Equivariant solutions to the optimal partition problem for the prescribed $Q$-curvature equation

A milestone in Geometric Analysis and Conformal Geometry is the Yamabe problem, which consists in finding a conformal metric on a given Riemannian manifold in order to obtain constant scalar curvature (see, for instance, the book [AubinBook] or the classic survey [LePa], and the references therein). There are many generalizations of this problem. One direction is to consider higher-order operators generalizing the conformal Laplacian. For instance, a fourth-order operator on Riemannian manifolds of dimension four, which is the analogue to the bilaplacian on the Euclidean space, was discovered by S. Paneitz [Paneitz1983] and later generalized to higher dimension by T.P Branson in [Branson1995], and by Branson and B. Ørsted in [BransonOrsted1991]. A systematic construction of conformally invariant operators of higher order was given by C. R. Graham, R. Jenne, J. Mason and G. A. J. Sparling in [GJMS] ($GJMS$-operators for short) and, more recently extended to fractional order conformally invariant operators by S.-Y. A. Chang and M. d. M. González in [ChangGonzalez2011]. This paper deals with optimal partition problems related to equations given by GJMS-operators.

In order to briefly describe these operators, let $M$ be a closed (compact without boundary) smooth manifold of dimension $N$ and let $m\in\mathbb{N}$ such that $2m<N$. For any Riemannian metric $g$ for $M$, there exists an operator $P_{g}:\mathcal{C}^{\infty}(M)\rightarrow\mathcal{C}^{\infty}(M)$, satisfying the following properties.

1. ($P_{1}$)

   $P_{g}$ is a differential operator and $P_{g}=(-\Delta_{g})^{m}+\emph{\text{lower order terms}}$, where $\Delta=\text{div}_{g}(\nabla_{g})$ is the Laplace Beltrami operator on $(M,g)$.

2. ($P_{2}$)

   $P_{g}$ is natural, i.e., for every diffeomorphism $\varphi:M\rightarrow M$, $\varphi^{\ast}P_{g}=P_{\varphi^{\ast}g}\circ\varphi^{\ast}$, where “$\!\!\!\phantom{K}{}^{\ast}\;$” denotes the pullback of a tensor.

3. ($P_{3}$)

   $P_{g}$ is self-adjoint with respect to the $L^{2}$-scalar product.

4. ($P_{4}$)

   $P_{g}$ is conformally invariant, that is, given any function $\omega\in\mathcal{C}^{\infty}(M)$, if we define the conformal metric $\widetilde{g}=e^{2\omega}g$, then the following identity holds true:

   (1)

   $$P_{\widetilde{g}}(f)=e^{-\frac{N+2m}{2}\omega}P_{g}\left(e^{\frac{N-2m}{2}\omega}f\right),\text{ for all }f\in\mathcal{C}^{\infty}(M).$$

There is a natural conformal invariant associated with $P_{g}$ given by

called the Branson $Q$-curvature, after Branson-Ørsted [BransonOrsted1991] and Branson [BransonBook, Branson1995], or simply the $Q$-curvature. When $m=1$, $P_{g}$ is the conformal Laplacian and $Q_{g}$ is the scalar curvature $R_{g}$, while for $m=2$, $P_{g}$ is the Paneitz-Branson operator and $Q_{g}$ is the usual $Q$-curvature [DjadliHebeyLedoux2000].

When considering conformal metrics $\widetilde{g}=u^{4/(N-2m)}g$ with $u>0$ in $\mathcal{C}^{\infty}(M)$, equation (1) is written as

where $2_{m}^{\ast}:=\frac{2N}{N-2m}$ is the critical Sobolev exponent of the embedding $H_{g}^{m}(M)\hookrightarrow L^{p}(M,g)$. Here $H_{g}^{m}(M)$ denotes the Sobolev space of order $m$, which is the closure of $\mathcal{C}^{\infty}(M)$ in $L^{2}_{g}(M)$ under the norm

Taking $\phi\equiv 1$ in (2), one obtains the prescribed $Q$-curvature equation

For $m=1$ and $Q_{\widetilde{g}}$ constant, we recover the Yamabe Problem. Some results about the existence and multiplicity of solutions to the prescribed $Q$-curvature problem can be found in [AzaizBoughazi2020, BaScWe, BenaliliBoughazi2016, DjadliHebeyLedoux2000, Ro, Tahri2020].

Given $\ell\in\mathbb{N}$, we will deal with the $\ell$-partition problem associated with (4) in the presence of symmetries. In order to describe the problem, let $\Gamma$ be a closed subgroup of Isom$(M,g)$ under suitable conditions (see hypotheses ($\Gamma 1$) and ($\Gamma 2$) below), and let $\Omega\subset M$ be an open and $\Gamma$-invariant subset, namely, if $x\in\Omega$, then $\gamma x\in\Omega$ for every isometry $\gamma\in\Gamma$. In what follows, $H_{0,g}^{m}(\Omega)$ denotes the closure of $\mathcal{C}_{c}^{\infty}(\Omega)$ under the Sobolev norm given by (3).

We consider the symmetric Dirichlet boundary problem

where $u:\Omega\rightarrow\mathbb{R}$ is said to be $\Gamma$-invariant if $u(x)=u(\gamma x)$ for every $x\in M$ and any $\gamma\in\Gamma$.

Denote by $c_{\Omega}^{\Gamma}$ the least energy among the nontrivial solutions, that is,

In the absence of symmetries, the lack of compactness due to the critical Sobolev exponent nonlinearity in equation (5), implies that this number is not achieved in general [StruweBook, Chapter III]. However, when the domain is smooth, $\Gamma$-invariant and the $\Gamma$-orbits have positive dimension, this number is attained (see Proposition 2.3 below). The $\ell$-partition problem consists in finding mutually disjoint and non empty $\Gamma$-invariant open subsets $\Omega_{1},\ldots,\Omega_{\ell}$ such that

where

The aim of this paper is to show that this problem has a solution on Einstein manifolds with positive scalar curvature, for every $m\geq 2$, and with less restrictive hypotheses on the metric, when $m=1$.

In order to state our main result, we need to impose some conditions over $(M,g)$ and its isometry group (conditions $(\Gamma 1)$ to $(\Gamma 3)$ below). For the reader convenience, we recall some basic facts about isometric actions (see Chapter 3 and Section 6.3 in [AlexBettiol] for a detailed exposition). For any $x\in M$, the $\Gamma$ orbit of $x$ is the set $\Gamma x:=\{\gamma x\;:\;\gamma\in\Gamma\}$, and the isotropy subgroup of $x$ is defined as $\Gamma_{x}:=\{\gamma\in\Gamma\;:\;\gamma x=x\}$. When $\Gamma$ is a closed subgroup of $\text{Isom}(M,g)$, the $\Gamma$-orbits are closed submanifolds of $M$ which are $\Gamma$-diffeomorphic to the homogeneous space $\Gamma/\Gamma_{x}$, meaning the existence of a $\Gamma$-invariant diffeomorphism between these two manifolds. An orbit $\Gamma x$ is called principal, if there exists a neighborhood $V$ of $x$ in $M$ such that for each $y\in V$, $\Gamma_{x}\subset\Gamma_{\gamma y}$ for some $\gamma\in\Gamma$. All points lying in a principal orbit have, up to conjugacy, the same isotropy group. Denote this group by $K$, called the principal isotropy group. We say that $\Gamma$ induces a cohomogeneity one action on $M$ if the principal orbits have codimension one. In presence of a cohomogeneity one action, each principal $\Gamma$-orbit is a closed hypersurface in $M$ diffeomorphic to $\Gamma/K$, and there are exactly two orbits of bigger codimension, called singular orbits. Denote them by $M_{-}$ and $M_{+}$. $M_{\pm}$ are closed submanifolds of codimension $\geq 2$, and all points lying in $M_{\pm}$ have, up to conjugacy, the same isotropy group. Denoting these groups by $K_{\pm}$, we have that $M_{\pm}$ is $\Gamma$-diffeomorphic to $\Gamma/K_{\pm}$.

We can now state the conditions on the group $\Gamma$. In what follows, $d$ will denote the geodesic distance between $M_{+}$ and $M_{-}$, that is, $d:=\text{dist}_{g}(M_{-},M_{+})$.

1. $(\Gamma 1)$

   $\Gamma$ is a closed subgroup of $\text{Isom}(M,g)$ inducing a cohomogeneity one action on $M$.

2. $(\Gamma 2)$

   $1\leq\dim\Gamma x\leq N-1$ for every $x\in M$.

3. $(\Gamma 3)$

   The metric $g$ on $M$ is a $\Gamma$-invariant metric of the form:

   $$g=dt^{2}+\sum_{j=1}^{k}f_{j}^{2}(t)\ g_{j},$$

   where $g_{t}:=\sum_{j=1}^{k}f_{j}^{2}(t)\ g_{j}$ is one parameter family of metrics on the principal orbit, for some positive smooth functions defined on the interval $I=[0,d]$, with suitable asymptotic conditions at $0$ and $d$. More precisely, the (smooth) metrics $g_{j}$ are defined on the principal orbit at $t$, for $t\in(0,d)$, and are defined around the singular orbits in such a way that $f_{j}$, $j=1,\dots,k$, satisfy appropriate smoothness conditions at the endpoints $t_{-}=0$ and $t_{+}=d$ of $I$, that ensure that $g$ can be extended to the singular orbits:

   (7)

   $$f_{1}(t_{\pm})=0,f_{1}^{\prime}(t_{\pm})=1;\ f_{j}(t_{\pm})>0,f_{j}^{\prime}(t_{\pm})=0\ \mbox{for $1<j\leq k$}.$$

   See [EscWang00, Section 1].

In Section 6, we give concrete examples where these hypotheses hold true.

We are now ready to state our main result, which describes the solution to the $\ell$-optimal partition problem in terms of the principal orbits of cohomogeneity one actions. The symbol “$\approx$” will stand for “$\Gamma$-diffeomorphic to”.

In Section 6 we give several examples of Einstein manifolds with positive scalar curvature satisfying conditions ($\Gamma 1$), ($\Gamma 2$) and ($\Gamma 3$). This result extends Theorem 1.1 in [CSS21] and Theorem 1.2 in [ClappFernandezSaldana2021] to more general manifolds and actions than the sphere with the $O(n)\times O(k)$-action, $n+k=N+1$. In the case of $m=1$, M. Clapp and A. Pistoia [ClappPistoia21] solved the $\Gamma$-invariant $\ell$-partition problem for actions with higher cohomogeneity and in [ClappPistoiaTavares21], the authors solved the $\ell$-partition problem without any symmetry assumption. However, neither the structure nor the domains solving the problem are explicit in these works.

Notice that in case $m=1$, the scalar curvature is $\Gamma$-invariant, since we are regarding isometric actions. An interesting application of Theorem 1.1 in this case, is the following result for the Yamabe problem.

This was proven in the case $\ell=2$ in [ClappPistoia21] for general manifolds in the presence of symmetries, and in [ClappPistoiaTavares21] in the non-symmetric case; for any $\ell\in\mathbb{N}$, the only manifold where this was known is the round sphere in the presence of symmetries given by isoparametric functions [FdzPetean20, CSS21].

This corollary clearly holds on Einstein manifolds with positive scalar curvature, for which the scalar curvature is constant. Then, a natural setting for extending its applications is Ricci soliton metrics. Recall that a Ricci soliton on a closed smooth manifold $M$ is a Riemannian metric $g$ satisfying the equation

for some constant $\mu=-1,0,$ or $1$, and some smooth function $f$, called the Ricci potential. Here, as usual, $Ric(g)$ denotes the Ricci curvature of $g$, and ${\rm Hess}_{g}(f)$ denotes the Hessian of $f$ with respect to $g$. A Ricci soliton is called steady, shrinking or expanding according to $\mu=0$, $\mu>0$ or $\mu<0$, respectively.

In these terms, a nontrivial explicit example that fulfills the hypotheses of Corollary 1.2 and that does not reduce to an Einstein metric, is the Koiso-Cao Ricci soliton [Koiso, Cao1]. It is known that this metric is a cohomogeneity one Kähler metric on $\mathbb{CP}^{2}\#\overline{\mathbb{CP}^{2}}$, with respect to the action of U$(2)$, the unitary group of dimension $2$. Here, as usual, $\#$ denotes the connected sum of smooth manifolds and $\overline{M}$ denotes a smooth manifold $M$ taken with reverse orientation. It is also known that it is non-Einstein with positive Ricci curvature. The singular orbits of the U$(2)$-action are both diffeomorphic to $\mathbb{S}^{2}$, and the principal orbits are diffeomorphic to $\mathbb{S}^{3}$. The associated fibrations, as a cohomogeneity one manifold, are both given by the Hopf fibration:

Here, for an orientable manifold $M$, we will denote by $\overline{M}$ to refer to the opposite orientation.

See [TOR17, Section 2] for details on the construction of the Koiso-Cao soliton from the Hopf fibration.

In order to establish the existence of a solution to the optimal partition problem (6), we follow the approach introduced in [ContiTerraciniVerzini2002], consisting in the study of a weakly coupled competitive system together with a segregation phenomenon. To this end, we will study the existence of $\Gamma$-invariant solutions to the following weakly coupled competitive $Q$-curvature system

where $\nu_{i}>0$, $\eta_{ij}=\eta_{ji}<0$ and $\alpha_{ij},\beta_{ij}>1$ are such that $\alpha_{ij}=\beta_{ji}$ and $\alpha_{ij}+\beta_{ij}=2_{m}^{\ast}$. We will say that a solution $\overline{u}=(u_{1},\ldots,u_{\ell})$ to the system (9) is fully nontrivial, if, for each $i=1,\ldots,\ell$, $u_{i}$ is non trivial.

To assure the existence of a fully nontrivial least energy $\Gamma$-invariant solution to the system (9) we will only assume that $\Gamma$ satisfies ($\Gamma 2$), allowing actions with bigger codimension of its principal orbits, and also we will assume that the operator $P_{g}$ is coercive, meaning the existence of a constant $C>0$ such that

We will say that a sequence of fully nontrivial elements $\overline{u}_{k}$ in the Sobolev space

is unbounded if $\|u_{k,i}\|_{H^{m}}\rightarrow\infty,$ as $k\rightarrow\infty$, for every $i=1,\ldots,\ell.$ In this direction, we state our next multiplicity result.

When $\ell=1$ and $m=1$, by a well-known argument given in [BenciCerami1991, Proof of Theorem A], the least energy solutions are positive and, hence, they give rise to a $\Gamma$-invariant solution to the Yamabe problem. Seeking for this kind of metrics is a classical problem posed by E. Hebey and M. Vaugon in [HebeyVaugon1993]. However, there is a gap in Hebey and Vaugon’s proof, for they used Schoen’s Weyl vanishing conjecture, which turns out to be false in higher dimensions by Brendle’s counterexample in [Brendle2008]. F. Madani solved the equivariant Yamabe problem in [Madani2012] assuming the positive mass theorem to construct good test functions. Here, the positive dimension of the orbits given by hypothesis ($\Gamma 2$), avoids these problems in higher dimensions. For $m\geq 2$, the Maximum Principle for the operator $P_{g}$ is not true in general, and the least energy solutions may change sign. In fact, it is not clear whether the components of the solutions to the system (9) are sign-changing or not. In case $m=2$ and $\ell=1$, by a recent result by J. Vétois [Vetois2022, Theorem 2.2], we can assure that an infinite number of the solutions to

must change sign when considering $(M,g)$ to be Einstein with positive scalar curvature, as we next state.

There are many examples of manifolds admitting an Einstein metric $g$ with positive scalar curvature, in order to ensure the coercivity of $P_{g}$. In Section 6, we describe explicit examples and some construction that provide large classes of examples. However, it is difficult to find nontrivial examples of non-Einstein manifolds for which the operator $P_{g}$ is coercive. Towards this direction, the next result for the Paneitz-Branson operator gives a sufficient condition for this to happen.

Again, it is complicated to compute the $Q$-curvature of an arbitrary manifold and the literature lacks examples, different from Einstein manifolds. We explore the possibility of obtaining positive $Q$-curvature and coercivity for more general manifolds, such as Ricci solitons. This is difficult to check because almost nothing is known about the Ricci curvature tensor in Ricci solitons. Our next result gives an explicit formula of the $Q$-curvature of these metrics.

We have the following immediate consequence of Theorems 1.3 and 1.6, together with Proposition 1.5.

Our work is organized as follows. In Section 2, we describe the variational setting in order to prove the existence of least energy $\Gamma$-invariant solutions to the problem (5). Next, in Section 3, we give the variational setting to study system (9) and prove Theorem 1.3 and Corollary 1.4. In Section 4, we describe how hypotheses $(\Gamma 1)$ and $(\Gamma 3)$ allow us to reduce the partition problem to a simpler one dimensional problem. In Section 5, we study the segregation phenomenon that gives the description of the domains and $\Gamma$-invariant functions that solve the optimal partition problem, showing Theorem 1.1. As an application of Theorem 1.1 with $m=1$, we prove Corollary 1.2. We give several examples where hypotheses $(\Gamma 1)$ to $(\Gamma 3)$ hold true in Section 6. Finally, in Section LABEL:Section:Q-curvatureRicciSolitons, we compute the $Q$-curvature of shrinking Ricci solitons, proving Theorem 1.6.

In this section, we study the existence of least energy solutions to problem (5).

From now on, $\Omega$ will denote either $M$ or an open, connected $\Gamma$-invariant subset of $M$ with smooth boundary, $\|\cdot\|_{p}$ will denote the usual norm in $L_{g}^{p}(\Omega)$, $p\geq 1$. For $u\in\mathcal{C}^{\infty}(M)$, the $k$-th covariant derivative of $u$ will be denoted by $\nabla^{k}u$ and we define its norm as the function $|\nabla^{k}u|_{g}:M\rightarrow\mathbb{R}$ given by

where we used the Einstein notation convention.

The Sobolev space $H_{0,g}^{m}(\Omega)$ is the closure of $\mathcal{C}_{c}^{\infty}(\Omega)$ under the norm

where, with some abuse of notation, $\|\nabla^{i}u\|_{2}:=\|\,|\nabla^{i}u|_{g}\|_{2}$ Notice that in case $\Omega=M$, then $\mathcal{C}_{c}^{\infty}(M)=\mathcal{C}^{\infty}(M)$ and $H_{0,g}^{m}(M)=H_{g}^{m}(M)$; if $\Omega\neq M$, then $H_{0,g}^{m}(\Omega)$ is a closed subspace of $H^{m}_{g}(\Omega)$.

Let $P_{g}$ be the corresponding GJMS-operator in $(M,g)$ of order $m$. For each $k\in\{0,1,\ldots,m-1\}$, there exists a symmetric $T^{0}_{2k}$-tensor field on $M$, which we will denote by $A_{(k)}(g)$, such that the operator $P_{g}$ can be written as

where the indices are raised via the musical isomorphism. In particular, for any $u,v\in\mathcal{C}_{c}^{\infty}(\Omega)$, integration by parts yields that

See [Ro, Proposition 1] for the details.

As a consequence, the bilinear form $(u,v)\mapsto\int_{\Omega}uP_{g}vdV_{g}$ can be extended to a continuous symmetric bilinear form on $H^{m}_{0,g}(\Omega)$. When $P_{g}$ is coercive, this bilinear form is actually a well-defined interior product on $H_{0,g}^{m}(\Omega)$ that induces a norm equivalent to $\|\cdot\|_{H^{m}}$ (see [Ro, Proposition 2]). We will denote this interior product and norm by $\langle\cdot,\cdot\rangle_{P_{g}}$ and $\|\cdot\|_{P_{g}}$ respectively, and endow $H_{g}^{m}(\Omega)$ with it in what follows. Notice that, by definition,

for every $u,v\in\mathcal{C}^{\infty}(\Omega)$.

The group $\text{Isom}(M,g)$ acts on $H_{g}^{m}(M)$ in the usual way:

Every element $\gamma\in\text{Isom}(M,g)$ induces a linear map

We next show that the norm is invariant under the action of isometries.

Every $\gamma\in\text{Isom}(M,g)$ also induces a linear map $\gamma:L^{p}_{g}(M)\rightarrow L^{p}_{g}(M)$, $p\geq 1$, given by $u\mapsto u\circ\gamma^{-1}$, which is also an isometry, thanks to (11).

Let $\Gamma$ be any closed subgroup of Isom$(M,g)$ such that $\dim\Gamma x\leq N-1$ for any $x\in M$. From now on, suppose that $\overline{\Omega}$ is $\Gamma$-invariant, namely, if $x\in\overline{\Omega}$, then $\Gamma x\subset\overline{\Omega}$. In this way, for any $u\in\mathcal{C}_{c}^{\infty}(\Omega)$ and every isometry $\gamma\in\Gamma$, it follows that

and $\gamma$ also induces linear isometries

for any $p\geq 1$.

We define the Sobolev space of $\Gamma$-invariant functions as

This is a closed subspace of $H_{0,g}^{m}(\Omega)$. In fact, if $\mathcal{C}_{c}^{\infty}(\Omega)^{\Gamma}$ denotes the space of smooth functions with compact support in $\Omega$ which are $\Gamma$-invariant, then $H_{0,g}^{m}(\Omega)$ coincides with the closure of this space under the Sobolev norm $\|\cdot\|_{P_{g}}$. As the dimension of any $\Gamma$ orbit is strictly less than $N$, the space $H_{0,g}^{m}(\Omega)^{\Gamma}$ is infinite dimensional, thanks to the existence of $\Gamma$-invariant partitions of the unity (Cf. [Palais1961, Theorem 4.3.1] and also [AlexBettiol, Claim 3.66]).

We will need the following Sobolev embedding result.

In what follows, we will suppose that $\Gamma$ satisfies condition ($\Gamma 2$). Under this hypothesis, the existence of least energy $\Gamma$-invariant solutions to (5) follows directly from standard variational methods using the Palais’ Principle of Symmetric Criticality [Palais1979] together with Lemma 2.2. For the reader’s convenience, we sketch the proof of a slightly more general result, namely, we show the existence of $\Gamma$-invariant solutions to the problem

where $p\in(2,2_{m}^{\ast}]$.

Consider the functional

Given that $\|\cdot\|_{P_{g}}$ is a well defined norm equivalent to the standard norm $\|\cdot\|_{H^{m}(\Omega)}$, Sobolev inequality yields that it is a $C^{1}$ functional for any $p\in(2,2_{m}^{\ast}]$. From (12), this functional is $\Gamma$-invariant, namely it satisfies that

Hence, due to Palais’ Principle of Symmetric Criticality [Palais1979], the critical points of $J_{\Omega}$ restricted to $H_{0,g}^{m}(\Omega)^{\Gamma}$ correspond to the $\Gamma$-invariant solutions to the problem (13). The nontrivial ones belong to the set

which is a $C^{1}$ codimension one Hilbert manifold in $H_{0,g}^{m}(\Omega)^{\Gamma}$. Notice that

Thanks to the Sobolev inequalities [AubinBook, Theorem 2.30], $\mathcal{M}_{\Omega}^{\Gamma}$ is closed and

We say that $J_{\Omega}$ satisfies condition $(PS)_{c}^{\Gamma}$ in $H_{0,g}^{m}(\Omega)^{\Gamma}$ if every sequence $u_{k}\in H_{0,g}^{m}(\Omega)^{\Gamma}$ such that $J_{\Omega}(u_{k})\rightarrow c$ and $\nabla J_{\Omega}(u_{k})\rightarrow 0$ in $H_{0,g}^{m}(\Omega)^{\Gamma}$ as $k\rightarrow\infty$, has a convergent subsequence.

As $\Gamma$ satisfies ($\Gamma 2$), then $\kappa\geq 1$ and $p\leq 2_{m}^{\ast}<2_{m,\Gamma}^{\ast}$. Hence, by Lemma 2.2, $J_{\Omega}$ satisfies condition $(PS)_{c_{\Omega}^{\Gamma}}^{\Gamma}$ and Theorem 7.12 in [AmbrosettiMalchiodiBook] yields that $c_{\Omega}^{\Gamma}$ is attained. Thus, there exists a least energy $\Gamma$-invariant solution to the problem (13).

We summarize this analysis in the following proposition.

To our knowledge, this is the first existence result of symmetric least energy solutions to the homogeneous Dirichlet boundary problem (13). Another result for non-homogeneous Dirichlet boundary conditions to problems involving the $GJMS$-operators, in the absence of symmetries, can be found in [BekiriBenalili2018, BekiriBenalili2019, BekiriBenalili2022].