K\"ahler-Ricci flow on horospherical manifold

TL;DR

This paper demonstrates that the Kähler-Ricci flow on smooth Fano horospherical manifolds converges to a Kähler-Ricci soliton, establishing existence through geometric flow analysis.

Contribution

It proves the existence of Kähler-Ricci solitons on all smooth Fano horospherical manifolds via flow convergence analysis.

Findings

Renormalized Kähler-Ricci flow converges in Cheeger-Gromov sense.

Limit of flow is a Kähler-Ricci soliton.

Existence of solitons on Fano horospherical manifolds established.

Abstract

In this paper, we prove the existence of a Kahler Ricci soliton on any smooth Fano horospherical manifold by a study of the Kahler-Ricci flow. Indeed, we prove that the renormalized Kahler Ricci flow converges in the sense of Cheeger Gromov and that this limit is a Kahler-Ricci soliton.

Equations

$$\mathrm{Ric}({\omega }_g)-{\omega }_g={\mathcal{L}}_X{\omega }_g,$$

$$\mathfrak{g}=\mathfrak{k}\oplus J\mathfrak{k},$$

$$\mathfrak{g}=\mathfrak{t}\oplus \underset{\alpha \in \Phi }{⨁}{\mathfrak{g}}_{\alpha }$$

$$\mathfrak{b}=\mathfrak{t}\oplus \underset{\alpha \in {\Phi }^{+}}{⨁}{\mathfrak{g}}_{\alpha }.$$

$$\mathfrak{p}=\mathfrak{t}\oplus \underset{\alpha \in {\Phi }^{+}\cup {\Phi }_I}{⨁}{\mathfrak{g}}_{\alpha }.$$

$$\mathfrak{q}=\mathfrak{t}\oplus \underset{\alpha \in {\Phi }^{-}\cup {\Phi }_I}{⨁}{\mathfrak{g}}_{\alpha }.$$

$$\mathfrak{l}=\mathfrak{t}\oplus \underset{\alpha \in {\Phi }_L}{⨁}{\mathfrak{g}}_{\alpha }.$$

$$2{\rho }_P:=\sum _{\alpha \in {\Phi }_P^{+}}\alpha .$$

$$G/H⟶G/P.$$

$$\mathfrak{h}={\mathfrak{t}}_0\oplus \underset{\alpha \in {\Phi }_P}{⨁}{\mathfrak{g}}_{\alpha },\text{ where }{\mathfrak{t}}_0=\mathfrak{t}\cap \mathfrak{h}.$$

$$(g,pH)\cdot xH=gx{p}^{-1}H.$$

$$\forall a\in \mathfrak{a},(t_{\chi },a)=⟨\chi ,a⟩.$$

$$\mathfrak{C}=\{\chi \in \Lambda :\forall \alpha \in {\Phi }^{+},⟨\chi ,{\alpha }^{\vee }⟩\ge \}.$$

$${\mathrm{Aut}}_G(X)\simeq {\mathrm{Aut}}_G(G/H)\simeq P/H.$$

$$(p,ph)\cdot \xi =\chi (p)\xi ,\forall \xi \in L_{eH}.$$

$$s:g\in G\mapsto g\cdot s(e)\in ({\pi }^{\ast }L{)}_g.$$

$$s_r:pH\in P/H\mapsto (p,eH)\cdot s^{\prime }(e),$$

$$s_l:pH\in P/H\mapsto (e,p^{-1}H)\cdot s^{\prime }(e),$$

$$\forall p\in P,s_r(pH)=\chi (p)s_l(pH).$$

$${\pi }^{\ast }L\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\forall x\in X,\forall \xi \in L_x,|\xi {|}_q^2=e^{-\psi (x)}|\xi {|}_{q^0}^2.$$

$${\omega }_{q^0}={\omega }_q+\sqrt{-1}\partial \overline{\partial }\psi .$$

$$\varphi :g\in G\mapsto -2\ln |s(g){|}_{{\pi }^{\ast }q}\in \mathbb{R}.$$

$$u:x\in {\mathfrak{a}}_1\mapsto -2\ln |s_r(\exp (x)H){|}_{{\iota }^{\ast }q}\in \mathbb{R}.$$

$$\varphi (k\exp (x)h)=u(x)-2\ln |\chi (\exp (x)h)|.$$

$$R\xi :pH\mapsto (H,p^{-1}H)\cdot \xi .$$

$${\omega }_{\exp (x)H}=\sum _{1\le j_1,j_2\le r}\frac{1}{4}\frac{{\partial }^{2}u}{\partial l_{j_1}\partial l_{j_2}}(x)\sqrt{-1}{\gamma }_{j_1}\wedge {\overline{\gamma }}_{j_2}+\sum _{\alpha \in {\Phi }_P^{+}}⟨\alpha ,\frac{1}{2}\nabla u(x)-t_{\chi }⟩\sqrt{-1}{\gamma }_{\alpha }\wedge {\overline{\gamma }}_{\alpha }$$

$${\omega }_{\exp (x)H}^n=\frac{{\mathrm{MA}}_{\mathbb{R}}(u)(x)}{4^r{2}^{\mathrm{Card}({\Phi }_P^{+}})}\prod _{\alpha \in {\Phi }_P^{+}}⟨\alpha ,\nabla u(x)-2t_{\chi }⟩\Omega ,$$

$$\Omega:=\Big{(}\bigwedge_{1\leq j\leq r}\gamma_{j}\wedge\overline{\gamma}_{j}\Big{)}\wedge\Big{(}\bigwedge_{\alpha\in\Phi^{+}_{P}}\gamma_{\alpha}\wedge\overline{\gamma}_{\alpha}\Big{)}.$$

$${\omega }_{\exp (x)H}^n=\frac{{\mathrm{MA}}_{\mathbb{R}}(u)(x)}{4^r{2}^{\mathrm{Card}({\Phi }_P^{+}})}\prod _{\alpha \in {\Phi }_P^{+}}⟨\alpha ,\nabla u(x)-2t_{\chi }⟩s_r^{-1}\wedge {\overline{s_r}}^{-1}.$$

$$S:=\Big{(}\bigwedge_{1\leq j\leq r}\gamma_{i}\Big{)}\wedge\Big{(}\bigwedge_{\alpha\in\Phi^{+}_{P}}\gamma_{\alpha}\Big{)},$$

Full text

Keywords: Kähler–Ricci soliton, horospherical manifold, Monge–Ampère equation, Kahler-Ricci flow.

AMS codes: 53C55, 58E11, 53C44, 14M27, 14J45

1. Introduction

The founding paper on the Kähler-Ricci solitons is Hamilton’s article [Ham88]. They are natural generalizations of Kähler-Einstein metrics and appear as fixed points of the Kähler-Ricci flow. On a Fano compact Kähler manifold $M$, a Kähler metric $g$ is a Kähler-Ricci soliton if its Kähler form $\omega_{g}$ satisfies :

[TABLE]

where $Ric(\omega_{g})$ is the Ricci form of $g$ and ${\mathcal{L}}_{X}\omega_{g}$ is the Lie derivative of $\omega_{g}$ along a holomorphic vector field $X$ on $M$. As usual, we denote the Kähler-Ricci soliton by the pair $(g,X)$ and $X$ is called the solitonic vector field. Note that if $X=0$ then $g$ is a Kähler-Einstein metric. When $X\neq 0$, we say that the Kähler-Ricci soliton is non-trivial. In order ti recover the original definition, we denote by $(\sigma_{t})_{t\in[0,+\infty[}$ the family of diffeomorphisms generated by $\frac{1}{2(1+t)}X$ and the Kähler form $\tilde{\omega}=(1+t)\,\sigma_{t}^{*}\omega_{g}$ satisfies the equation of the Ricci flow $\frac{\partial}{\partial t}\,\tilde{\omega_{t}}=-\operatorname{Ric}(\tilde{\omega}_{t})$.

The first study of the solitonic vector field $X$ was done in the paper [TZ00, TZ02]. Thanks to the Futaki function, the authors discovered an obstruction to the existence of Kähler-Ricci soliton and proved that $X$ is in the center of a reductive Lie subalgebra $\eta_{r}(M)$ of Lie algebra $\eta(M)$ of all holomorphic vector fields. This study also gives us a uniqueness result about Kähler-Ricci soliton (Theorem 0.1 in [TZ00]). Subsequently, the study was developped by Wang and Zhu in [WZ04] where they show the existence of Kähler-Ricci solitons on toric manifolds using the continuity method. Another wau to prove the existence of a Kähler-Ricci soliton is by a study of the Kähler-Ricci flow. Indeed, in [TZ06], it was proved that if a Fano compact Kähler manifold admits a Kähler-Ricci soliton then the renormalized Kähler-Ricci flow converges in the sense of Cheeger-Gromov to a Kähler-Ricci soliton. Zhu showed in [Zhu12] that the Kähler-Ricci flow converges to a Kähler-Ricci soliton on a toric manifold without the a priori assumption of existence of Kähler-Ricci soliton. Recently, this technique has been also extended in [Hua17] to toric fibration. In this paper, we extend this result to horospherical manifolds.

Theorem 1.1**.**

Let $M$ be a Fano horosphercal manifold with associated horospherical homegeneous space $G/H$ where $G$ is the complexification of a maximal compact subgroup $K$. The solution $\omega_{t}$ defined for $t\in[0,+\infty[$ of the Kähler-Ricci flow with inital value a $K$-invariant metric $\omega_{0}$ converges in the Cheeger-Gromov sense to a Kähler form $\omega_{\infty}$ when $t$ tends to $+\infty$ where $\omega_{\infty}$ is a Kähler-Ricci soliton.

Our paper is divided into three sections. The first one makes horospheric geometry reminders, it is inspired [Del16]. The second one gives reminders about the Kähler-Ricci solitons and proves the existence of a solitonic vector field whose Futaki invariant vanishes on every horospherical manifold. Finally, in a last part, the main result of the article is proved by studying the Kähler-Ricci flow.

The author would like to thank F. Paulin for his help for the redaction of this paper.

2. Horospherical Varieties

In this section, we give some reminders on the theory of algebraic groups and on horospherical varieties. A good reference for the group theory part is [Spr98]. For the notion of horospherical variety, we refer to [Pas09, Tim11].

2.1. Reductive group

Let $G$ be a reductive connected linear complex algebraic group. We denote by ${\mathfrak{g}}$ its Lie algebra. If $K$ is a maximal compact subgroup of $G$ with Lie algebra ${\mathfrak{k}}$ then

[TABLE]

where $J$ is the complex structure of ${\mathfrak{g}}$. Fix a maximal torus $T$ of a Borel subgroup $B$ of $G$. Denote by $\Phi\subset{\mathfrak{X}}(T)$ the root system of $(G,T)$ where ${\mathfrak{X}}(T)$ is the group of algebraic characters of $T$. We have the root space decomposition:

[TABLE]

where for any $\alpha\in{\mathfrak{X}}(T)$, ${\mathfrak{g}}_{\alpha}:=\{x\in{\mathfrak{g}}~{}:~{}\forall h\in{\mathfrak{t}}~{}~{}\operatorname{ad}(h)(x)=\alpha(h)x\}$, so that ${\mathfrak{g}}_{\alpha}$ is a complex line if and only if $\alpha\in\Phi$.

Let $\Phi^{+}:=\Phi^{+}(B)$ be the set of positive roots (associated with $B$) so that the Lie algebra ${\mathfrak{b}}$ of $B$ satisfies

[TABLE]

We then define the negative roots $\Phi^{-}:=\Phi^{-}(B)=-\Phi^{+}$ of $\Phi$ associated with $B$ so that $\Phi=\Phi^{+}\sqcup\Phi^{-}.$ Let $B^{-}$ be the unique Borel subgroup of $G$ called the opposite Borel subgroup of $B$ with respect to $T$ verifying $B\cap B^{-}=T$. Note that $\Phi^{+}(B^{-})=\Phi^{-}(B)$.

Denote by $\Sigma$ the set of simple roots as the set of roots in $\Phi^{+}$ that cannot be written as the sum of two elements of $\Phi^{+}$. For any subset $I$ of $\Sigma$, if we give a subset $I$, let $\Phi_{I}$ be the subset of $\Phi$ generated by the roots contained in $I$. The parabolic group containing $B$ with respect to $I$, denoted by $P:=P_{I}$, is the connected closed subgroup of $G$ whose Lie algebra is

[TABLE]

Let $\Phi_{P}:=\Phi^{+}\cup\Phi_{I}.$ The parabolic subgroup opposed to $P$, denoted by $Q:=Q_{I}$, is the parabolic subgroup associated with $I$ for the Borel subgroup $B^{-}$ i.e. the connected closed Lie subgroup with Lie algebra

[TABLE]

Moreover $L=P\cap Q$ is a Levi subgroup of $P$ and we define $\Phi_{L}$ by

[TABLE]

. Let $\Phi^{+}_{P}$ be the set of roots that are not in $\Phi_{Q}=\Phi^{-}\cup\Phi_{I}$ so that $\Phi^{+}_{P}\sqcup\Phi_{Q}=\Phi.$ Note that $\Phi^{+}_{P}$ is the set of roots of the unipotent radical $U$ of $P$, that $\Phi=\Phi^{+}_{P}\sqcup\Phi_{I}\sqcup\Phi_{Q}^{+},$ and that $\Phi^{+}_{Q}=-\Phi^{+}_{P}.$ Finally, we define

[TABLE]

2.2. Horospherical subgroups and homogeneous horospherical spaces

If $H$ a closed connected algebraic subgroup of $G$, then $H$ is said to be a horospherical subgroup of $G$ if $H$ contains the unipotent radical $U$ of a Borel subgroup $B$. We can build horospherical subgroups using parabolic subgroups. Let $N_{G}(H):=\{g\in G~{}:~{}gHg^{-1}=H\}$ be the normalizer of $H$ in $G$.

Proposition 2.1** ([Pas06]).**

Let $H$ be a horospherical subgroup of $G$. Then $P:=N_{G}(H)$ is a parabolic subgroup of $G$ containing the Borel subgroup $B$, and the quotient $P/H=T/(T\cap H)$ is a torus. Conversely, if $H$ is a connected closed algebraic subgroup of $G$ such that $N_{G}(H)$ is a parabolic subgroup $P$ of $G$ and $P/H$ is a torus, then $H$ is horospherical. The fibration

[TABLE]

is a torus fibration over a generalized flag manifold with fiber $P/H$.

We obtain the following decomposition of the Lie algebra ${\mathfrak{h}}$ of $H$:

[TABLE]

If $H$ is a horospherical subgroup of $G$, then $G/H$ is called a homogeneous horospherical space. On $G/H$, the normalizer $P:=N_{G}(H)$ acts by right multiplication by the inverse and $H$ acts trivially. We define an action of $G\times P/H$ on $G/H$ by

[TABLE]

Note that the isotropy group of $eH$ is $\{(p,pH)\in G\times P/H~{}:~{}p\in P\}$. This group is called $diag(P)$ and is isomorphic to $P$ by the first projection.

2.3. Character groups and one parameter subgroup

Let ${\mathfrak{a}}={\mathfrak{t}}\cap J{\mathfrak{k}}.$ We have an identification between ${\mathfrak{N}}(T)\otimes_{\mathbb{Z}}{\mathbb{R}}$ and ${\mathfrak{a}}$, where ${\mathfrak{N}}(T)$ is the group of algebraic one-parameter subgroups $\lambda:{\mathbb{C}}^{\times}\rightarrow T$ of $T$, given by the derivative at point $1$ of the restriction of $\lambda$ to ${\mathbb{R}}^{+}_{*}$. Since, $T\cap H$ is a subtorus of $T$, the image ${\mathfrak{N}}(T\cap H)$ defines a sublattice of ${\mathfrak{N}}(T)$ corresponding to the one-parameter subgroups having values in $T\cap H$. With ${\mathfrak{a}}_{0}={\mathfrak{N}}(T\cap H)\otimes_{\mathbb{Z}}{\mathbb{R}}$, we have ${\mathfrak{t}}_{0}={\mathfrak{a}}_{0}\oplus J{\mathfrak{a}}_{0}.$

Recall that the Killing form $\kappa$ of ${\mathfrak{g}}$ defines a scalar product $(\cdot,\cdot)$ on ${\mathfrak{a}}\cap[{\mathfrak{g}},{\mathfrak{g}}]$. In addition, $\kappa$ is egal to zero on ${\mathfrak{z}}({\mathfrak{g}})$. Thus we can define a global scalar product on ${\mathfrak{a}}$ by taking a scalar product invariant by the Weyl group $W$ on ${\mathfrak{a}}\cap{\mathfrak{z}}({\mathfrak{g}})$ and assuming that ${\mathfrak{a}}\cap{\mathfrak{z}}({\mathfrak{g}})$ and ${\mathfrak{a}}\cap[{\mathfrak{g}},{\mathfrak{g}}]$ are orthogonal. Let ${\mathfrak{a}}_{1}$ be the orthogonal of ${\mathfrak{a}}_{0}$ for the scalar product $(\cdot,\cdot)$ so that ${\mathfrak{t}}={\mathfrak{a}}_{0}\oplus{\mathfrak{a}}_{1}\oplus J{\mathfrak{a}}_{0}\oplus J{\mathfrak{a}}_{1}$ and ${\mathfrak{p}}/{\mathfrak{h}}\simeq{\mathfrak{a}}_{1}\oplus J{\mathfrak{a}}_{1}$.

Finally, we recall that there is a natural pairing $\langle\cdot,\cdot\rangle$ between ${\mathfrak{N}}(T)$ and ${\mathfrak{X}}(T)$ defined by $\chi\circ\lambda(z)=z^{\langle\lambda,\chi\rangle}$ for all $\lambda\in{\mathfrak{N}}(T),\chi\in{\mathfrak{X}}(T)$ and $z\in{\mathbb{C}}^{\times}$. In addition, the natural pairing between $\Lambda={\mathfrak{X}}(T)\otimes_{\mathbb{Z}}{\mathbb{R}}$ and ${\mathfrak{N}}(T)\otimes_{\mathbb{Z}}{\mathbb{R}}$ obtained by ${\mathbb{R}}$-linearity can be seen as $\langle\chi,a\rangle=\ln\chi(\exp a)$ for all $\chi\in{\mathfrak{X}}(T)$ and $a\in{\mathfrak{a}}\simeq{\mathfrak{N}}(T)\otimes_{\mathbb{Z}}{\mathbb{R}}$. We can also identify $\Lambda$ with ${\mathfrak{a}}^{*}$. Since $(\cdot,\cdot)$ is a scalar product on ${\mathfrak{a}}$, for $\chi\in{\mathfrak{X}}(T)$, we denote by $t_{\chi}$ the unique element of ${\mathfrak{a}}$ such that

[TABLE]

For every $\alpha\in\Phi$, let $e_{\alpha}$ be a generator of the complex line ${\mathfrak{g}}_{\alpha}$ such that $[e_{\alpha},e_{-\alpha}]=t_{\alpha}$. Let us end this section by recalling the polar decomposition.

Proposition 2.2** ([Del16]).**

The image of ${\mathfrak{a}}_{1}$ in $G$ under the exponential is a fundemental domain for the action of $K\times H$ on $G$, where $K$ acts by multiplication on the left and $H$ by multiplication on the right by the inverse. As a consequence, the set $\{\exp(a)H~{}:~{}a\in{\mathfrak{a}}_{1}\}$ is a fundamental domain for the action of $K$ on $G/H$.

2.4. Horospherical variety

Recall that a complex algebraic $G$-variety is a reduced finite type scheme over ${\mathbb{C}}$ with an algebraic action of $G$. A normal complex algebraic $G$-variety $X$ will be said to be $G$-spherical if it admits an open and dense $B$-orbit. Note that $X$ is then connected. A $G$-spherical variety $X$ will be called horospherical if the stabilizer $H$ in $G$ of a point $x$ in the open and dense $B$-orbit is horospherical. We then will say that $(X,x)$ is a horospherical embedding. In particular, $X$ has $G/H$ as a dense open subset. Two horospherical embeddings $(X,x)$ and $(X^{\prime},x^{\prime})$ are *isomorphic * if there is an $G$-equivariant isomorphism from $X$ to $X^{\prime}$ sending $x$ to $x^{\prime}$. In this paper, we will assume that $X$ is smooth. Thanks to the GAGA theorems (see [Ser56]), the variety $X$ is therefore a Fano projective complex manifold and in particular $X$ is a connected compact Fano Kähler manifold for the metric induced by the Fubini-Study metric of the projective space.

We fix a homogeneous horospherical space $G/H$. Recall that there is a unique element $\alpha^{\vee}\in{\mathfrak{X}}(T)^{*}\simeq{\mathfrak{N}}(T)$ such that $\langle\alpha,\alpha^{\vee}\rangle=2$. With our previous notations, we have $\alpha^{\vee}=\frac{2}{\|t_{\alpha}\|^{2}}\,t_{\alpha}.$ Define $a_{\alpha}\in{\mathbb{Z}}$ by $a_{\alpha}=\langle 2\rho_{P},\alpha^{\vee}\rangle$ and * Weyl’s dominant closed chamber* ${\mathfrak{C}}$ by

[TABLE]

and the open dominant Weyl chamber as the interior of the closed dominant Weyl chamber. The semigroup of dominant weights is defined by $\Lambda^{+}:={\mathfrak{X}}(T)\cap{\mathfrak{C}}$.

We have the following definition introduced in [Pas06].

Definition 2.3**.**

Let $G/H$ be a homogeneous horospherical space. A convex polytope ${\mathcal{Q}}$ of ${\mathfrak{a}}\simeq{\mathfrak{N}}(T)\otimes_{\mathbb{Z}}{\mathbb{R}}$ is said to be $G/H$-reflective if we have the following three conditions:

1. (1)

${\mathcal{Q}}$ has its vertices in ${\mathfrak{N}}(T)\cup\left\{\frac{\alpha^{\vee}}{a_{\alpha}}~{}:~{}\alpha\in\Phi^{+}_{P}\right\}$ and contains [math] in its interior, 2. (2)

the dual polytope ${\mathcal{Q}}^{*}$ has its vertices in ${\mathfrak{X}}(T)$, 3. (3)

for all $\alpha\in\Phi^{+}_{P}$, we have $\cfrac{\alpha^{\vee}}{a_{\alpha}}\in{\mathcal{Q}}$.

The moment polytope $\Delta^{+}\subset{\mathfrak{a}}^{*}$ with respect to the Borel subgroup $B$ of the horospherical manifold $X$ is the Kirwan’s moment polytope of the Kähler manifold $(X,\omega)$ for the action of a maximum compact subgroup $K$ of $G$, where $\omega$ is a $K$-invariant Kähler form in $2\pi\,c_{1}(X)$ (see [Bri87, KMG84] for more details). We then have the following result.

Proposition 2.4** ([Pas06]).**

Let $G/H$ be a homogeneous horospherical space. There exists a bijection between theset of Fano horospherical embedding of $G/H$ and the set of $G/H$-reflective polytopes ${\mathcal{Q}}$ in ${\mathfrak{a}}$. In addition, the polytope $2\rho_{P}+{\mathcal{Q}}^{*}$ is the moment polytope with respect the Borel subgroup $B$ of the horospherical embedding. The assumption $(3)$ of Definition 2.3 is then equivalent to the fact that $\Delta^{+}=2\rho_{P}+{\mathcal{Q}}^{*}$ is included in ${\mathfrak{C}}$. In particular, we see that $0\in\operatorname{Int}({\mathcal{Q}}^{*})$ and that therefore $2\rho_{P}\in\operatorname{Int}(\Delta^{+})$.

We fix a horospherical embedding $(X,x)$ of $G/H$ and we denote $P:=N_{G}(H)$. By taking the restriction to the open $B$-orbit, we have an isomorphism between the $G$-equivariant automorphisms of $X$ and those of $G/H$ :

[TABLE]

One can consult [Kno91] which deals with the problem in the more general context of spherical varieties.

2.5. Associated linearized line bundle

In this section, we introduce associated linearize line bundle over homogeneous horospherical space. They were first introduced by Delcroix in [Del16]. Let $H$ a be horospherical subgroup of $G$ and $P=N_{G}(H)$.

Definition 2.5**.**

A line bundle $\Pi:L\rightarrow G/H$ is $(G\times P/H)$-linearized if there exists an action $\Theta:(G\times P/H)\times L\rightarrow L$ denoted by $(x,y)\mapsto\Theta_{x}(y)$ such that

• $\bullet$

$\Pi:L\rightarrow G/H$* is a $(G\times P/H)$-equivariant morphism,*

• $\bullet$

the application induced by the action of $\Theta_{{(e,pH)}}$ between the fibers is linear.

Note that to any $(G\times P/H)$-linearized line bundle, we can associate a character of $P$. Indeed, for any $p\in P$, the action of $(p,pH)\in diag(P)$ is trivial on the trivial class in $G/H$. Thus, $\Theta_{(p,pH)}$ induces a linear isomorphism between $L_{eH}$ and $L_{eH}$ and therefore a linear representation of dimension 1 of $diag(P)\simeq P$ i.e. there is a character $\chi\in{\mathfrak{X}}(P)$ such that

[TABLE]

Let us consider the projection $\pi:G\longrightarrow G/H$. We can define the pulled-back line bundle $\pi^{*}L$ over $G$. Since $\pi$ is $G$-equivariant, the line bundle $\pi^{*}L$ admits a $G$-linearization. In particular, we define a global section $s$ on $\pi^{*}L$ by chosing an element $s(e)\in(\pi^{*}L)_{e}$ and setting

[TABLE]

Now let us consider the inclusion $\iota:P/H\longrightarrow G/H$. We can define the restriction $\iota^{*}L$ of the line bundle $L$. Since $\iota$ is equivariant for the action of $P\times P/H$, we obtain that $\iota^{*}L$ is a $(P\times P/H)$-linearized bundle. Two global sections of $\iota^{*}L$ can also be defined:

[TABLE]

where the element $s^{\prime}(e)\in(\iota^{*}L)_{e}$ is such that $s^{\prime}(e)$ and $s(e)$ are mapped to the same element of $L_{eH}$ by the canonical applications $\pi^{*}L\rightarrow L$ and $\iota^{*}L\rightarrow L$. Note that these sections are linked by the formula:

[TABLE]

We have the following commutative diagram

[TABLE]

Given a Hermitian metric $q$ on a complex line bundle $L$ over a complex manifold $M$ and a local trivialisation $s$ of $L$ over an open subset $U$ of $M$, we define the local potential of $q$ with respect to $s$ by $\varphi:x\in U\mapsto-\ln a_{x}$ where $a_{x}=|s(x)|_{q}^{2}$. We can associate, to any Hermitian metric $q$, a $(1,1)$-form $\omega_{q}$ called the curvature of $q$ by $\omega_{q}|_{U}=\sqrt{-1}\partial\overline{\partial}\varphi$ where $\varphi$ is the local potential. One checks that $\omega_{q}|_{U}$ does not depend on the local trivialisation and therefore defines a $(1,1)$-global form. In addition, one can prove that $\omega_{q}\in 2\pi\,c_{1}(L)$. We will also say that $L$* has positive curvature* if there is a metric $q$ such that $\omega_{q}$ is a Kähler form. Fix a reference Hermitian metric $q^{0}$ on $L$ and for any Hermitian metric $q$, we define a smooth function $\psi$ on $X$, called the global potential of $q$ with respect to $q^{0}$ by

[TABLE]

By definition of the $(1,1)$-forms $\omega_{q}$ and $\omega_{q^{0}}$ and by computing in local charts, we see that the function $\psi$ satisfies the following relation:

[TABLE]

We refer to [Dem] for more details.

Let $G/H$ be a homogeneous horospherical space, $L$ a $(G\times P/H)$-linearized line bundle over $G/H$ and $q$ a Hermitian $K$- invariant metric on $L$. We can then consider the Hermitian metric $\pi^{*}q$ on $\pi^{*}L$ and define the local potential $\phi$ (which is actually defined on the whole $G$) with respect to the section $s$ of Equation (4) :

[TABLE]

We also define the potential $u:{\mathfrak{a}}_{1}\rightarrow{\mathbb{R}}$ with respect to tje section $s$ of $\iota^{*}L$ defined in Equation (4):

[TABLE]

We have the following relation between these two potentials, using the character $\chi$ defined in equation (3).

Proposition 2.6** (Propostion 2.7 in [Del16]).**

For all $k\in K$, $x\in{\mathfrak{a}}_{1}$ and $h\in H$, we have

[TABLE]

2.6. Curvature in horospherical case

Let us now, recall Delcroix’s computation of the curvature of a Hermitian metric on a line bundle over $G/H$ in an adapted basis. The first step is to define this basis. For this, we identify the tangent space of $G/H$ at $eH$ with ${\mathfrak{g}}/{\mathfrak{h}}\simeq\oplus_{\alpha\in\Phi^{+}_{P}}{\mathfrak{a}}_{1}\oplus J{\mathfrak{a}}_{1}\oplus{\mathbb{C}}e_{-\alpha}$ . We get a complex basis of the tangent space $T_{eH}(G/H)$ as the concatenation of a real basis $(l_{j})_{1\leq j\leq r}$ of ${\mathfrak{a}}_{1}$ with $(e_{-\alpha})_{\alpha\in\Phi^{+}_{P}}$. On $P/H$, we can define for $\xi\in T_{eH}(G/H)$ the holomorphic vector field:

[TABLE]

We then have a complex basis of $T^{1,0}(P/H)$ given by $(Rl_{j}-\sqrt{-1}JRl_{j})/$2 for $1\leq j\leq r$ and $(Re_{-\alpha}-\sqrt{-1}JRe_{-\alpha})/$2 for all $\alpha\in\Phi^{+}_{P}$, and we denote by $(\gamma_{j})_{1\leq j\leq r}\cdot(\gamma_{\alpha})_{\alpha\in\Phi^{+}_{P}}$ the dual basis.

Theorem 2.7** (Theorem 2.8 in [Del16]).**

Let $\omega$ be the curvature $(1,1)$-form of a $K$-invariant Hermitian metric $q$ on a $(G\times P/H)$-linearized line bundle $L$ over $G/H$, whose associated character is denoted by $\chi$. The form $\omega$ is determined by its restriction to $P/H$, given for any $x\in{\mathfrak{a}}_{1}$ by

[TABLE]

where $\nabla u(x)\in{\mathfrak{a}}_{1}$ is the gradient of the function $u$ defined by Equation (7) for the scalar product $(\cdot,\cdot)$. In addition, with $\operatorname{MA}_{\mathbb{R}}(u)=\det(\operatorname{Hess}(u))$,

[TABLE]

where $n$ is the dimension of $G/H$ and

[TABLE]

Moreover, by choosing $L=K_{G/H}^{-1}$ $s(e)$ appropriately, with $s_{r}(e)$ defined in equation (5), we have

[TABLE]

In order to explain the choice of $s(e)$, let

[TABLE]

which therefore defines a section of the line bundle $\iota^{*}K_{G/H}$. In particular, we have $S|_{eH}\in(\iota^{*}K_{G/H})_{eH},$ and using the following isomorphisms

[TABLE]

we denote via these isomorphisms $s(e):=S|_{eH}$. We then obtain, by the definition of $\Omega$ and $s_{r}$ (Equations (8) and (5)), that

[TABLE]

and we can conclude. We will constantly use this choice afterwards.

In this section, we consider a horospherical manifold $X$ with reductive group $G$ and horospherical subgroup $H$. Note $P=N_{G}(H)$ the normalizer of $H$ in $G$. Recall that $P$ is a parabolic subgroup and that there is another parabolic subgroup $Q$ such that there is a Levi subgroup $L$ with $P\cap Q=L$. From now, we consider that the parabolic subgroup $P$ contains the opposite Borel subgroup $B^{-}$ i.e. $\Phi^{+}_{P}=\Phi^{-}\backslash\Phi_{L}=-(\Phi^{+}\backslash\Phi_{L})$. Recall that we introduced the moment polytope $\Delta^{+}$ of $X$ with respect to the Borel subgroup $B$. Now, we define

[TABLE]

and the support function $v_{2\Delta}$ by

[TABLE]

where $(\cdot,\cdot)_{e}$ is the usual euclidian scalar product on ${\mathfrak{a}}^{*}$ (for any basis of ${\mathfrak{a}}_{1}$). This function satisfies the following properties:

• $\bullet$

$\forall x\in{\mathfrak{a}}_{1}^{*},~{}~{}\forall\alpha\in{\mathbb{R}}^{+}_{*},~{}~{}v_{2\Delta}(\alpha\,x)=\alpha v(x)$

• $\bullet$

$\forall(x,y)\in({\mathfrak{a}}_{1}^{*})^{2},~{}~{}v_{2\Delta}(x+y)\leq v_{2\Delta}(x)+v_{2\Delta}(y)$

• $\bullet$

$\forall x\in{\mathfrak{a}}_{1}^{*},~{}~{}v(x)\leq d\,\|x\|$ where $d=\sup_{p\in 2\Delta}\|p\|$ and $\|\cdot\|$ the norm on ${\mathfrak{a}}$ associated with the scalar product $(\cdot,\cdot)_{e}$

In addition, if $x\in{\mathfrak{a}}_{1}^{*}$ is such that $\|x\|=1$, then $2\Delta$ is contained in the half-space $\{y\in{\mathfrak{a}}_{1}^{*}~{}:~{}(x,y)\leq v_{2\Delta}(x)\}$ and at least one point of $2\Delta$ is in the border of this half-space i.e. in $\{y\in{\mathfrak{a}}_{1}^{*}~{}~{}:~{}(x,y)=v_{2\Delta}(x)\}$.

Proposition 2.8** (Proposition $4.2$ in [Del16]).**

Let $q$ be $K$-invariant Hermitian metric with positive curvature on the line bundle $K_{X}^{-1}$ and we denote $\chi$ the character of the restriction of $K_{X}^{-1}$ in $G/H$ and let $u:{\mathfrak{a}}_{1}\rightarrow{\mathbb{R}}$ be the convex potential defined by Equation (7). Then $u$ is a smooth and strictly convex function such that the application $du:x\mapsto d_{x}u\in{\mathfrak{a}}_{1}^{*}$ verifies $\text{im}(du)=2\Delta$ and the function $u-v_{2\Delta}$ is bounded on ${\mathfrak{a}}_{1}$. In particular, the polytope $2\,\Delta$ is independent of the chosen metric $q$.

Since $2\rho_{Q}=-2\rho_{P}$ belongs to $\operatorname{Int}(\Delta^{+})$ by Proposition 2.4, we have

[TABLE]

Recall that $\Delta^{+}\subset{\mathfrak{C}},$ where ${\mathfrak{C}}$ is the Weyl chamber for the Borel subgroup $B$ (see Proposition 2.4). Thus

[TABLE]

The latter can still be written

[TABLE]

Since $\Phi^{+}_{P}\subset\Phi^{+}(B^{-})=-\Phi^{+}(B)$, we then have

[TABLE]

Hence by compactness of $\Delta^{+}$, there exists $f^{\prime}>0$ such that

[TABLE]

3. Kähler-Ricci solitons in the horospherical case

3.1. Definition

Let $(M,\omega)$ be a compact Kähler manifold. Recall that the Kähler metric $g$ and the Kähler form $\omega$ can be written locally

[TABLE]

[TABLE]

Now, the Ricci form is the real $(1,1)$-form defined locally by

[TABLE]

Recall that it can be written globally as

[TABLE]

and it satisfies

[TABLE]

Recall that a compact Kähler manifold is called Fano if its first Chern Class is positive i.e. $c_{1}(M)>0$. On a Fano compact Kähler manifold $M$, a Kähler metric $g^{\prime}$ is a Kähler-Ricci soliton if its Kähler form $\omega_{g^{\prime}}$ satisfies :

[TABLE]

where ${\mathcal{L}}_{X}\omega_{g^{\prime}}$ is the Lie derivative of $\omega_{g^{\prime}}$ along a holomorphic vector field $X$ on $M$. Usually, we denote the Kähler-Ricci soliton by the pair $(g^{\prime},X)$ and $X$ is called the solitonic vector field. We immediately note that if $X=0$ then $g^{\prime}$ is a Kähler-Einstein metric. When $X\neq 0$, we say that the Kähler-Ricci soliton is non-trivial. By abuse, we will say that $g^{\prime}$ or $\omega_{g^{\prime}}$ is a Kähler-Ricci soliton if there exists a holomorphic vectors field on $M$ such that $(g,X)$ is Kähler-Ricci soliton.

3.2. Horospherical case

Let us fix a connected Fano compact Kähler manifold $M$. Let us recall (see for instance [Gau] for details on real and complex vector fields) that the group of complex automorphisms $\operatorname{Aut}(M)$ of $M$ is a finite dimensional Lie group whose Lie algebra is the set, denoted $\eta^{\mathbb{R}}(M)$, of holomorphic real vector fields (see Theorem 1.1 in Chapter $3$ of [Kob12]). If $K$ is a maximal compact subgroup of the connected identity component $\operatorname{Aut}^{\circ}(M)$ of $\operatorname{Aut}(M)$ then we have, see [Fuj78], that

[TABLE]

where $\operatorname{Aut}_{r}(M)$ is a reductive subgroup of $\operatorname{Aut}^{\circ}(M)$ and the complexification of $K$ and $R_{u}$ the unipotent radical of $\operatorname{Aut}^{\circ}(M)$. In addition, if we denote by $\eta^{\mathbb{R}}(M)$, $\eta_{r}^{\mathbb{R}}(M)$, $\eta_{u}^{\mathbb{R}}(M)$ and ${\mathfrak{k}}$ the Lie algebras of $\operatorname{Aut}(M),\operatorname{Aut}_{r}(M),R_{u}$ and $K$ respectively, then we have

[TABLE]

Recall that $X\in\eta^{\mathbb{R}}(M)$ if and only if ${\mathcal{L}}_{X}\omega$=0. Moreover, if we denote by $\eta(M)$ the Lie algebra of the complex holomorphic vector fields i.e. the holomorphic sections of the complex vector bundle $T^{1,0}M$, then there is an isomorphism between $\eta^{\mathbb{R}}(M)$ and $\eta(M)$ given by $X\mapsto X^{1,0}$. If we denote by $\eta_{r}(M)$ and $\eta_{u}(M)$ the image of $\eta_{r}^{\mathbb{R}}(M)$ and $\eta_{u}^{\mathbb{R}}(M)$ respectively by the previous isomorphism, we then obtain a decomposition

[TABLE]

Assume that $(M,x)$ is a horospherical embedding under the action of the reductive group $\operatorname{Aut}_{r}(M)=:G$ and that $x\in M$ is such that its isotropy group $H$ in $G$ is a horospherical subgroup in $G$ containing the unipotent radical of the opposite Borel subgroup $B^{-}$. Let $P:=N_{G}(H)$. Using the decompositions in section 2, with ${\mathfrak{g}}=\eta^{{\mathbb{R}}}_{r}(M)$ the Lie algebra of $G$, we have

[TABLE]

In particular, the Lie algebra of the Lie group $\operatorname{Aut}_{G}(M)$ of the $G$-equivariant automorphisms of $M$ is identified with the Lie algebra of $P/H$ which corresponds to the factor ${\mathfrak{a}}_{1}\oplus J{\mathfrak{a}}_{1}$ in the previous decomposition. Note that, by definition of $\operatorname{Aut}_{G}(M)$, this Lie algebra also identifies with the center ${\mathfrak{z}}({\mathfrak{g}})$ of ${\mathfrak{g}}$.

We fix a Riemanian metric $g_{0}$ with Kähler form $\omega_{0}\in 2\pi\,c_{1}(M)$ on M. By the $\partial\overline{\partial}$-lemma, there is a unique function $h$ in ${\mathcal{C}}^{\infty}(M,{\mathbb{R}})$ such that

[TABLE]

In addition, if we fix a Hermitian metric $m^{0}$ on $K_{M}^{-1}$ such that $\omega_{m^{0}}=\omega_{g^{0}}$, then we can define a volume $dV$ on $M$ given in a local trivialisation $s$ of $K_{M}^{-1}$ by

[TABLE]

where $\varphi$ is the local potential with respect to the trivialization $s$. Up to an additive constant, $h$ is the logarithm of the potential of $dV$ with respect to $\omega^{n}_{g^{0}}$, so we renormalize in order to have

[TABLE]

Indeed, by writing locally Equality (19), we get that

[TABLE]

where $\varphi$ is the local potential in this open set. This last equation can finally be written globally

[TABLE]

We conclude by using the maximum principle.

3.3. Determination of the solitonic vector field

The first step in order to prove the existence of a Kähler-Ricci soliton is to determine the solitonic vector field. To do this, we use the Futaki invariant. We have the following result (see proposition $2.1$ in [TZ02]):

Proposition 3.1**.**

There exists a unique complex holomorphic vector field $X\in\eta_{r}(M)$ with $\operatorname{Im}(X)\in{\mathfrak{k}}$ such that the Futaki invariant of $X$, noted $F_{X}:\eta(M)\rightarrow{\mathbb{C}}$, vanishes on $\eta_{r}(M)$. Moreover, $X$ is equal to zero or belongs to the center of $\eta_{r}(M)$, and we have

[TABLE]

In particular $F_{X}(\cdot)$ is a Lie character on $\eta_{r}(M)$.

By applying this theorem to the horospherical case, we obtain the following result

Proposition 3.2**.**

The vector field $X\in\eta_{r}(M)={\mathfrak{g}}^{1,0}$ given by proposition 3.1 has the following form:

[TABLE]

Proof.

See proposition 3.2 of [Del17] ∎

4. Convergence of the Kähler-Ricci Flow

4.1. Preliminaries

Let $M$ a connected compact Kähler manifold. A familly $(\omega_{t})_{t\in I}$ of Kähler form on $M$ defined on a interval $I$ of ${\mathbb{R}}$ containing[math] is called a solution of the Kähler-Ricci flow with initial condition $\omega_{0}$ if

[TABLE]

If we assume that $M$ is a Fano manifold i.e. $c_{1}(M)>0$ then we renormalize the Kähler-Ricci flow by the change of variables $\tilde{\omega}_{t}=e^{t}\,\omega_{1-e^{-t}}$ and we obtain the renormalized Kähler-Ricci flow :

[TABLE]

This renormalization enables the flow to verify interesting properties

Lemma 4.1**.**

Let $M$ a connected Fano compact Kähler manifold. If $(\omega_{t})_{t\in[0,T[}$ a solution of the renormalized Kähler-Ricci flow (22) with initial value $\omega_{0}\in 2\pi\,c_{1}(M)$ on a interval $[0,T[$ where $T>0$ then $\omega_{t}\in 2\pi\,c_{1}(M)$ for all $t\in[0,T[$. Moreover, we have

• $\bullet$

for all $t\in[0,T[$,

[TABLE]

• $\bullet$

for all $t\in[0,T[$, there exists a function $\varphi_{t}\in{\mathcal{C}}^{\infty}(M,{\mathbb{R}})$ such that

[TABLE]

Proof.

By taking the cohomology class of Equation (22), we obtain the ordinary first order differential equation

[TABLE]

with the intial value $[\omega(0)]=2\pi\,c_{1}(M)$ so solving this equation gives us $[\omega(t)]=2\pi\,c_{1}(M)$. The first point is hence a consequence of the Stokes theorem and the last point is a consequence of the $\partial\overline{\partial}$-lemma. ∎

Let $g_{0}$, $\omega_{0}$ and $\eta(M)$ be as in section 3. By the Hodge theory, for all $X\in\eta(M)$ there exists a unique function $\theta_{X}\in{\mathcal{C}}^{\infty}(M,{\mathbb{C}})$ satisfies

[TABLE]

and so, by the Cartan formula,

[TABLE]

Lemma 4.2** (Proof of Proposition 2.1 in [TZ02]).**

Let $\varphi\in{\mathcal{C}}^{\infty}(M,{\mathbb{R}})$ such that $\omega_{\varphi}:=\omega_{0}+\sqrt{-1}\partial\overline{\partial}\varphi$ is a Kähler form. The function $\theta_{X}(\omega_{\varphi})$ associated with the vector field $X\in\eta(M)$ for the metric $\omega_{\varphi}$ satisfies

[TABLE]

i.e. we get

[TABLE]

For the rest of the paper, in order to simplify the equations, we write $\theta_{X}$ for $\theta_{X}(\omega_{0})$ and we will precise the metric if the situation requires it. Moreover, we have the well-known following result (for example [BEG13] for a proof).

Proposition 4.3**.**

Let $(M,\omega_{0})$ be a connected Fano compact Kähler manifold. A familly $(\omega_{t})_{t\in[0,T[}$ is a solution of the renormalized Kähler-Ricci flow (22) if and only if there exists a familly $(\varphi_{t})_{t\in[0,T[}$ of smooth functions on $M$ satisfies

[TABLE]

such that

[TABLE]

Moreover, the solution of (22) (and equivalently for (27)) exists for $t\in[0,+\infty[$ and are unique.

We observe that $\frac{\partial\varphi_{t}}{\partial t}|_{t=0}=h$. Let $h_{t}:M\rightarrow{\mathbb{C}}$ be defined by Equation (19) with $\omega_{0}$ replaced by $\omega_{t}$. Now, we recall a deep estimate due to Perelman. We can consult [TZ06, ST08] for a proof.

Lemma 4.4**.**

Let $(\varphi_{t})_{t\in[0,+\infty[}$ a familly of solutions of the equation (27). By assuming the constant $c_{t}$ such that $h_{t}=-\frac{\partial\varphi_{t}}{\partial t}+c_{t}$ satisfies

[TABLE]

there exists a constant $A$ independent of the time $t$ such that

[TABLE]

Now, we assume that $M$ is as in Section 3 and that $\omega_{0}$ is a $K$-invariant Kähler form. If we denote by $m_{0}$ a $K$-invariant Hermitian form on $K_{M}^{-1}$ such that $\omega_{0}=\omega_{m^{0}}$, then there exists a convex potentials $u_{0}\in{\mathcal{C}}^{\infty}({\mathfrak{a}}_{1},{\mathbb{R}})$ (see Equation (7)) i.e we have

[TABLE]

Moreover, because the Kähler-Ricci flow perserves the $K$-invariance, we obtain that $\omega_{t}$ admit also a convex potentials $u_{t}\in{\mathcal{C}}^{\infty}({\mathfrak{a}}_{1},{\mathbb{R}})$ which satisfies

[TABLE]

Hence, by $K$-invariance and using Theorem 2.7 (in particular, Equation (9)) and the normalisation (20), we can reduce Equation (22) to the following real Monge-Ampère equation for the familly of convex potential $(u=u_{t})_{t\in[0,+\infty[}$ belonging to ${\mathcal{C}}^{\infty}({\mathfrak{a}}_{1},{\mathbb{R}})$ :

[TABLE]

Thanks to Proposition 4.3, there exists a family $(u_{t})$ solution of (28) for all $t\in[0,+\infty[$.

4.2. Study of the solutions $(u_{t})_{t\in[0,+\infty[}$ of Equation (28)

We recall a useful lemma.

Lemma 4.5** ([Guz75]).**

Let $\Omega$ be a convex bounded domain of ${\mathbb{R}}^{n}$. Then there is a unique ellipsoid $E$, called the minimal ellipsoid of $\Omega$, whose volume is minimal among ellipsoids containing $\Omega$. In addition, $E$ satisfies

[TABLE]

Let $T$ be an affine transformation preserving the center $x_{0}$ of $E$ i.e. $T(x)=T_{vect}(x-x_{0})+x_{0}$ for a matrix $T_{vect}\in\operatorname{SL_{n}}({\mathbb{R}})$ and such that $T(E)$ is a ball $B_{R}$ with center $x_{0}$ for a certain $R>0$ (depending on $E$). In particular, we have $B_{R/n}\subset T(\Omega)\subset B_{R}$.

We set $\overline{u}_{t}=u_{t}-c_{t}$ where $c_{t}$ is the constant defined in Lemma 4.4.

Lemma 4.6**.**

The function $\overline{u}_{t}$ attains its minimum $m_{t}$ at a point $x_{t}\in{\mathfrak{a}}_{1}$ .

Proof.

Recall that convex function on ${\mathfrak{a}}_{1}$ that admits a critical point admits a global minimum. Moreover, we can work with the function $u_{t}$ because it differs with $\overline{u}_{t}$ by a constant. Note that $u_{t}$ is a convex function thanks to Proposition 2.8. In order to conclude, it is therefore sufficient to prove that $0\in\nabla u_{t}({\mathfrak{a}}_{1})$ and it this follow from Equation (12) and Proposition (2.7). ∎

Lemma 4.7**.**

We have

[TABLE]

Proof.

First we show the lower bound. By Lemma 4.4, one have

[TABLE]

so, because $\varphi_{t}|_{P/H}(\exp(x)H)=u_{t}(x)$, we get

[TABLE]

Using Equation 28, we obtain

[TABLE]

and finally we get

[TABLE]

We integrate the previous equation in order to get

[TABLE]

and by using Proposition 2.8, we finally have

[TABLE]

Now we recall, thanks to Proposition 2.8 that $|\nabla\overline{u}_{t}(x)|\leq d_{0}:=\sup_{x\in 2\Delta}\|x\|$and so, by using the mean value theorem,

[TABLE]

By Equation (30), there exists a time independent constant $c_{2}$ such that

[TABLE]

By doing the change of variable $y=x-x_{t}$, the left term is independent of $t$ and

[TABLE]

Hence we can conclude that there exists a constant $C>0$ such that for every $t\in[0,+\infty[$,

[TABLE]

Now, we show the upper bound. The proof is inspired by the proof of Lemma 2.1 in [WZ04]. We set

[TABLE]

Since $\overline{u}$ is convex and $m_{t}<+\infty$, we have

• $\bullet$

$A_{k}$ is bounded for all $k\geq 0$ and $\cup_{k\in{\mathbb{N}}}A_{k}={\mathfrak{a}}_{1}$

• $\bullet$

$x_{t}\in A_{0}$.

• $\bullet$

$\cup_{i=0}^{k}A_{i}$ is convex for all $k\geq 0$.

Moreover, there exists a constant $c>0$ independent of the time such that

[TABLE]

Indeed, thanks to Equation (28), we have

[TABLE]

by the Equation (29), where

[TABLE]

Hence, we get

[TABLE]

By using Lemma 4.5, there exists a affine transformation $y=T(x)$ such that the vectorial part has a determinant egal to $1$ and preserving the center of the minimal ellipsoid of $A_{0}$ and satisfying

[TABLE]

and

[TABLE]

Moreover, we clain

[TABLE]

Indeed, let

[TABLE]

where $y_{t}$ is the center of the minimal ellipsoid of $A_{0}$. A direct computation gives us

[TABLE]

and $v\geq\nu$ on $\partial T(A_{0})$ hence on $T(A_{0})$ thanks to the comparison principle. In particular, we get

[TABLE]

Moreover, by the convexity of $\overline{u}_{t}$, we obtain

[TABLE]

where $(k+1)\cdot A_{0}$ is the dilation of $A_{0}$ of factor $(k+1)$. Moreover, thanks to the equation (32), we get

[TABLE]

Now, if we denote by $\omega_{r}$ the volume of the unit ball of ${\mathfrak{a}}_{1}$ then we have by Equation (34)

[TABLE]

where $B>0$ is a constant independent of $t$. Finally we get, by using Equations (28) and (29) and Proposition 2.8,

[TABLE]

where $\tilde{C}$ is a constant independent of $t$. Finally, we get

[TABLE]

where $C>0$ is a constant independent of $t$. ∎

Now, we set

[TABLE]

and

[TABLE]

Remark that $\overline{\varphi}_{t}$ extends to a smooth function on $M$ which is the potential of $\exp(x_{t})^{*}\omega_{t}$ with respect to the Kähler form $\omega_{0}$, that so

[TABLE]

Lemma 4.8**.**

There exists a constant $C^{\prime}>0$ independent of $t$ such that for every $t\in[0,+\infty[$

[TABLE]

Proof.

By $K$-invariance and density of $G/H$, it is sufficient to prove the result on $P/H$ i.e. where the function $\overline{\varphi}_{t}$ has the following expression

[TABLE]

By convexity, we have

[TABLE]

and so, by definition of the support function $v_{2\Delta}$ and by Proposition (2.8),

[TABLE]

Hence, by Proposition 4.7, there exists $F$ independant of $t$ such that

[TABLE]

For the lower bound, using Equation (28), the function $\overline{\overline{u}}_{t}$ satisfies the following real Monge-Ampère equation on ${\mathfrak{a}}_{1}$ :

[TABLE]

By density since $\overline{\overline{u}}_{t}=\overline{\varphi}_{t}+u_{0}$ and $u_{t}=\varphi_{t}+u_{0}$, we see that $\overline{\varphi}_{t}$ satisfies the following complex Monge-Ampère equation on $M$:

[TABLE]

Now, since $h$ is a continuous function on a compact manifold and thanks to Lemma 4.4, there exists a constant $B^{\prime}>0$ independent of $t$ such that

[TABLE]

and hence, by integrating,

[TABLE]

By density we can reduce the integration on $G/H$ and doing an integration along the fiber, there exits a constant $C^{\prime\prime}$ independent of $t$ such that

[TABLE]

So, by doing the change of variables $y=\nabla\overline{\overline{u}_{t}}(x)+4\tau_{\rho_{P}}$ and applying Proposition 2.8, there exist two constants time independent $c_{1}$ and $c_{2}$ such that

[TABLE]

Now, since $M$ is compact, this implies that there exists a time independent constant $C_{0}^{\prime}$ such that

[TABLE]

We can conclude, thanks to Lemma 4.7, as wanted

[TABLE]

∎

Let us end this section with the following lemma.

Lemma 4.9**.**

There exists a constant $C_{0}>0$ such that

[TABLE]

where

[TABLE]

where we denote

[TABLE]

Proof.

The proof is stated in [Zhu12] and inspired at the origin by Proposition 3.1 of [TZ06]. ∎

4.3. ${\mathcal{C}}^{0}$-estimate of $\overline{\varphi}_{t}$

This section recalls general results which are extracted to [Zhu12]. In particular, we recall (and modify when necessary) Propositions 3.1 and 3.2 and Corollary 3.1

Recall (see Proposition 3.2) that we have a vector field $X\in{\mathfrak{z}}({\mathfrak{g}})$ such that

[TABLE]

Now, we consider the one paremeter subgroup of automorphims $\sigma_{t}=\exp(t\,X)$ generated by $X$. We define the familly of potentials $(\varphi_{t}^{\prime})_{t\in[0,+\infty[}$ by the formula :

[TABLE]

Now, thanks to the equation (28), we have

[TABLE]

Moreover, by the maximum principle, we show that the previous equation is equivalent modulo a constant to the following equation

[TABLE]

Now, we introduce the following space of potentials

[TABLE]

and we define the generalized K$-$energy functional associated to $X$ defined in $\mathcal{M}_{X}(\omega_{0})$ by

[TABLE]

where $(\psi_{t})_{t\in[0,1]}$ is a path connecting 0 to $\varphi$ in $\mathcal{M}_{X}(\omega_{0})$ and $\dot{\psi}_{t}=\frac{\partial\psi_{t}}{\partial t}$.

Lemma 4.10**.**

Let $\sigma\in Aut_{r}(M)$. If we define the potential $\varphi_{\sigma_{t}}$ by

[TABLE]

We have $\varphi_{\sigma_{t}}\in\mathcal{M}_{X}(\omega_{0})$ and $\tilde{\mu}(\varphi)=\tilde{\mu}(\varphi_{\sigma_{t}})$.

Proof.

Because $X\in{\mathfrak{z}}({\mathfrak{g}})$, we have

[TABLE]

and so

[TABLE]

Now, if $Y\in{\mathfrak{g}}$ then $Y$ induce a one-parameter subgroup in $\operatorname{Aut}_{r}(M)$ and so, by the above, $\rho_{t}\in\mathcal{M}_{X}(\omega_{0})$. Moreover, we can prove (see Computation $2.13$ in [PSSW11]) that

[TABLE]

Moreover, we have $F_{X}(Y)=0$ (see Equation (39)) and so $\frac{d\tilde{\mu}(\varphi_{\rho_{t}})}{dt}=0$ and hence

[TABLE]

∎

Lemma 4.11**.**

For all $t\in[0,+\infty[$, we have

[TABLE]

Proof.

First, $\varphi_{t}^{\prime}\in\mathcal{M}_{X}(\omega_{0})$ (see Lemma 4.10) and so we can compute $\tilde{\mu}(\varphi_{t}^{\prime})$. Nowe we get, thanks to Equation (40) and doing a integration by part :

[TABLE]

Hence, we get

[TABLE]

We can conclude, thanks to Lemma 4.10 and Equation (42),

[TABLE]

∎

With theses lemmas, we can prove the following ${\mathcal{C}}^{0}$-estimate for $\overline{\varphi}_{t}$.

Proposition 4.12**.**

There exists a constant $C>0$ independant of time $t$ such that

[TABLE]

Proof.

Firstly, we recall (see [TZ02] the definition functional $\tilde{F}$ on $\mathcal{M}_{X}(\omega_{0})$):

[TABLE]

with $\tilde{J}$ is defined by:

[TABLE]

where $\\ psi_{t}$ is a path connecting [math] to $\varphi$ in $\mathcal{M}_{X}(\omega_{g})$ and $\dot{\psi}_{t}$ is the derivative of $\psi_{t}$ with respect to $t$. We can show that $\tilde{J}$ is independant of the path $\psi$ and that $\tilde{J}(\varphi)>0$ (voir [WZ04]). We also prove ( see Lemma 5.1 of [TZ02] or Formula 67 of [DR17]) that there exits a constant $C>0$ such that :

[TABLE]

By using Lemma 4.11 and the definition of $\overline{\varphi}$, we get

[TABLE]

and hence by using Equations (43) and (37) and Lemmas 4.7 and 4.8

[TABLE]

Now, if we set

[TABLE]

then we know that, thanks to [CTZ05], there exists a uniform constant $c>0$ such that

[TABLE]

By using, the preivous inegality, we get

[TABLE]

Now Lemma 4.9 tell us that

[TABLE]

and,by Equation (38),

[TABLE]

Lemma 4.7 allows to conclude. ∎

4.4. Uniforme estimate of $c_{t}$ and $\partial\varphi_{t}/\partial t$

Now, we prove in this section that modulo a renormalisation of $h_{t}$, we can get uniform bounds of $c_{t}$ and $\partial\varphi_{t}/\partial t$. Before, we prove a lemma about $K$-energy functionnal :

Lemma 4.13**.**

Let $(\varphi_{t})_{t\in[0,+\infty[}$ be a solution of Equation 22. We have

[TABLE]

Proof.

Indeed, we have, thanks to Equation (43), Lemma 4.7, Proposition 4.12 and because $\tilde{J}(\overline{\varphi})>0$,

[TABLE]

∎

There exists a constant $C>0$ independant of time such that

[TABLE]

Indeed, by using Equation 41, we have

[TABLE]

To conclude, it is sufficient to remark (see Lemma 4.10) that

[TABLE]

and, thanks to Lemma 4.13 and 4.11, we have that

[TABLE]

and, by taking the limit, the result is proved. Moreover, we get immediately

[TABLE]

Lemma 4.14**.**

If we renormalise the function $h$ by adding a constant such that

[TABLE]

then

[TABLE]

In particular, there exists a constant $C>0$ independant of time $t$ such that

[TABLE]

Proof.

The proof on is inspired by Lemma $4.2$ of [TZ06] but this lemma assume the existence of a Kähler-Ricci soliton. We set

[TABLE]

We must prove $\lim_{t\rightarrow+\infty}a_{t}=0$. First, we have

[TABLE]

and hence

[TABLE]

Moreover, thanks to Equation (40), we get

[TABLE]

hence we obtain

[TABLE]

and, by using the hypothesis of renormalisation,

[TABLE]

Now, we remark, by using Equation (45),

[TABLE]

And we obtain, by using Equation (46),

[TABLE]

To conclude, it is sufficient to remark that

[TABLE]

because the integral $\leq\int_{0}^{+\infty}(\int_{M}\|\overline{\partial}\cfrac{\partial\varphi_{s}^{\prime}}{\partial s}\|^{2}e^{\theta_{X}+X(\varphi_{s}^{\prime})}\omega^{n}_{\varphi_{s}^{\prime}})ds$ is convergent (see Equation 44). ∎

Proposition 4.15**.**

If we renormalize the function $h_{t}$ by adding a constant such that

[TABLE]

then there exists two constant $C>0$ and $\tilde{C}>0$ such that

[TABLE]

Proof.

If $(\varphi_{t}^{\prime})_{t\in[0,+\infty[}$ is a solution of (40) then

[TABLE]

Hence, by Lemma 4.14, there exist a uniform constant $C>0$ such that

[TABLE]

Moreover, we know that there exists a uniform bound for $|X(\psi)|$ for all $\psi\in\mathcal{M}_{X}(\omega_{0})$ (see Corollary 5.3 in [Zhu00]) and the function $\theta_{X}$ is bounded so there exists a constant $C_{1}>0$ independent of time $t$ such that

[TABLE]

but $h_{t}=-\partial\varphi_{t}/\partial t+c_{t}$ (see Lemma 4.4) so $|\int_{M}\left(h_{t}-c_{t}\right)\,\omega^{n}_{\varphi_{t}^{\prime}}|\leq C_{1}.$ Finally, we get

[TABLE]

Now, thanks to Equation (23) and by definition of $\varphi_{t}^{\prime}$, we have $\operatorname{Vol}(M,\omega_{\varphi_{t}^{\prime}})=V,$ hence, by Lemma 4.4,

[TABLE]

and

[TABLE]

∎

4.5. Modified Kähler-Ricci Flow

In this section, the points $x_{t}$ are modified to points $x_{t}^{\prime}$ in order to verify larger smoothness assumptions and Equation (40) is modified accordingly. Finally we will then study the convergence of the Kähler-Ricci flow thus modified.

We start by proving the following lemma that will allow us to modify the $x_{t}$ family later.

Lemma 4.16**.**

We have

[TABLE]

Proof.

By Propositions 2.8 and 4.12, we have

[TABLE]

We get also

[TABLE]

Moreover, by using Lemma 4.7 and Proposition 4.15, we have

[TABLE]

By using the previous inegalities, we have

[TABLE]

If we take $x=x_{t}$ then

[TABLE]

This means that

[TABLE]

To conclude, it is enough to remark that $2\Delta$ contains a ball of the form $B(0,\delta)$ with $\delta>0$. This is due to the fact that we have an isomorphism (given by $\nabla u$) between ${\mathfrak{a}}_{1}\simeq{\mathbb{R}}^{r}$ and $2\Delta$ (see Proposition 2.8). Taking as point $p$ the intersection point between the line ${\mathbb{R}}\,(x_{t}-x_{t^{\prime}})$ and the ball $B(0,\delta)$, we therefore obtain thanks to the case of equality of the Cauchy-Schwarz theorem:

[TABLE]

hence

[TABLE]

∎

Now, we want to build a family $x_{t}^{\prime}$ for $t\in[0,+\infty[$ such that there exists two constants $D$ et $\tilde{D}$ independent of time $t$ satisfying

[TABLE]

In order to realize it, we consider the following family

[TABLE]

We remark

[TABLE]

and if $t$ is written $t=i+\epsilon$ with $i\in{\mathbb{N}}$ and $\epsilon\in]0,1[$ then, by using Lemma 4.16,

[TABLE]

Moreover, we have

[TABLE]

this is a direct result of Lemma 4.16 and that the function $x_{t^{\prime}}$ is defined per piece on the intervals $[i,i+1]$. So we get $\tilde{D}=C$ where $C$ is the constant of Lemma 4.16. To conclude, it is necessary to modify the family $x_{t}^{\prime}$ to obtain a ${\mathcal{C}}^{1}$-interpolation at integers points. In order to do this, we remark, thanks to Lemma 4.16, that

[TABLE]

We define the polynomial function $P_{i}=(P^{1}_{i},\cdots,P^{r}_{i})$ where the function $P^{j}$ is defined by

[TABLE]

where $\left(\cdot\right)_{j}$ refers to the $j$-th coordinate of the vector in the paranthesis. We note

[TABLE]

So we define the function $f_{i}(t):[i-\delta,i+\delta]\rightarrow{\mathfrak{a}}_{1}$ by

[TABLE]

We see that

[TABLE]

In addition, by choosing $\delta$, we have $|x_{i-\delta}-x_{i+\delta}|\leq 1$ and recall that the function $dx_{t}^{\prime}/dt$ is uniformly bounded to non-integer points, we therefore obtain by that there are two constants $A$ and $\tilde{A}$ independent of $i$ and $t$ such that

[TABLE]

To conclude, it is therefore sufficient to replace for any $t\in[i-\delta,i+\delta]$ the $x_{t}^{\prime}$ function by the $f_{i}(t)$ function. We then have the desired result.

We set

[TABLE]

Let us start by noting that the function $\tilde{\varphi}$ extends by density to a global function on $M$ which is the potential of $\exp(x_{t})^{*}\omega_{\varphi}$ with respect to $\omega_{0}$ so $\omega_{\tilde{\varphi}}$ is a Kähler form. In addition, we have

[TABLE]

and so

[TABLE]

Now, we notice that $\tilde{u}_{t}$ satisfies the following equation on ${\mathfrak{a}}_{1}$ :

[TABLE]

and by density, we then show that $\tilde{\varphi}$ satisfies the following equation on $M$ :

[TABLE]

où $\tilde{X}_{t}:=dx_{t}^{\prime}/dt$ is the holomorphic vector field given by $\tilde{X}_{t}:x\in M\mapsto\partial/\partial s|_{s=0}\,\exp(s\,dx^{\prime}_{t}/dt)\cdot x.$ Equation (49) is the Kähler-Ricci flow modified by $\tilde{X}_{t}$. In particular, it can be written in the form

[TABLE]

Let us finish this section, by noting that $\theta_{\tilde{X}}=\tilde{X}(u_{0})$, we get, by Proposition 4.15, that there exists a constant $C>0$ independent of $t$ such that

[TABLE]

4.5.1. Proof of the convergence

We can now prove of the main theorem. Before we will prove the following result:

Theorem 4.17**.**

There exists a sequence of positives reals $(t_{i})_{i\in{\mathbb{N}}}$ such that $t_{i}\rightarrow_{i\rightarrow+\infty}+\infty$ and the subsequence $\tilde{\varphi}_{t_{i}}$, extracted to the solution $\tilde{\varphi}_{t}$ of the Käher Ricci flow modified by $X_{t}$ (Equation (50)), converge to a potential $\overline{\varphi}_{\infty}$ such that $(\omega_{0}+\sqrt{-1}/2\pi\,\partial\overline{\partial}\overline{\varphi}_{\infty},X)$ is a Kähler-Ricci soliton where $X$ is the solitonic vector field defined by Proposition 3.2.

Proof.

The proof is divided into several points:

• $\bullet$

Using the $\mathcal{C}^{2}$ estimates of Yau found in [Yau78] for some Monge-Ampère equations, we show that there is a constant $C_{2}>0$ independent of time $t$

[TABLE]

Moreover, using Calabi’s computations in [Yau78], we finally obtain that there is a time-independent constant $C_{3}>0$ such that:

[TABLE]

Finally, using the regularization theorem of parabolic equations, we obtain:

[TABLE]

So for any sequence $t_{n}$ of real such that $t_{i}\rightarrow_{i\rightarrow+\infty}+\infty$, we can find an extracter $\gamma$ such as the subsequence $\tilde{\varphi}_{t_{\gamma(i)}}$ converges to a smooth function $\tilde{\varphi}_{\infty}$.

• $\bullet$

Recall that $X$ is the solitonic vector fiel defined in Proposition 3.2, we denote $\sigma_{t}:=\exp(tX)$ is one parameter subgroup of automorphisms induced by $X$. We consider multiplying by $\exp(x_{t})$ on $G/H$, because we have a action of $G$ on $M$, this application extends into an automorphism $\rho_{t}$ on $M$. In addition, we can see directly that $\rho_{t}$ checks $\rho_{t}^{*}\tilde{\varphi}_{t}=\varphi_{t}.$ We set $\rho^{\prime}_{t}=\rho_{t}\circ\sigma_{t}^{-1}$. We get

[TABLE]

In addition, we know that there is a uniform bound for $|X(\psi)|$ for all $\psi\in\mathcal{M}_{X}(\omega_{0})$ (see Corollary 5.3 of [Zhu00]) and that the function $\theta_{X}$ is bounded since continuous on a compact variety so we obtain that there exists a constant $C_{1}>0$ independent of $t$ such that

[TABLE]

In particular, since we have

[TABLE]

we get that the function $t\mapsto\tilde{\mu}_{\omega_{0}}(\varphi_{t}^{\prime})$ is decreasing and so admits a finite limit at $+\infty$. Moreover, there exits a sequence of reals $(t_{i})_{i\in{\mathbb{N}}}$ such that

[TABLE]

Indeed, we define the sequence $(t_{i})_{i\in{\mathbb{N}}}$ such that

[TABLE]

and so, thanks to Equations (52), its satsifies

[TABLE]

We conlude by remarking that the right terme converge to [math] when $i$ tends to $+\infty$ since $\tilde{\mu}_{\omega_{0}}(\varphi^{\prime}_{t})$ admits a finite limit when $t$ tends to $+\infty$.

• $\bullet$

By using Equation (49), we get that

[TABLE]

it implies, thanks to the equation 51, that $(\sigma_{t}^{\prime})^{*}\cfrac{\partial\varphi_{t}^{\prime}}{\partial t}$ is uniformly bounded in a $C^{k}$ way for all $k\in{\mathbb{N}}$. So according to Arzela-Ascoli’s theorem, we can extract a convergent subsequence $(\sigma_{t}^{\prime})^{*}\cfrac{\partial\varphi^{\prime}}{\partial t}(t_{i},\cdot)$. Moreover, by using Equation (53) and Lemma 4.14, we obtain, even if it means extracting a subsequence again, that $(\sigma_{t}^{\prime})^{*}\cfrac{\partial\varphi_{t}^{\prime}}{\partial t}(t_{i},\cdot)$ converges to [math] in a ${\mathcal{C}}^{k}$ way for all $k\in{\mathbb{N}}$. Using the equation (54) and using the first point of the proof to extract a convergent subsequence, we obtain that $\omega_{\tilde{\varphi}_{t_{i}}}$ converges to the Kähler-Ricci soliton defined by

[TABLE]

∎

Corollary 4.18**.**

There exists a Kähler-Ricci soliton $(g_{KRS},X_{KRS})$ on every horospherical manifold $M$.

If we use the result of uniqueness (theorem $3.2.$ of [TZ02]), we obtain that there is an automorphism $\beta\in\operatorname{Aut}_{r}(M)$ such that

[TABLE]

Then, for simplicity, we can assume that $\beta=Id$. Now, we can complete Theorem 4.17.

Theorem 4.19**.**

With the previous notation, the solution $\omega_{\tilde{\varphi}_{t}}$ to the Kähler-Ricci modified by $X_{t}$ converge to $\omega_{\tilde{\varphi}_{\infty}}$ when $t\rightarrow+\infty$.

Proof.

To prove this theorem, thanks to Arzela-Ascoli theorem, it is sufficient to show that for any sequence of real positives $(t_{i})_{n\in{\mathbb{N}}}$ such that $t_{i}\rightarrow+\infty$, there is a subsequence of extractors $\gamma$ such as $\omega_{\tilde{\varphi}_{t_{\gamma(i)}}}$ converge to $\omega_{\tilde{\varphi}{\infty}}$. We consider a sequence of real $(t_{i})_{i\in{\mathbb{N}}}$ such that $t_{i}\rightarrow+\infty$ and set $\epsilon>0$. We are going to divide the proof into several points:

• $\bullet$

The first step is to show that there is a sequence of real $(s_{i})_{i\in{\mathbb{N}}}$ such that

[TABLE]

and such that

[TABLE]

In particular, by taking up the approach of the proof of Theorem 4.17, we also have, even if it means extracting, that $\omega_{\varphi_{s_{i}}}$ converges to $\omega_{\varphi_{s_{\infty}}}$ which is a Kähler-Ricci soliton and in particular

[TABLE]

Moreover, thanks to the theorem of uniqueness (Theorem $3.2.$ of [TZ02]), there exists $\beta\in Aut_{r}(M)$ such that $\beta^{*}\omega_{\varphi_{s_{\infty}}}=\omega_{KRS}.$ Moreover, thanks to uniqueness of Kähler-Ricci solitons ( Theorem $3.2.$ of [TZ02]), we know that there exists $\beta\in Aut_{r}(M)$ such that $\beta^{*}\omega_{\varphi_{s_{\infty}}}=\omega_{KRS}.$ For simplicity, even if it means changing $\omega_{KRS}$, we can assume that $\beta=Id$.

[TABLE]

and, thanks to Equation (52),

[TABLE]

We then conclude by noting that the right term converges to [math] when $i$ tends to $+\infty$ since $\tilde{\mu}_{\omega_{0}}(\varphi^{\prime}_{t})$ admits a finite limit when $t$ tends to $+\infty$.

• $\bullet$

The second step is to show that the sequence $(\tilde{\varphi}_{t_{i}}-\tilde{\varphi}_{s_{i}})_{i\in{\mathbb{N}}}$ checks from a certain rank that

[TABLE]

where $\delta(\epsilon)$ satisfies $\delta(\epsilon)\longrightarrow_{\epsilon\rightarrow 0}0.$ To do this, we will use the theory of parabolic equations. Indeed, by using Equation (49), we prove that $\left(\tilde{\varphi}_{t}-\tilde{\varphi}_{s_{i}}\right)$ is solution for $t\in[t_{i}-\epsilon,t_{i}+\epsilon]$ of Monge-Ampère flow :

[TABLE]

où

[TABLE]

and

[TABLE]

We notice that from a certain rank, thanks to Equation (48) and (56), we have that there exists a constant $C>0$ independent of $t$ such as $\|G\|_{{\mathcal{C}}^{k}}\leq C\epsilon$. To conclude, it is then sufficient to use the theorem of implicit functions.

• $\bullet$

We then conclude by showing that $\tilde{\varphi}_{t_{i}}$ converges to $\varphi_{KRS}$ in a ${\mathcal{C}}^{k}$ way for all $k>0$. Indeed, we have

[TABLE]

Using the two previous points, we obtain that from a certain rank, we have

[TABLE]

Since the result is true for any $\epsilon>$0, we get the desired result.

∎

By definition of convergence in the Cheeger-Gromov sense, we directly obtain the main theorem that we recall here

Theorem 4.20**.**

Let $M$ a Fano horosphercal manifold of horospherical homegeneous space $G/H$ where $G$ is the complexification of a maximal compact subgroup $K$. The solution $\omega_{\varphi_{t}}$ defined for $t\in[0,+\infty[$ of the Kähler-Ricci flow with inital value a $K$-invariant metric $\omega_{0}$ converge in the Cheeger-Gromov sense to a Kähler form $\omega_{\infty}$ when $t$ tends to $+\infty$ where $\omega_{\infty}$ is a Kähler-Ricci soliton.