Average area ratio and normalized total scalar curvature of hyperbolic n-manifolds

Let $(M,h_{0})$ be a closed hyperbolic $3$-manifold, and let $N$ be a closed $3$-manifold with Riemannian metric $g$ whose scalar curvature satisfies $R_{g}\geq-6$. $F:N\rightarrow M$ is a smooth map with degree $d$. The average area ratio of $F$ is as follows.

$$\text{Area}_{F}(g/h_{0})=\int_{(y,P)\in Gr_{2}(M)}\sum_{x\in F^{-1}(y)}\lim_{\delta\rightarrow 0}\frac{\text{area}_{g}((dF_{x})^{-1}(D_{\delta}))}{\delta}\,d\mu_{h_{0}},$$

where $D_{\delta}$ is a subset of the totally geodesic disc of $\mathbb{H}^{n}$ which is tangential to $P$ at $x$, $D_{\delta}$ has an area equal to $\delta$, and $\mu_{h_{0}}$ stands for the unit volume measure on $Gr_{2}(M)$ with respect to the metric induced by $h_{0}$. We refer to Section 2 for a more detailed definition. Gromov proved the following inequality using stability (see page 73 of [8]).

$$\text{Area}_{F}(g/h_{0})\geq\frac{d}{3}.$$

The proof indicates that if the equality holds, then $(N,g)$ should be hyperbolic, and in the universal covers, the preimage of any totally geodesic disc is totally geodesic as well. However, it means the inequality is not sharp, and therefore, Gromov conjectured that the lower bound could be replaced by $d$. Moreover, the higher dimensional case may share the same property.

Recently, Lowe-Neves (Corollary 1.3 of [19]) verified the special case where $n=3$ and $F$ is a local diffeomorphism. We will generalize this conclusion in two aspects in Theorem 1.2 and Theorem 1.4.

Let $\mathbb{H}^{n}$ denote the hyperbolic $n$-space, where $n\geq 3$. In the Poincaré ball model, the asymptotic boundary $\partial_{\infty}\mathbb{H}^{n}$ can be considered as the $(n-1)$-unit sphere $S^{n-1}_{\infty}$. A homeomorphism $f:S^{n-1}_{\infty}\rightarrow S^{n-1}_{\infty}$ is called $K$-quasiconformal if $K(f)=\underset{x\in S^{n-1}_{\infty}}{\text{ess sup}}K_{f}(x)\leq K$, where the dilatation of $f$ is

$$K_{f}(x)=\underset{r\rightarrow 0}{\lim\sup}\,\underset{y,z}{\sup}\frac{|f(x)-f(y)|}{|f(x)-f(z)|},$$

the supremum is taken over $y,z$ so that $r=|x-y|=|x-z|$. A $K$-quasicircle in $S^{n-1}_{\infty}$ is the image of a round circle under a $K$-quasiconformal map.

Let $M=\mathbb{H}^{n}/\pi_{1}(M)$ be a closed orientable $n$-manifold $(n\geq 3)$ that admits a hyperbolic metric $h_{0}$, A closed surface immersed in $M$ with genus at least $2$ is said to be essential if the immersion is $\pi_{1}$-injective, and the image of its fundamental group in $\pi_{1}(M)$ is called a surface subgroup. Let $S(M,g)$ denote the set of surface subgroups of genus at most $g$ up to conjugacy, and let the subset $S(M,g,\epsilon)\subset S(M,g)$ consist of the conjugacy classes whose limit sets are $(1+\epsilon)$-quasicircles. Moreover,

$$S_{\epsilon}(M)=\underset{g\geq 2}{\cup}S(M,g,\epsilon).$$

Suppose $h$ is an arbitrary Riemannian metric on $M$. For any $\Pi\in S(M,g)$, we set

$$\text{area}_{h}(\Pi)=\inf\{\text{area}_{h}(\Sigma):\Sigma\in\Pi\}.$$

Then the minimal surface entropy with respect to $h$ is defined as follows.

(1.1)

$$E(h)=\underset{\epsilon\rightarrow 0}{\lim}\,\underset{L\rightarrow\infty}{\lim\inf}\,\dfrac{\ln\#\{\text{area}_{h}(\Pi)\leq 4\pi(L-1):\Pi\in S_{\epsilon}(M)\}}{L\ln L}.$$

According to [4] and [13], for odd-dimensional manifolds, and among metrics with sectional curvature less than or equal to $-1$, $E(h)$ attains the minimum at the hyperbolic metric $h_{0}$, and $E(h_{0})=2$. On the other hand, Theorem 1.1 of [19] shows that when $n=3$, $E(h)$ is maximized at $h_{0}$ among all metrics with scalar curvature greater than or equal to $-6$.

In the following results, we assume $(M,h_{0})$ is a closed hyperbolic manifold of dimension $n\geq 3$.

Note that such a neighborhood is “cylindrical”, that is, if $h\in\mathcal{U}$, then any metric $h^{\prime}$ in the conformal class of $h$ with $R_{h^{\prime}}\geq-n(n-1)$ also belongs to $\mathcal{U}$.

Let $(N,g_{0})$ be another closed hyperbolic manifold of the same dimension $n$, the theorem leads to the following corollary.

In particular, when $n=3$, we prove a more general result for any smooth map between homeomorphic 3-manifolds which is not necessarily a diffeomorphism. Let $(M,h_{0})$ and $(N,g_{0})$ be closed hyperbolic 3-manifolds.

In a different situation, suppose that $\pi_{1}(N)$ is isomorphic to an index $d$ subgroup $G<\pi_{1}(M)$. Let $\tilde{M}$ denote the covering space of $M$ with $\pi_{1}(\tilde{M})=G$ and let $p:\tilde{M}\rightarrow M$ denote the covering map, the hyperbolic metric on $\tilde{M}$ is still represented by $h_{0}$. We have the following corollary.

Besides, when the dimension of $M$ is an odd number, Hamenstädt [9] verified the existence of surface subgroups, and based on the result, the author discussed the corresponding minimal surface entropy and proved that $E(h_{0})=2$ in [13]. As a result, we can extend Theorem 1.2 of [19] to any odd dimensions at least 3.

This inequality can be considered as an analogue of the following comparison related to the volume entropy and geodesic stretch (Theorem 1.2 of [16]). More generally, suppose that $(M,h_{0})$ is a compact space of any dimension with negative curvature, and $h$ is another metric on $M$. It says that

$$I_{\mu_{0}}(h/h_{0})E_{vol}(h)\geq E_{vol}(h_{0}),$$

where $\mu_{0}$ is the Bowen-Margulis measure of $h_{0}$ which is the unique measure with maximal measure theoretic entropy. The equality holds if and only if the geodesic flows of $h$ and $h_{0}$ are time preserving conjugate after rescaling.

We briefly mention a similar result. Besson-Courtois-Gallot [3] showed that for any compact hyperbolic manifold (it also holds for compact locally symmetric spaces) $(M,h_{0})$ with dimension $n\geq 3$,

$$\big{(}\frac{\text{vol}_{h}(M)}{\text{vol}_{h_{0}}(M)}\big{)}^{\frac{1}{n}}E_{vol}(h)\geq E_{vol}(h_{0}),$$

the inequality is sharp unless $h$ is isometric to $h_{0}$ after rescaling. Calegari-Marques-Neves [4] conjectured the minimal surface analogue for closed hyperbolic 3-manifolds (also see [13] for higher odd-dimensional closed hyperbolic manifolds and locally symmetric spaces).

$$\big{(}\frac{\text{vol}_{h}(M)}{\text{vol}_{h_{0}}(M)}\big{)}^{\frac{2}{n}}E(h)\geq E(h_{0})=2,$$

the equality holds if and only if $h$ and $h_{0}$ are isometric up to scaling.

The author believes that Theorem 1.2 and Theorem 1.6 also hold for the other compact locally symmetric spaces of rank one, which include complex hyperbolic manifolds, quaternionic hyperbolic manifolds, and the Cayley plane. To check the details, we refer the readers to the construction of the surface subgroups of locally symmetric spaces and the discussion of the equidistribution property in [14] and the corresponding definition of the minimal surface entropy in Section 5 of [13].

The organization of this paper is as follows. Firstly, in Section 2, we introduce notations and definitions that are used throughout this paper. In Section 3, we discuss the average area ratio for dimension $n\geq 3$ and prove Theorem 1.2. In Section 4, we establish the equidistribution property and average area ratio formula which will be used in the last two sections. And in Section 5, we consider the special case for 3-manifolds and give the proof for Theorem 1.4. Finally, Section 6 concentrates on the minimal surface entropy defined on odd-dimensional manifolds, and the proof of Theorem 1.6 will be given.

Let $(M,h_{0})$ be a closed hyperbolic manifold of dimension $n\geq 3$, and let $(N,g)$ be another closed Riemannian manifold of the same dimension. Suppose that $F:(N,g)\rightarrow(M,h_{0})$ is a smooth map. For any $(x,L)$ in the Grassmannian bundle $Gr_{2}(N)$, $|\Lambda^{2}F(x,L)|_{g}$ denotes the Jacobian of $dF_{x}$ at plane $L$. Take an arbitrary regular value $y\in M$ of $F$, for $(y,P)\in Gr_{2}(M)$, we let

(2.1)

$$|\Lambda^{2}F|^{-1}_{g}(y,P)=\sum_{x\in F^{-1}(y)}\frac{1}{|\Lambda^{2}F(x,(dF_{x})^{-1}(P)|_{g}}.$$

This definition ([19]) is equivalent to Gromov’s definition ([8]):

$$|\Lambda^{2}F|^{-1}_{g}(y,P)=\sum_{x\in F^{-1}(y)}\lim_{\delta\rightarrow 0}\frac{\text{area}_{g}((dF_{x})^{-1}(D_{\delta}))}{\delta},$$

where $D_{\delta}$ is a subset of the totally geodesic disc $D\subset\mathbb{H}^{n}$ which is tangential to $P$ at $x$, and $D_{\delta}$ has area equal to $\delta$.

The average area ratio of $F$ is defined in [8] by

(2.2)

$$\text{Area}_{F}(g/h_{0})=\int_{(y,P)\in Gr_{2}(M)}|\Lambda^{2}F|^{-1}_{g}(y,P)\,d\mu_{h_{0}},$$

where $\mu_{h_{0}}$ stands for the unit volume measure on $Gr_{2}(M)$ with respect to the metric induced by $h_{0}$.

Let $(M,h_{0})$ be a closed hyperbolic manifold of dimension $n\geq 3$, and let $\mathcal{M}$ be the space of Riemannian metrics on $M$. The total scalar curvature (or Einstein-Hilbert functional) $\tilde{\mathcal{E}}:\mathcal{M}\rightarrow\mathbb{R}$ is

$$\tilde{\mathcal{E}}(h)=\int_{M}R_{h}\,dV_{h}.$$

It is a Riemannian functional in the sense that it’s invariant under diffeomorphisms, but it’s not scale-invariant. To resolve this issue, we consider

$$\mathcal{E}(h)=(\text{vol}_{h}(M))^{\frac{2}{n}-1}\int_{M}R_{h}\,dV_{h}.$$

This is called the normalized total scalar curvature (or normalized Einstein-Hilbert functional) of $M$. Under conformal deformations, the first variation of $\mathcal{E}$ is

$$\mathcal{E}^{\prime}(h)\cdot l=\frac{n-2}{2n}\text{vol}_{h}(M)^{\frac{2}{n}-1}\int_{M}\langle R_{h}-\fint_{M}R_{h}\,dV_{h},\,l\rangle_{h}\,dV_{h}.$$

It equals to zero provided that $(M,h)$ has constant scalar curvature. Assuming $R_{h}$ is constant, we can simplify the full variation to

$$\mathcal{E}^{\prime}(h)\cdot l=\text{vol}_{h}(M)^{\frac{2}{n}-1}\int_{M}\langle\frac{1}{n}R_{h}-Ric_{h},l\rangle_{h}\,dV_{h}.$$

Thus, a metric $h$ is critical if and only if $(M,h)$ is Einstein. In particular, the hyperbolic metric $h_{0}$ is a critical point for $\mathcal{E}$. Furthermore, since $\mathcal{E}$ is scale-invariant, from now on we may assume that for $h_{t}=h_{0}+tl$,

$$\int_{M}\frac{d}{dt}|_{t=0}(\sqrt{\text{det}_{h_{0}}(h_{t})})\,dV_{h_{0}}=\frac{1}{2}\int_{M}\text{tr}_{h_{0}}l\,dV_{h_{0}}=0.$$

Taking this into account, we obtain the second variation of $\mathcal{E}$ at $h_{0}$ as follows.

(2.3)		$\displaystyle\mathcal{E}^{\prime\prime}(h_{0})(l,l)=$	$\displaystyle\text{vol}_{h_{0}}(M)^{\frac{2}{n}-1}\int_{M}\frac{d^{2}}{dt^{2}}|_{t=0}R_{h_{t}}-2(n-1)\frac{d^{2}}{dt^{2}}|_{t=0}(\sqrt{\text{det}_{h_{0}}(h_{t})})$
		$\displaystyle+2\frac{d}{dt}|_{t=0}R_{h_{t}}\,\frac{d}{dt}|_{t=0}(\sqrt{\text{det}_{h_{0}}(h_{t})})\,dV_{h_{0}},$

Substituting the formulas

$$\frac{d^{2}}{dt^{2}}\sqrt{\text{det}_{h_{0}}(h_{t})}=\big{(}\frac{1}{4}(\text{tr}_{h_{t}}l)^{2}-\frac{1}{2}\text{tr}_{h_{t}}(l^{2})\big{)}\sqrt{\text{det}_{h_{0}}(h_{t})},$$

$$\frac{d}{dt}R_{h_{t}}=-\Delta_{h_{t}}(\text{tr}_{h_{t}}l)+\delta_{h_{t}}^{2}l-\langle Ric_{h_{t}},l\rangle_{h_{t}},$$

$$\frac{d}{dt}Ric_{h_{t}}=-\frac{1}{2}\Delta_{h_{t}}l+\frac{1}{2}Ric_{h_{t}}(l)+\frac{1}{2}l(Ric_{h_{t}})-Rm_{h_{t}}*l-\delta^{*}_{h_{t}}(\delta_{h_{t}}l)-\frac{1}{2}\nabla_{h_{t}}^{2}(\text{tr}_{h_{t}}l),$$

where $(Rm_{h_{t}}*l)_{ij}=R_{ikjm}l^{km}$, we obtain that

	$\displaystyle\mathcal{E}^{\prime\prime}(h_{0})(l,l)=$	$\displaystyle\text{vol}_{h_{0}}(M)^{\frac{2}{n}-1}\int_{M}\langle\frac{1}{2}\Delta_{h_{0}}l+Rm_{h_{0}}*l+\delta^{*}_{h_{0}}(\delta_{h_{0}}l)$
		$\displaystyle+\frac{n-1}{2}(\text{tr}_{h_{0}}l)h_{0}-\frac{1}{2}(\Delta_{h_{0}}(\text{tr}_{h_{0}}l))h_{0}+(\delta_{h_{0}}^{2}l)h_{0}\,,l\rangle_{h_{0}}\,dV_{h_{0}}.$

According to Ebin’s Slice Theorem (see [6]), for any $h\in\mathcal{M}$ lying in a small neighborhood of $h_{0}$, there exist $\phi\in\text{Diff}(M)$, $f\in C^{\infty}(M)$, and a transverse-traceless tensor $l_{TT}$, i.e., $\delta_{h_{0}}(l_{TT})=0$ and $\text{tr}_{h_{0}}(l_{TT})=0$, such that $\phi^{*}h=h_{0}+fh_{0}+l_{TT}$. Then we can simplify the second variation.

(2.4)

$$\mathcal{E}^{\prime\prime}(h_{0})(fh_{0},fh_{0})=-\frac{(n-1)(n-2)}{2}\text{vol}_{h_{0}}(M)^{\frac{2}{n}-1}\int_{M}\langle\Delta_{h_{0}}f-nf,\,f\rangle_{h_{0}}\,dV_{h_{0}}.$$

And we also get

(2.5)

$$\mathcal{E}^{\prime\prime}(h_{0})(l_{TT},l_{TT})=\text{vol}_{h_{0}}(M)^{\frac{2}{n}-1}\int_{M}\langle\frac{1}{2}\Delta_{h_{0}}l_{TT}+Rm_{h_{0}}*l_{TT},l_{TT}\rangle_{h_{0}}\,dV_{h_{0}}.$$

Thus,

	$\displaystyle\mathcal{E}(h)$	$\displaystyle=\mathcal{E}(\phi^{*}h)=\mathcal{E}(h_{0}+fh_{0}+l_{TT})$
		$\displaystyle=\mathcal{E}(h_{0})+\mathcal{E}^{\prime\prime}(h_{0})(fh_{0},fh_{0})+\mathcal{E}^{\prime\prime}(h_{0})(l_{TT},l_{TT})+\text{higher order variations}.$

Taking the estimates of (2.4) and (2.5) into consideration, we can use this expansion to discuss the local behavior of $\mathcal{E}$. In particular, as discussed in [2], $\mathcal{E}$ reaches a local maximum at $h_{0}$, and there exists $C>0$, so that for any metric $h$ in a small neighborhood of $h_{0}$ in $\mathcal{M}$,

$$\mathcal{E}(h_{0})-\mathcal{E}(h)\geq C\,d(h,h_{0})^{2},\quad\text{where }d(h,h_{0})=\underset{\phi\in\text{Diff}(M)}{\inf}|\phi^{*}h-h_{0}|_{H^{1}(M,h_{0})}.$$

The inequality is sharp unless $h$ and $h_{0}$ are isometric via some $\phi\in\text{Diff}(M)$.

Suppose that the hyperbolic manifold $M$ has an odd dimension $n\geq 3$. According to the construction by Hamenstädt [9], for any small number $\epsilon>0$, there is an essential surface $\Sigma_{\epsilon}$ in $M$ which is sufficiently well-distributed and $(1+\epsilon)$-quasigeodesic (i.e. the geodesics on the surface with respect to intrinsic distance are $(1+\epsilon,\epsilon)$-quasigeodesics in $M$). And as discussed in [13], $\Sigma_{\epsilon}$ determines an $(1+O(\epsilon))$-quasiconformal map on $S_{\infty}^{n-1}$, and thus associated with an element in $S_{\epsilon}(M)$.

Furthermore, let $G(M,g,\epsilon)$ denote the subset of $S(M,g,\epsilon)$ consisting of homotopy classes of finite covers of $\Sigma_{\epsilon}$ that have genus at most $g$. It has cardinality comparable to $g^{2g}$. Moreover, let $S_{i}$ denote the minimal representative of an element in $G(M,g_{i},\frac{1}{i})$, then it is homotopic to an $(1+\frac{1}{i})$-quasigeodesic surface $\Sigma_{i}$. From Lemma 4.3 of [4], for any continuous function $f$ on $M$, the unit Radon measure induced by integration over $S_{i}$ satisfies

(2.6)

$$\underset{i\rightarrow\infty}{\lim}\,\frac{1}{\text{area}_{h_{0}}(S_{i})}\int_{S_{i}}f\,dA_{h_{0}}=\nu(f),$$

where the limiting measure $\nu$ is positive on any non-empty open set of $M$.

Notice that the measure $\nu$ is not necessarily identified with the unit Radon measure on $M$ induced by integration over $M$, the latter measure is denoted by $\mu$. However, in this paper, we need to find a sequence of minimal surfaces whose Radon measures defined above converge to $\mu$. To solve this problem, we introduce Labourie’s construction [17] in Section 4. And we stress that both methods of Hamenstädt and Labourie require that $M$ has an odd dimension.

Firstly, given a metric $h$ on $M$, we look at the conformal class of $h$. Since $M$ admits a hyperbolic metric, every conformal class $[h]$ must be scalar negative, i.e., it has a metric with negative scalar curvature. And according to the Yamabe problem, after rescaling, there exists a unique metric $\bar{h}\in[h]$ with constant scalar curvature $c<0$. In Theorem 1.2, we assume that $c\geq-n(n-1)$. Set $\bar{h}^{\prime}=\frac{c}{-n(n-1)}\bar{h}$, so $R_{\bar{h}^{\prime}}\equiv-n(n-1)$, and for any surface or 2-dimensional subset $S$ in $M$, we have

$$\text{area}_{\bar{h}^{\prime}}(S)=\frac{c}{-n(n-1)}\text{area}_{\bar{h}}(S)\leq\text{area}_{\bar{h}}(S).$$

Therefore, to prove the theorem, we may assume that $R_{\bar{h}}\equiv-n(n-1)$.

Let $\mathcal{M}$ be the space of all Riemannian metrics on $M$, and let $\mathcal{M}_{R}$ be the subset consisting of metrics with constant scalar curvature $-n(n-1)$. From the previous section, it remains to consider metrics in $\mathcal{M}_{R}$.

Define a functional $\mathcal{A}$ from the space of Riemannian metrics on $M$ to $\mathbb{R}$ as follows.

	$\displaystyle\mathcal{A}(h)=$	$\displaystyle\int_{(x,P)\in Gr_{2}M}R_{h}(x)\,|\Lambda^{2}\text{Id}|^{-1}_{h}(x,P)\,d\mu_{h_{0}}$
	$\displaystyle=$	$\displaystyle\lim_{\delta\rightarrow 0}\int_{(x,P)\in Gr_{2}M}R_{h}(x)\,\frac{\text{area}_{h}(D_{\delta}(P))}{\delta}\,d\mu_{h_{0}}$
	$\displaystyle=$	$\displaystyle\lim_{\delta\rightarrow 0}\int_{(x,P)\in Gr_{2}M}R_{h}(x)\,\fint_{D_{\delta}(P)}\sqrt{\text{det}_{h_{0}}(h|_{D_{\delta}(P)})}\,dA_{h_{0}}\,d\mu_{h_{0}},$

where $D_{\delta}(P)$ is a subset of the totally geodesic disc in $\mathbb{H}^{n}$ that is tangential to $P$ at $x$, and $D_{\delta}(P)$ has hyperbolic area equal to $\delta$, $\mu_{h_{0}}$ is the unit volume measure on $Gr_{2}M$ with respect to the metric induced by $h_{0}$. Notice that the metric $h\in\mathcal{M}_{R}$ satisfies

	$\displaystyle\mathcal{A}(h)$	$\displaystyle=-n(n-1)\,\int_{(x,P)\in Gr_{2}M}|\Lambda^{2}\text{Id}|^{-1}_{h}(x,P)\,d\mu_{h_{0}}$
		$\displaystyle=\mathcal{A}(h_{0})\,\int_{(x,P)\in Gr_{2}M}|\Lambda^{2}\text{Id}|^{-1}_{h}(x,P)\,d\mu_{h_{0}}.$

Therefore, from the definition of the average area ratio (2.2), to deduce that if $h$ is in a small neighborhood of $h_{0}$ in $\mathcal{M}_{R}$, then

$$\text{Area}_{\text{Id}}(h/h_{0})=\int_{(x,P)\in Gr_{2}M}|\Lambda^{2}\text{Id}|^{-1}_{h}(x,P)\,d\mu_{h_{0}}\geq 1,$$

we only need to show that $\mathcal{A}$ attains a local maximum at $h_{0}$.

To see this, we’ll discuss the first and second variations of $\mathcal{A}$ in detail. Before start, we notice that the functional $\mathcal{A}$ is scale-invariant, so it suffices to assume $\text{vol}_{h_{0}}(M)=1$, and the symmetric tensor $l=h-h_{0}$ satisfies $\int_{M}\text{tr}_{h_{0}}l\,dV_{h_{0}}=0$. Let $h_{t}=h_{0}+tl$, we rewrite $\mathcal{A}(h_{t})$ as

$$\mathcal{A}(h_{t})=\int_{x\in M}R_{h_{t}}(x)\,a_{h_{t}}(x)\,dV_{h_{0}},$$

where

$$a_{h_{t}}(x)=\lim_{\delta\rightarrow 0}\fint_{P\in Gr_{2}M_{x}}\fint_{D_{\delta}(P)}\sqrt{\text{det}_{h_{0}}(h_{t}|_{D_{\delta}(P)})}\,dV_{h_{0}}d\nu_{h_{0}}.$$

Then we have

(3.5)

$$\frac{d}{dt}\mathcal{A}(h_{t})=\int_{M}\frac{d}{dt}R_{h_{t}}\,a_{h_{t}}+R_{h_{t}}\frac{d}{dt}a_{h_{t}}\,dV_{h_{0}}.$$

In addition,

(3.6)

$$\frac{d}{dt}R_{h_{t}}=-\Delta_{h_{t}}(\text{tr}_{h_{t}}l)+\delta_{h_{t}}^{2}l-\langle Ric_{h_{t}},l\rangle_{h_{t}},$$

where $\Delta_{h_{t}}$ is the rough Laplacian with negative eigenvalues, and $\delta_{h_{t}}^{2}$ is the double divergence operator. In particular, when $t=0$, applying the Stokes’ theorem, we have

$$\int_{M}\frac{d}{dt}|_{t=0}\,R_{h_{t}}\,a_{h_{0}}\,dV_{h_{0}}=\int_{M}-\langle Ric_{h_{0}},l\rangle_{h_{0}}\,dV_{h_{0}}=(n-1)\int_{M}\text{tr}_{h_{0}}l\,dV_{h_{0}}.$$

On the other hand, since

$$\frac{d}{dt}\sqrt{\text{det}_{h_{0}}(h_{t}|_{D_{\delta}(P)})}=\frac{1}{2}\text{tr}_{h_{t}}l|_{D_{\delta}(P)}\,\sqrt{\text{det}_{h_{0}}(h_{t}|_{D_{\delta}(P)})},$$

let $\lambda_{1},\cdots,\lambda_{n}$ be the eigenvalues of the matrix $l$ at point $x\in M$ with respect to $h_{0}$, we obtain by computation that

	$\displaystyle\frac{d}{dt}|_{t=0}\,a_{h_{t}}$	$\displaystyle=\lim_{\delta\rightarrow 0}\frac{1}{2}\fint_{P\in Gr_{2}M_{x}}\fint_{D_{\delta}(P)}\text{tr}_{h_{0}}l|_{D_{\delta}(P)}\,dA_{h_{0}}d\nu_{h_{0}}$
		$\displaystyle=\frac{1}{2}\,\frac{\sum_{i<j}\lambda_{i}+\lambda_{j}}{\binom{n}{2}}=\frac{1}{2}\,\frac{(n-1)\sum_{i=1}^{n}\lambda_{i}}{\binom{n}{2}}$
		$\displaystyle=\frac{1}{n}\text{tr}_{h_{0}}l.$

It follows that for any symmetric 2-tensor $l$,

$$\mathcal{A}^{\prime}(h_{0})\cdot l=(n-1)\int_{M}\text{tr}_{h_{0}}l\,dV_{h_{0}}-n(n-1)\frac{1}{n}\int_{M}\text{tr}_{h_{0}}l\,dV_{h_{0}}=0,$$

thus $h_{0}$ is a critical point of $\mathcal{A}$.