ))\nonumber \\\le & {} C I(u, v)^{1/2^{n-1}}\left( I(u, \psi )^{1-\frac{1}{2^{n-1}}}+I(v, \psi )^{1-\frac{1}{2^{n-1}}}\right) . \end{aligned}$$

(6.32)

Now we suppose $I(u, v)\le I(u, \psi )+I(v, \psi )$. Taking $\phi =\frac{u+v}{2}$, we consider

$$\begin{aligned} b_p:=\int _M \mathrm{{d}}(u-v)\wedge \mathrm{{d}}^c_B(u-v)\wedge \omega ^p_\psi \wedge \omega ^{n-p-1}_\phi \wedge \eta . \end{aligned}$$

By (6.30), $b_0\le I(u, v)$ and $b_{n-1}=\Vert \mathrm{{d}}(u-v)\Vert _{\psi }^2$. We claim that, $p=0, \cdot , n-2$,

$$\begin{aligned} b_{p+1}\le b_p+4\sqrt{b_p I(\psi , \phi )}. \end{aligned}$$

(6.33)

We compute

$$\begin{aligned} b_{p+1}-b_p=&\int _M \mathrm{{d}}(u-v)\wedge \mathrm{{d}}^c_B(u-v)\wedge \mathrm{{dd}}^c_B (\psi -\phi )\omega ^p_\psi \wedge \omega ^{n-p-2}_\phi \wedge \eta \\ =&-\int _M \mathrm{{d}}(u-v)\wedge \mathrm{{dd}}^c_B(u-v)\wedge \mathrm{{d}}^c_B (\psi -\phi )\omega ^p_\psi \wedge \omega ^{n-p-2}_\phi \wedge \eta \\ =&-\int _M \mathrm{{d}}(u-v)\wedge (\omega _u-\omega _v)\wedge \mathrm{{d}}^c_B (\psi -\phi )\omega ^p_\psi \wedge \omega ^{n-p-2}_\phi \wedge \eta . \end{aligned}$$

Using Cauchy–Schwarz inequality, we compute

$$\begin{aligned}&\left| \int _M \mathrm{{d}}(u-v)\wedge \omega _u\wedge d (\psi -\phi )\omega ^p_\psi \wedge \omega ^{n-p-2}_\phi \wedge \eta \right| \\&\quad \le \left( \int _M \mathrm{{d}}(u-v)\wedge \mathrm{{d}}^c_B(u-v)\wedge \omega _u \wedge \omega ^p_\psi \wedge \omega ^{n-p-2}_\phi \wedge \eta \right) ^{1/2}\\&\qquad \times \left( \int _M d(\psi -\phi )\wedge \mathrm{{d}}^c_B(\psi -\phi )\wedge \omega _u \wedge \omega ^p_\psi \wedge \omega ^{n-p-2}_\phi \wedge \eta \right) ^{1/2}\le 2 \sqrt{b_p I(\psi , \phi )}, \end{aligned}$$

where we have used that $\omega _u\le 2\omega _\phi$ and (6.24). We can get the same estimate for

$$\begin{aligned} \left| \int _M \mathrm{{d}}(u-v)\wedge \omega _v\wedge d (\psi -\phi )\omega ^p_\psi \wedge \omega ^{n-p-2}_\phi \wedge \eta \right| . \end{aligned}$$

This establishes (6.33). By Proposition 6.8, we know that

$$\begin{aligned} I(\psi , \phi )\le (n+1)J(\psi , \phi )\le \frac{n}{2}(I(\psi , u)+I(\psi , v)). \end{aligned}$$

Denote $a=(I(\psi , u)+I(\psi , v))$. We write (6.33) as

$$\begin{aligned} b_{p+1}\le b_p+4\sqrt{b_p a}, p=0, \ldots , n-2. \end{aligned}$$

Note that $b_0=I(u, v)\le a$, hence it is evident that $b_p\le C a$. Hence it follows that, for $p=0, \ldots , n-2$,

$$\begin{aligned} b_{p+1}\le C \sqrt{b_p a}. \end{aligned}$$

A direct computation gives that,

$$\begin{aligned} b_{n-1}\le C b_0^{1/2^{n-1}} a^{1-\frac{1}{2^{n-1}}}. \end{aligned}$$

This completes the proof. $\square$

More generally, we have the following [30, Proposition 3.48]

Proposition 6.9

Suppose $C>0$ and $\phi , \psi , u, v\in \mathcal {E}_1(M, \xi , \omega ^T)$ satisfies

$$\begin{aligned} -C\le \mathbb {I}(\phi ), \mathbb {I}(\psi ), \mathbb {I}(u), \mathbb {I}(v), \sup _M \phi , \sup _M \psi , \sup _M u, \sup _M v\le C. \end{aligned}$$

Then there exists a continuous function $f_C:\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$ depending only on C with $f_C(0)=0$ such that

$$\begin{aligned}&\left| \int _M \phi (\omega _u^n-\omega _v^n)\wedge \eta \right| \le f_C(I(u, v))\nonumber \\&\left| \int _M (u-v)(\omega _\phi ^n-\omega _\psi ^n)\wedge \eta \right| \le f_C(I(u, v)). \end{aligned}$$

(6.34)

Proof

The proof is similar in philosophy as Lemma 6.7 and follows almost identically as in Kähler setting, see [30, Proposition 3.48]. Hence we skip the details. $\square$

As a consequence, we have the following [30, Theorem 3.46]:

Theorem 6.5

Suppose $u_k, u\in \mathcal {E}_1(M, \xi , \omega ^T)$. Then the following holds:

• (1) $d_1(u_k, u)\rightarrow 0$ if and only if $\int _M |u_k-u|\omega ^n_T\wedge \eta \rightarrow 0$ and $\mathbb {I}(u_k)\rightarrow \mathbb {I}(u)$.

• (2) If $d_1(u_k, u)\rightarrow 0$, then $\omega ^n_{u_k}\wedge \eta \rightarrow \omega _u^n\wedge \eta$ weakly and $\int _M |u_k-u|\omega ^n_v\wedge \eta \rightarrow 0$ for $v\in \mathcal {E}_1(M, \xi , \omega ^T)$.

Proof

If $d_1(u_k, u)\rightarrow 0$, then Propositions 6.6 and 6.9 imply (1) and (2). For the reverse direction in (1), it follows almost identically as in Kähler setting, see [30, Proposition 3.52], using Proposition 6.9 and approximation argument. We sketch the process. First we have

$$\begin{aligned} \int _M u_k \omega _u^n\wedge \eta \rightarrow \int _M u \omega _u^n\wedge \eta . \end{aligned}$$

And then one argues that

$$\begin{aligned} I(u, u_k)\le (n+1)\left( \mathbb {I}(u_k)-\mathbb {I}(u)-\int _M (u-u_k)\omega _u^n\wedge \eta \right) \end{aligned}$$

Hence this shows that $I(u, u_k)\rightarrow 0$. Using Proposition 6.9 and Lemma 6.6, one can then show

$$\begin{aligned} \int _M |u_k-u|\omega _u^n\wedge \eta , \int _M |u_k-u|\omega _{u_k}^n\wedge \eta \rightarrow 0, k\rightarrow \infty . \end{aligned}$$

This gives the desired convergence $d_1(u_k, u)\rightarrow 0$. $\square$

As an application of results established above, we have the following compactness result in Sasaki setting, following [30, Theorem 4.45].

Theorem 6.6

Let $u_j\in \mathcal {E}_1(M, \xi , \omega ^T)$ be a $d_1$-bounded sequence for which the entropy

$$\begin{aligned} \sup _jH(u_j)<\infty . \end{aligned}$$

Then $\{u_j\}$ contains a $d_1$-convergence sequence.

Proof

We sketch the proof for completeness; for details see [30, Theorem 4.45]. First $d_1$ bounded implies that $\mathbb {I}$ and $\sup u$ are both bounded. Together with Proposition 3.4, this implies that $d_1$ bounded set is precompact in $L^1$. That is, there exists $u\in \mathcal {E}_1(M, \xi , \omega ^T)$ such that after passing by subsequence,

$$\begin{aligned} \int _M |u_k-u|(\omega ^T)^n\wedge \eta \rightarrow 0. \end{aligned}$$

Moreover, we have (see [30, Proposition 4.14, Corollary 4.15])

$$\begin{aligned} \limsup \mathbb {I}(u_k)\le \mathbb {I}(u). \end{aligned}$$

Since all elements in $\mathcal {E}_1(M, \xi , \omega ^T)$ have zero Lelong number, we apply Zeriahi’s uniform version of the famous Skoda integrability theorem [59] (we apply Zeriahi’s theorem in each foliation chart) to obtain: for any $p\ge 1$, there exists $C=C(p)$ such that

$$\begin{aligned} \int _M e^{-p u_j} (\omega ^T)^n\wedge \eta \le C. \end{aligned}$$

Since $\sup u_j\le C$, we have

$$\begin{aligned} \int _M e^{p |u_j|} (\omega ^T)^n\wedge \eta \le C. \end{aligned}$$

Now we need to use the assumption that $H(u_j)$ is uniformly bounded above. We proceed as in the proof of [30, Theorem 4.45] to conclude

$$\begin{aligned} \int _M |u_j-u|\omega _{u_j}^n\wedge \eta \rightarrow 0. \end{aligned}$$

By Proposition 6.26 (which holds for $\mathcal {E}_1$), we can then conclude that $\liminf \mathbb {I}(u_j)\ge \mathbb {I}(u)$. This gives $\lim \mathbb {I}(u_j)=\mathbb {I}(u)$. Hence $d_1(u_j, u)\rightarrow 0$, as a consequence of Theorem 6.5. $\square$

Finally we have the extension of $\mathcal {K}$-energy, see [7, Theorem 1.2] for Kähler setting.

Theorem 6.7

The $\mathcal {K}$-energy can be extended to a functional $\mathcal {K}: \mathcal {E}_1\rightarrow \mathbb {R}\cup \{+\infty \}$. Such a $\mathcal {K}$-energy in $\mathcal {E}^1$ is the greatest $d_1$-lsc extension of $\mathcal {K}$-energy on $\mathcal {H}$. Moreover, $\mathcal {K}$-energy is convex along the finite-energy geodesics of $\mathcal {E}^1$.

Proof

As in Kähler setting [19], we can write the $\mathcal {K}$-energy as the following:

$$\begin{aligned} \mathcal {K}(\phi )=H(\phi )+\mathbb {J}_{\omega ^T, -Ric}(\phi ), \end{aligned}$$

where $H(\phi )$ is the entropy part and $\mathbb {J}$ is the entropy part, taking the formula, respectively,

$$\begin{aligned} H(\phi )=&\int _M \log \frac{\omega _\phi ^n\wedge \eta }{\omega ^n_T\wedge \eta } dv_\phi \\ \mathbb {J}_{-Ric}(\phi )=&\frac{n\underline{R}}{(n+1)!}\int _M \phi \sum _{k=0}^n \omega ^k_T\wedge \omega _\phi ^{n-k}\wedge \eta -\frac{1}{n!}\int _M \phi \sum _{k=0}^{n-1}Ric\wedge \omega ^k_T\wedge \omega _\phi ^{n-1-k}\wedge \eta . \end{aligned}$$

As a direct consequence of this formula, $\mathcal {K}(\phi )$ is well defined for $\phi \in \mathcal {H}_\Delta$. More importantly, for $\phi _0, \phi _1\in \mathcal {H}$, and $\phi _t\in \mathcal {H}_\Delta$ being the geodesic connecting $\phi _0, \phi _1$, $\mathcal {K}(\phi _t)$ is convex with respect to $t\in [0, 1]$.

Now we extend $H(\phi )$ and $\mathbb {J}_{-Ric}$ to $\mathcal {E}_1$ separately. As in [7], the extension of $\mathbb {J}_{-Ric}$ to $\mathcal {E}_1$ is $d_1$-continuous, while since $d_1(u_k, u)\rightarrow 0$ implies that $\omega _{u_k}^n\wedge \eta \rightarrow \omega _u^n\wedge \eta$ weakly (Theorem 6.5), this implies that the extension of $\phi \rightarrow H(\phi )$ to $\mathcal {E}_1$ is $d_1$ lsc. Moreover, by [49, Lemma 5.4], the extension of $\mathcal {K}$ is the greatest lsc extension. In the end, the convexity of the extended $\mathcal {K}$-energy along the finite-energy geodesic segments follows exactly as in [7, Theorem 4.7]. $\square$