Toric non-archimedean $\mu$-entropy and thermodynamical structure

We prepare some terminologies convenient for our arguments. Let $V$ be an $n$-dimensional affine space over $\mathbb{R}$. A half space of $V$ is a subset of the form $\ell^{-1}([0,\infty))$ for some non-zero affine function $\ell:V\to\mathbb{R}$. The face of a half space $\ell^{-1}([0,\infty))$ is the hyperplane $\ell^{-1}(0)$.

We call a subset $P\subset V$ a polytope if it is a compact subset with non-empty interior and can be expressed as the intersection of finitely many half spaces. A subset $F\subset P$ is called a face of $P$ if it is the intersection of $P$ and the face of a half space which contains $P$. A face is again a polytope of a lower dimensional affine subspace $W\subset V$. The relative interior of a face is the interior in such $W$. A facet of $P$ is a face of $P$ which gives a polytope of a codimension one affine subspace. A vertex of $P$ is a face of $P$ with one element. A simple polytope is a polytope for which every vertex can be written as the intersection of precisely $n$-facets. In this article, we restrict our main interest to simple polytopes. In toric geometry, such (rational) polytopes represent toric orbifolds.

It is well-known that a complete parallel translation invariant measure on $V$ (with the usual $\sigma$-algebra) which puts positive finite mass on relatively compact open sets is a constant multiple of the Lebesgue measure under an affine identification $V\cong\mathbb{R}^{n}$. A flat measure on a polytope (resp. a face of a polytope) is the restriction of such translation invariant measure on $V$ (resp. such measure on the affine subspace $W$ which contains the face as a polytope). We note the topological boundary $\partial P=P\setminus P^{\circ}$ is the union of facets. A flat measure on $\partial P$ is a measure whose restriction to each facet is flat.

For a polytope $P$ and $p\geq 1$, we consider the following spaces:

which are independent of the choice of the flat measure $d\mu$ on $P$. Here lsc lc is short for lower semi-continuous log convex. Namely, we have $\liminf u(x_{i})\geq u(x_{\infty})$ for $x_{i}\to x_{\infty}\in P$ and $u((1-t)x_{0}+tx_{1})\leq u^{1-t}(x_{0})u^{t}(x_{1})$. For two lsc convex functions $q,q^{\prime}$, the condition $q=q^{\prime}$ almost everywhere with respect to $d\mu$ is equivalent to the condition $q=q^{\prime}$ everywhere. In particular, the boundary value $q|_{\partial P}$ can be recovered from interior values $q|_{P^{\circ}}$.

We endow $\mathcal{M}_{\mathrm{NA}}^{\exp,p}(P)$ with the $L^{p}$-topology and call its element (nonequilibrium) state. For $q\in\mathcal{E}_{\mathrm{NA}}^{\exp,p}(P)$, we can assign a state

For $q\in\mathcal{E}_{\mathrm{NA}}^{\exp,1}(P)$, we have $q\in\mathcal{E}_{\mathrm{NA}}^{\exp,p}(P)$ if and only if $u(q)\in\mathcal{M}_{\mathrm{NA}}^{\exp,p}(P)$. Later we explain a Kähler geometric background of the space $\mathcal{E}_{\mathrm{NA}}^{\exp,p}(P)$.

Now let us consider a triple $\mathcal{P}=(P,d\mu,d\sigma)$ of a simple polytope $P$, a flat measure $d\mu$ on $P$ and a flat measure $d\sigma$ on $\partial P$, which we call a simple non-archimedean Hamiltonian system or just a system. When we refer to $\mathcal{P}$ with general non-simple polytope $P$, we call it a general system.

For $u\in\mathcal{M}_{\mathrm{NA}}^{\exp,1}(P)$, we introduce

which we call the (nonequilibrium) entropy and (non-archimedean nonequilibrium) internal $\mu$-energy, respectively. For $T\in\mathbb{R}$, which we call temperature, we further introduce

We call this functional (non-archimedean nonequilibrium) free $\mu$-energy. We will see $S_{\mathcal{P}}(u)=-\infty$ implies $U_{\mathcal{P}}(u)=\infty$ and this definition makes the functional $F_{\mathcal{P}}(T,u(q))$ continuous along increasing sequence $q_{i}\nearrow q$. As we explain later, there is a Kähler geometric background for these functionals.

In section 4, we newly unveil there is a stochastic thermodynamical interpretation of these functionals, which gives us a strong motivation to our terminologies of thermodynamical flavor. We note some references (cf. [3]) in the same field use similar terminologies like entropy and free energy, but these are not directly related to ours. We are mostly interested in K-unstable case, while the reference focuses on K-stable case.

Now we state our main result. Its output to Kähler geometry is explained later.

This main theorem summarizes Theorem 3.3, Theorem 3.5 and Theorem 3.18. To show the existence, we establish the following compactness result, which is proved in section 2.2. The proof works not only for lsc lc functions but for general non-negative convex functions.

As an interest in Kähler geometry, we also prove the following. It summarizes Theorem 3.11 and Theorem 3.13. We explain terminologies in the next section.

We further prove some results in section 4 motivated by thermodynamics, using our existence result. To appreciate the results, we must recall the origin of $\mu$-cscK metric and the enigmatic way our parameter $T$ is introduced in the theory. We explain these in section 4. Though we present our main result as the existence and uniqueness of minimizer for the free $\mu$-energy $F_{\mathcal{P}}(T,\bullet)$, the observation in section 4 would be much deeper. It should be explored further in future study.

Let $T\cong\mathbb{G}_{m}^{\times n}$ be an algebraic torus and $T_{\mathrm{cpt}}\subset T$ be the maximal compact torus. For the character lattice $M=\mathrm{Hom}(T,\mathbb{G}_{m})=\mathrm{Hom}(T_{\mathrm{cpt}},U(1))$, we put $\mathfrak{t}^{\vee}:=M\otimes_{\mathbb{Z}}\mathbb{R}$. We denote the dual vector space by $\mathfrak{t}$ and identify it with the Lie algebra of $T_{\mathrm{cpt}}$.

Ler $(X,L)\circlearrowleft T$ be a polarized toric variety of dimension $n$. As usual in toric geometry, we assign a general system $\mathcal{P}=(P,d\mu,d\sigma)$ as follows. We put

Here $H^{0}(X,L)_{\mu}$ denotes the eigenspace $\{s\in H^{0}(X,L)~{}|~{}s.t=\mu(t)s~{}\forall t\in T\}$. The lattice $M$ defines a unique flat measure on $\mathfrak{t}^{\vee}$ by setting $\mu(B(m_{1},\ldots,m_{n}))=1$ for the box

generated by a $\mathbb{Z}$-basis $m_{1},\ldots,m_{n}\in M$. Since $|\det A|=1$ for $A\in GL(M)$, the flat measure is independent of the choice of $m_{1},\ldots,m_{n}$. We denote its restriction to $P$ by $d\mu$. On the other hand, since the affine subspace $W\subset\mathfrak{t}^{\vee}$ spanned by a facet $Q$ of $P$ is a rational subspace, the intersection $W\cap M$ gives a lattice spanning $W$. Then we similarly define a unique flat measure on each facet $Q$ compatible with the lattice $W\cap M$ and a flat measure $d\sigma$ on $\partial P$.

The flat measure $d\sigma$ on $\partial P$ represents the anti-canonical divisor $-K_{X}$. In general, flat signed measures on $\partial P$ are in one to one correspondence with $T$-invariant $\mathbb{R}$-Weil divisors: each facet $Q\subset\partial P$ represents a $T$-invariant prime divisor $E_{Q}$ and a flat signed measure $d\sigma^{\prime}$ on $\partial P$ represents the divisor $\sum_{Q}\frac{d\sigma^{\prime}(Q)}{d\sigma(Q)}E_{Q}$. For the flat signed measure $d\sigma^{\prime}$ representing a $T$-invariant $\mathbb{R}$-Weil divisor $\Delta=\sum_{Q}a_{Q}E_{Q}$ on $X$, we put

which represents the $\mathbb{R}$-divisor $-(K_{X}+\Delta)$. From thermodynamical perspective, the measure $d\sigma$ plays the role of Hamiltonian of the system and its variation can be interpreted as thermodynamical work.

A triple $(X,\Delta,L)$ of a toric variety $X$, a $T$-invariant $\mathbb{R}$-divisor $\Delta$ with coefficient $a_{Q}\in[0,1)$ and a $T$-equivariant polarization $L$ is called a polarized log toric variety. By the above construction, we can assign a general system $\mathcal{P}=(P,d\mu,d\sigma_{\Delta})$ for a polarized log toric variety.

The associated polytope $P$ is known to be simple if and only if $X$ has at most orbifold singularity. See [13, Theorem 3.1.19].

Let $X$ be a compact Kähler manifold and $L\in H^{2}(X,\mathbb{R})$ be a Kähler class. We consider a pair $(\omega,f)$ of a Kähler metric (form) $\omega$ in $L$ and a smooth real valued function $f$ on $X$. For $\lambda\in\mathbb{R}$, a pair $(\omega,f)$ is called $\mu^{\lambda}$-cscK metric if it satisfies the following two conditions:

1. (1)

   The complex vector field $\partial^{\sharp}_{\omega}f=g^{i\bar{j}}f_{,\bar{j}}\partial_{i}$ is holomorphic.

2. (2)

   The $\mu^{\lambda}$-scalar curvature $s^{\lambda}(\omega,f):=s(\omega)+\Delta_{\omega}f-|\partial_{\omega}^{\sharp}f|^{2}-\lambda f$ is constant.

When the imaginary part $\xi=-2\mathrm{Im}\partial^{\sharp}_{\omega}f$ is specified, we call the metric $\omega$ $\mu^{\lambda}_{\xi}$-cscK metric. Since $i_{\xi}\omega=-2\mathrm{Im}i_{\partial^{\sharp}_{\omega}f}\omega=-2\mathrm{Im}g(J\partial^{\sharp}_{\omega}f,\cdot)=-df$, $f$ is the Hamiltonian potential of $\xi$. The imaginary part $\xi$ preserves the metric $g$, so it is a Killing vector field. This implies we can always take a closed torus $T_{\mathrm{cpt}}$ acting on $(X,L)$ so that $\xi\in\mathfrak{t}=\mathrm{Lie}(T_{\mathrm{cpt}})$. We note there is a subtle but an important difference between $\mu^{\lambda}$-cscK metric and $\mu^{\lambda}_{\xi}$-cscK metric: when we refer to $\mu^{\lambda}$-cscK metric, the vector $\xi$ is not fixed.

The notion unifies two frameworks of canonical metrics: constant scalar curvature Kähler (cscK) metric and Kähler–Ricci soliton. Indeed, when $X$ is a Fano manifold (i.e. the anti-canonical class $-K_{X}=c_{1}(X)$ is positive), a Kähler metric $\omega$ in the anti-canonical Kähler class $T^{-1}K_{X}$ ($T<0$) is $\mu^{-2\pi T}$-cscK metric if and only if it is Kähler–Ricci soliton: $\mathrm{Ric}(\omega)-\mathscr{L}_{\partial^{\sharp}f}\omega=-2\pi T\omega$. Kähler–Ricci soliton is defined only for Fano manifolds due to the a priori constraint

while $\mu^{\lambda}$-cscK metric makes sense for general polarized manifold.

A fundamental question on $\mu^{\lambda}$-cscK metric is the existence and uniqueness modulo translation by automorphisms. When we fix $\xi$, the theory of $\mu^{\lambda}_{\xi}$-cscK metric is enclosed in the theory of weighted cscK metric. It is then proved by [28] that $\mu^{\lambda}_{\xi}$-cscK metric is unique modulo translation by aumorphisms preserving $\xi$. As for the uniqueness of $\mu^{\lambda}$-cscK metric, we must show the uniqueness of $\xi$ (modulo translation) which admits a $\mu^{\lambda}_{\xi}$-cscK metric. We confirm this uniqueness for toric manifolds with $\lambda\leq 0$.

On the other hand, fixing $\xi$, the existence of $\mu^{\lambda}_{\xi}$-cscK metric implies an algebro-geometric condition of $(X,L)$ called $\mu^{\lambda}_{\xi}$K-stability.

The notion can be defined for general polarized scheme, but here for simplicity we only explain a toric version of the notion. It can be simply described in terms of convex functions.

When $\xi=0$, the invariant is independent of $\lambda$:

It is called the Donaldson–Futaki invariant.

We note the $\mu^{\lambda}_{\xi}$-Futaki invariant is well-defined for general integrable convex function $q$ on general system $\mathcal{P}$. For a set $\mathcal{Q}$ of convex functions, we call $\mathcal{P}$ toric $\mu^{\lambda}_{\xi}$K-semistable (resp. $\mu^{\lambda}_{\xi}$K-polystable) with respect to $\mathcal{Q}$ if $\mathrm{Fut}^{\lambda}_{\xi}(q)\geq 0$ for every $q\in\mathcal{Q}$ (resp. if further $\mathrm{Fut}^{\lambda}_{\xi}(q)>0$ for non-affine $q$). We call $\mathcal{P}$ K-unstable if it is not $\mu_{0}$K-semistable. We note even if $X$ is K-unstable, it may be $\mu^{\lambda}_{\xi}$K-semistable with respect to some $\lambda$ and non-trivial $\xi\neq 0$.

By the result of [1] (see also [27, 23]), it is known that a polarized manifold $(X,L)$ is $\mu^{\lambda}_{\xi}$K-polystable if it admits a $\mu^{\lambda}_{\xi}$-cscK metric. When $\xi=0$, the case of cscK metric, it is also known that a uniform version of toric K-polystability implies the existence of cscK metric for toric manifold by [19, 10, 29]. As far as the author knows, this implication for general $\xi\neq 0$ is still unkown. It is natural to expect the same method for $\xi=0$ would work. For general polarized manifold, it is still a conjecture even for cscK metric, which is known as YTD conjecture, but recently there are great progress (cf. [9, 29, 30, 7]). We do not pursue this direction in this article as we are rather interested in K-instability.

Now we return to $\mu$-cscK metric. Originally, $\mu$-cscK metric is introduced in [22] based on a moment map picture on Kähler–Ricci soliton observed in [21]. The moment map picture heuristically explains why the existence of $\mu^{\lambda}_{\xi}$-cscK metric is related to $\mu^{\lambda}_{\xi}$K-stability and moduli problem, so this perspective is important in stability aspect. We review this later in order to explain how the parameter $\lambda$ appears.

On the other hand, it turns out later in [24] that $\mu^{\lambda}$-cscK metric is also characterized by Perelman’s entropy. This perspective illuminates “instability aspect” of the theory of $\mu$-cscK metric and $\mu$K-stability, which is the main interest of this article.

Now we introduce Perelman’s functionals. For a Kähler metric $\omega$ and a real valued function $f$ normalized as $\int_{X}e^{f}\mathrm{vol}_{\omega}=\int_{X}\mathrm{vol}_{\omega}$ for $\mathrm{vol}_{\omega}:=\omega^{n}/n!$, we put

which we call Perelman’s $\mathcal{W}$-entropy and $\mu$-entropy, respectively. We note the above convention differs from Perelman’s original one by sign and multiplication of constants, which is off course not essential. Perelman considered general Riemannian metric for $\mathcal{W}^{\lambda}$, but we restrict our interest to Kähler metric in the context of $\mu^{\lambda}$-cscK metric, which is in turn essential for our variational result.

It is proved in [24] that we can characterize $\mu^{\lambda}$-cscK metrics $(\omega,f)$ as the critical points of $\mathcal{W}^{\lambda}$. We note here we must restrict metrics to the space $\mathcal{H}(X,L)$ of Kähler metrics in $L$. Moreover, for $\lambda\leq 0$, we can also characterize $\omega$ of $\mu^{\lambda}$-cscK metrics $(\omega,f)$ as minimizers of $\bm{\mu}^{\lambda}_{\mathrm{Per}}$. This characterization is nice as our differential geometric interest is in the metric $\omega$ and not so much in the function $f$ or the vector field $\xi=-2\mathrm{Im}\partial_{\omega}f$.

Recall we fix vector $\xi$ to define $\mu^{\lambda}_{\xi}$K-stability, while now we have a vector-free characterization of $\mu^{\lambda}$-cscK metric. To match up, it is better to have an algebro-geometric criterion for $\xi$ which tells if $(X,L)$ could be $\mu^{\lambda}_{\xi}$K-semistable. However, in general, there may be no such vector. In such case, we are interested in finding “most destabilizing $q$” in a suitable sense. It is revealed in [25] that the nature of these two questions are actually the same. We can formalize and solve both questions in terms of toric non-archimedean $\mu$-entropy, which is essentially the functional $F_{\mathcal{P}}(T,\bullet)$ in our main theorem.

Let $\mathcal{P}$ be the general system associated to a polarized toric variety $(X,L)$. For a lsc convex function $q\in\mathcal{E}_{\mathrm{NA}}^{\exp,1}(P)$, we put

For $\lambda\in\mathbb{R}$, we consider

and call it the toric non-archimedean $\mu$-entropy.

These are essentially the functionals we already introduced:

The differential of toric non-archimedean $\mu$-entropy $\bm{\check{\mu}}_{\mathrm{NA}}^{\lambda}$ gives the minus of $\mu^{\lambda}_{\xi}$-Futaki invariant:

This implies that if $\bm{\check{\mu}}_{\mathrm{NA}}^{\lambda}$ is maximized at a vector $\ket{\xi}$, then $(X,L)$ is toric $\mu^{\lambda}_{\xi}$K-semistable.

It is proved in [24] that there is an inequality between $\bm{\check{\mu}}_{\mathrm{NA}}^{\lambda}$ and $\bm{\mu}_{\mathrm{Per}}^{\lambda}$ for $\lambda\in\mathbb{R}$:

Moreover if $\lambda\leq 0$ and there exists a $\mu^{\lambda}_{\xi}$-cscK metric $\omega$, then the above equality is achieved by $\bm{\check{\mu}}_{\mathrm{NA}}^{\lambda}(\ket{\xi})=\bm{\mu}_{\mathrm{Per}}^{\lambda}(\omega)$. This implies if there exists a $\mu^{\lambda}_{\xi}$-cscK metric for $\lambda\leq 0$, then $\ket{\xi}$ maximizes $\bm{\check{\mu}}_{\mathrm{NA}}^{\lambda}$, which in particular gives another proof for the $\mu^{\lambda}_{\xi}$K-semistability of $\mu^{\lambda}_{\xi}$-cscK manifold with $\lambda\leq 0$. It is conjectured the equality holds for $\lambda\leq 0$ even when there is no $\mu^{\lambda}$-cscK metric.

The non-archimedean $\mu$-entropy $\bm{\check{\mu}}_{\mathrm{NA}}^{\lambda}$ is essentially introduced in [23] for $q\in\mathcal{H}_{\mathrm{NA}}(P)$ and more generally for a test configuration $(\mathcal{X},\mathcal{L})\in\mathcal{H}_{\mathrm{NA}}(X,L)$ of a non-toric polarized variety $(X,L)$. The paper [25] introduces a completion $\mathcal{E}_{\mathrm{NA}}^{\exp}(X,L)$ of the space $\mathcal{H}_{\mathrm{NA}}(X,L)$ of test configuraitons in terms of non-archimedean pluripotential theory developed in [5]. The space $\mathcal{E}_{\mathrm{NA}}^{\exp}(X,L)$ consists of functions on the Berkovich analytification of $X$, which is the reason for the word “non-archimedean”. In the toric setup, the $T$-invariant part of the space $\mathcal{E}_{\mathrm{NA}}^{\exp}(X,L)$ corresponds to the intersection $\bigcap_{p\geq 1}\mathcal{E}_{\mathrm{NA}}^{\exp,p}(P)$. It takes considerable pages to extend the non-archimedean $\mu$-entropy to $\mathcal{E}_{\mathrm{NA}}^{\exp}(X,L)$ as an upper semi-continuous functional, which in our toric case turns extremely easy to show. In the toric case, $\bm{\check{\mu}}_{\mathrm{NA}}^{\lambda}(q)$ is continuous along increasing sequence by the monotone convergence theorem, so we may replace $\sup_{q\in\mathcal{H}_{\mathrm{NA}}(X,L)}\bm{\check{\mu}}_{\mathrm{NA}}^{\lambda}(q)$ in (1.17) with $\sup_{q\in\mathcal{E}_{\mathrm{NA}}^{\exp,1}(P)}\bm{\check{\mu}}_{\mathrm{NA}}^{\lambda}(q)$.

Theorem 1.1 confirms that there always exists a maximizer $q\in\mathcal{E}_{\mathrm{NA}}^{\exp,\frac{n}{n-1}}(P)$ of $\bm{\check{\mu}}_{\mathrm{NA}}^{\lambda}$, even when the toric variety $(X,L)$ is $\mu^{\lambda}$K-unstable. If $q$ is a bounded convex function, we can assign a filtration $\mathcal{F}_{q}=\mathcal{F}_{\varphi_{q}}$ on the graded ring $R(X,L)=\bigoplus_{m}H^{0}(X,L^{\otimes m})$ as in [25]. If further the filtration is finitely generated, which is the case for instance when $q\in\mathcal{H}_{\mathrm{NA}}(P)$, it produces an algebro-geometric degeneration $(\mathcal{X},\mathcal{L})/B_{\sigma}$ of $(X,L)$ called polyhedral configuration in [25]. Then we can show that the central fibre $(\mathcal{X}_{0},\mathcal{L}_{0})$ of such degeneration is $\mu^{\lambda}_{\xi}$K-semistable with respect to the vector $\xi$ generated by the degeneration. See [25] for further detail.

One ultimate goal of our study is to construct a degeneration of $\mu$K-unstable polarized variety to a $\mu$K-semistable “space” (see Remark 3.17), which is often referred to as optimal degeneration. The non-archimedean $\mu$-entropy is not the unique quantity measuring K-instability. We briefly review other works on K-instability as reference.

$\bullet$ Optimal degeneration in the context of Kähler–Ricci soliton

The first pioneering work on optimal degeneration in the context of Kähler–Ricci soliton on Fano manifold would be Chen–Wang’s work [12]. They constructed a metric geometric degeneration along Kähler–Ricci flow. Chen–Sun–Wang [11] gives an algebro-geometric description of the degeneration, in which the idea of using filtration and two step degeneration is utilized. Dervan–Székelyhidi [14] introduced a quantity called $H$-invariant and showed Chen–Sun–Wang’s filtration minimizes the $H$-invariant. Han–Li [18] pursued an algebro-geometric aspect of $H$-invariant and proved the uniqueness of Chen–Sun–Wang’s degeneration. Finally, Blum–Liu–Xu–Zhuang [4] constructed Chen–Sun–Wang’s degeneration in a purely algebro-geometric way, which allows them to extend the result to Fano variety with singularity. It is known by Wang–Zhu’s work [34] that every toric Fano manifold admits Kähler–Ricci soliton. This result can also be regarded as the explicit description of optimizer in toric case: the optimizer for $H$-invariant is given by the vector associated to Kähler–Ricci soliton. In [25], it turns out that the non-archimedean $\mu$-entropy also characterizes Chen–Sun–Wang’s degeneration.

$\bullet$ Optimal degeneration in the context of extremal metric

Optimal degeneration problem of polarized variety would be firstly considered by Donaldson [16] in the context of extremal metric. The quantity for K-instability in this context is called normalized Donaldson–Futaki invariant. In toric case, Székelyhidi [33] proved the existence of optimal destabilizing $L^{2}$-integrable convex function. Xia [35] pursued the problem from pluripotential theoretic perspective and proved the existence of an optimal destabilizing geodesic ray in a suitable completion $\mathcal{E}^{2}(X,L)$ of the space of Kähler metrics. It turns out by Li’s analysis [29] that the geodesic ray can be interpreted as a non-archimedean metric. Compared to Kähler–Ricci soliton, much less is known on the regularity of optimal destabilizer. We speculate this context can be treated as the limit $\lambda\to-\infty$ of our framework.

$\bullet$ Optimal degeneration in the context of Mabuchi soliton:

Optimal destabilization problem for Mabuchi soliton would be firstly considered by Hisamoto [20]. The quantity for K-instability in this context is called normalized Ding invariant. In toric case, Yao [36] constructed an optimal destabilizing convex function for normalized Ding invariant. He also proved the crucial regularity result: the optimizer is piecewise affine. This is much better than the current knowledge on the optimizer for normalized Donaldson–Futaki invariant and non-archimedean $\mu$-entropy.

$\bullet$ Still many other works…

There are still many other works on optimal degeneration. Among all, the structure of the above three frameworks is closest to that of our framework. For instance, we have Donaldson type inequality (cf. [16, 14, 20, 35, 24]) which can be interpreted as minimax theorem on some functional (cf. [24, 25]). Thus here we do not give further comments on those other works in the context of $\delta$-invariant and the normalized volume.

This work is supported by RIKEN iTHEMS Program. We used Wolfram Mathematica to create some graphs.

We put

For $i=1,\ldots,p$, we put

and consider the $(n-1)$-dimensional Lebesgue measure $d\bm{x}^{\prime}$.