Soliton to the fractional Yamabe flow

Nonlinear Analysis 139 (2016) 211–217

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Nonlinear Analysis www.elsevier.com/locate/na

Soliton to the fractional Yamabe flow Pak Tung Ho Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

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Article history: Received 6 September 2015 Accepted 29 February 2016 Communicated by Enzo Mitidieri

In this paper, we show that soliton to the fractional Yamabe flow must have constant fractional order curvature. © 2016 Elsevier Ltd. All rights reserved.

MSC: primary 35R11 53C44 secondary 53A30 53C21 Keywords: Fractional Yamabe problem fractional Yamabe flow Soliton

1. Introduction Suppose that X is an (n + 1)-dimensional smooth manifold with smooth boundary M , where n ≥ 3. A function ρ is a defining function on the boundary M in X if ρ>0

in X,

ρ=0

on M,

dρ ̸= 0

on M.

We say that a Riemannian metric h+ on X is conformally compact if, for some defining function ρ, the metric h = ρ2 h+ extends smoothly to X. This induces a conformal class of metrics g = h|T M on M as defining function vary. The manifold (M, [ g ]) equipped with the conformal class [ g ] is called the conformal + infinity of (X, h ). A metric h+ is called asymptotically hyperbolic if it is conformally compact and its sectional curvature approaches −1 at infinity, which is equivalent to |dρ|h = 1 on M . If Ric(h+ ) = −nh+ , then we call (X, h+ ) a conformally compact Einstein manifold. In these setting, given a representative g of the conformal infinity, there exists a unique defining function ρ such that in a tubular neighborhood near M such that the metric

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.na.2016.02.026 0362-546X/© 2016 Elsevier Ltd. All rights reserved.

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P.T. Ho / Nonlinear Analysis 139 (2016) 211–217

h+ has the normal form h+ =

dρ2 + gρ ρ2

(1.1)

where gρ is a one-parametric family of metric of metrics on M satisfying g0 = g. g The conformal fractional Laplacian Pγ is constructed as the Dirichlet-to-Neumann operator for the + scattering problem for (X, h ). In particular, it follows from [17,30] that given f ∈ C ∞ (M ), for all but a discrete set of values s ∈ C, the generalized eigenvalue problem − ∆h+ u − s(n − s)u = 0

in X

(1.2)

has a solution of the form u = F ρn−s + Gρs ,

F, G ∈ C ∞ (X),

F |ρ=0 = f.

(1.3)

The scattering operator on M is defined as S (s)f = G|M , g and it is a meromorphic family of pseudo-differential operators in the whole complex plane. The conformal fractional Laplacian on (M, g) is defined as n  Γ (γ) g Pγ = 22γ S + γ . Γ (−γ) g 2 g With this normalization, the principal symbol of the operator Pγ is equal to that of the fractional Laplacian γ  g (−∆ ) . The operator Pγ satisfies the following property: under a conformal change of metric g 4

g = u n−2γ g,

u > 0,

we have n+2γ g (uφ) Pγg (φ) = u− n−2γ Pγ

(1.4)

g for all smooth functions φ. As proven in [13,17], when h+ is Poincar´e–Einstein, P1 is the conformal Laplacian,  g  g P is the Paneitz operator, and P , where k are positive integers, are the GJMS operator discovered in [16]. 2

k

One can then define the fractional order curvature g  g Q γ = Pγ (1).

Hence, for the case when γ = 1, the fractional order curvature is the scalar curvature. As an analogy to the Yamabe problem, one can consider the fractional Yamabe problem: Find a metric g conformal to g such that its fractional order curvature Qgγ is constant. We refer the readers to [8,14,15,27,31] and references therein for results related to the fractional Yamabe problem. See also [7,24,25] for results related to the fractional Nirenberg problem of prescribing fractional order curvature. Inspired by the Yamabe flow (see [1,2,9,32,33] for results related to the Yamabe flow, and also [6,5,18,20,21] for results related to the CR Yamabe flow), which is a geometric flow introduced to study the Yamabe problem, we consider the fractional Yamabe flow on M . This is defined as the evolution of the metric g = g(t): ∂g = −(Qgγ − Qgγ )g, ∂t

g|t=0 = g,

where Qgγ is the average of the fractional order curvature Qgγ given by  Qg dVg g  γ Qγ = M . dVg M

(1.5)

(1.6)

P.T. Ho / Nonlinear Analysis 139 (2016) 211–217

213

We remark that the fractional Yamabe flow has been studied in [26] for the case when M is the sphere. We will discuss the existence and convergence of the fractional Yamabe flow in a forthcoming paper. In this paper, we study solitons to the fractional Yamabe flow, which is a self-similar solution of (1.5). More precisely, we say that g(t) is a soliton of the fractional Yamabe flow if there exists a one-parameter family of diffeomorphism {ψt } on X and a one-parameter family of real-valued functions σ(t) defined on X such that g = σ(t)ψt∗ ( g)

(1.7)

satisfies (1.5) on M , with ρ(0) = 1 and ψ0 = idX . We note that Yamabe soliton, which is a self-similar solution of the Yamabe flow, has been studied by many authors. See [3,4,10–12,22,23,28,29] and the references therein. We prove the following: Theorem 1.1. Any soliton of the fractional Yamabe flow must have constant fractional order curvature. We remark that the corresponding result for the Yamabe flow was proved by di Cerbo and Disconzi in [12] and by Hsu in [22] independently. More precisely, they proved that the compact Yamabe soliton must have constant scalar curvature. The corresponding result for the CR Yamabe flow was proved in [19]: the compact CR Yamabe soliton must have constant Webster scalar curvature. In fact, the proof of Theorem 1.1 follows the strategy in [12,19]. 2. Proof 4

Suppose g is a solution of the fractional Yamabe flow (1.5). If we write g = u n−2γ g, then its volume form is given by 2n

dVg = u n−2γ dV , g

(2.1)

and its fractional order curvature can be written as n+2γ g (u) Qgγ = Pγg (1) = u− n−2γ Pγ

(2.2)

4

by (1.4). Since g = u n−2γ g, the flow (1.5) is equivalent to n − 2γ g ∂u =− (Qγ − Qgγ )u. ∂t 4

(2.3)

∂ n dVg = − (Qgγ − Qgγ )dVg ∂t 2

(2.4)

We have the following: Proposition 2.1. There holds

along the fractional Yamabe flow (1.5). Proof. It follows from (2.1) and (2.3) that  n+2γ ∂u 2n 2n ∂ ∂  n−2γ 2n n dVg = u dV = u n−2γ dV = − (Qgγ − Qgγ )u n−2γ dV g g g ∂t ∂t n − 2γ ∂t 2 n g = − (Qγ − Qgγ )dVg . 2 This proves the assertion. 

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P.T. Ho / Nonlinear Analysis 139 (2016) 211–217

Lemma 2.2. Along the fractional Yamabe flow (1.5), the fractional order curvature satisfies ∂ g n − 2γ g g n + 2γ g g Qγ = Qγ (Qγ − Qgγ ) − Pγ (Qγ − Qgγ ). ∂t 4 4 Proof. Differentiate (2.2) with respect to t and using (1.4) and (2.3), we obtain   n+2γ n+2γ ∂ g n + 2γ − n−2γ −1 ∂u  g ∂u Qγ = − u Pγg (u) + u− n−2γ Pγ ∂t n − 2γ ∂t ∂t  g  n+2γ n+2γ n + 2γ g n − 2γ g − n−2γ − g g (Qγ − Qγ )u u n−2γ Pγ = Pγ (u) − (Qγ − Qgγ )u 4 4 n + 2γ g g n − 2γ g g Qγ (Qγ − Qgγ ) − Pγ (Qγ − Qgγ ), = 4 4 as required. 

(2.5)

Lemma 2.3. Along the fractional Yamabe flow (1.5), we have    d n − 2γ g Qγ dVg = − (Qgγ − Qgγ )2 dVg . dt 2 M M Proof. We have      d ∂ g ∂ Qgγ dVg = Qγ dVg + Qgγ dVg dt ∂t ∂t M M  M   n − 2γ g g n n + 2γ g g Qγ (Qγ − Qgγ ) − Pγ (Qγ − Qgγ ) dVg − Qg (Qg − Qgγ )dVg = 4 4 2 M γ γ M   n − 2γ n − 2γ g g g =− (Qγ − Qγ )Pγ (1)dVg − Qgγ (Qgγ − Qgγ )dVg 4 4 M M  n − 2γ Qgγ (Qgγ − Qgγ )dVg (2.6) =− 2 M where we have used (2.2), (2.4), (2.5) and the fact that Pγg is self-adjoint. It follows from (1.6) that  Qgγ (Qgγ − Qgγ )dVg = 0. M

Now the assertion follows by combining this and (2.6).



Now we are ready to prove Theorem 1.1. Proof of Theorem 1.1. It follows from (1.6) and Proposition 2.1 that     d ∂ n dVg = dVg = − (Qg − Qgγ )dVg = 0. dt 2 M γ M M ∂t Combining this with Lemma 2.3, we get     g d  M Qgγ dVg  n − 2γ M (Qgγ − Qγ )2 dVg = − .   n−2γ dt  dV  n−2γ 2 n n dV M

g

M

(2.7)

g

On the other hand, if g is a soliton to the fractional Yamabe flow given by (1.7), we have   n−2γ n ψt∗ ( g) g g − n+2γ 4 4 dVg = σ(t) 2 dVψ∗ ( and Q = P (1) = σ(t) P σ(t) γ γ γ t g)

(2.8)

P.T. Ho / Nonlinear Analysis 139 (2016) 211–217

215

by (1.4). This implies that σ(t)

Qgγ dVg M

 

M

dVg

n−2γ 4

=  n−2γ n

P t M γ

 σ(t)

n−2γ 4

n

σ(t) 2

2

M

dVψ∗ ( M g)

n−2γ 4

dVψ∗ ( g)

M

ψ ∗ ( g) Qγ t dVψ∗ ( M t g)

=  

σ(t)



dVψ∗ ( g) t



σ(t)  n−2γ M

σ(t)



 n−2γ n

t

 n−2γ n

t

n−2γ 4



=



ψ ∗ ( g)



ψ ∗ ( g)

Pγ t

dVψ∗ ( g)

(1)dVψ∗ ( t g)  n−2γ n

t

 g g ψt∗ (Q Q γ )dVψ ∗ ( γ dV M g t g) =  = n−2γ  n   n−2γ  n dVψ∗ ( dV M M g t g) 

M

(2.9)

ψ ∗ ( g) where the first equality follows from (2.8), the second equality follows from the fact that Pγ t is self-adjoint, the fourth equality follows from Lemma 2.4 that we will prove below, and the last equality follows from the assumption that ψt is diffeomorphism. Now, by combining (2.7) and (2.9), we can conclude that the fractional order curvature Qgγ of g is constant. This proves Theorem 1.1. 

It remains to prove the following: Lemma 2.4. There holds ∗ g) g Qγψ ( = ψ ∗ (Q γ)

for any diffeomorphism ψ on X. Proof. By the definition of the fractional order curvature, it suffices to show that  g  ∗ g) Pγψ ( (1) = ψ ∗ Pγ (1) . To show this, first we note that if the metric h+ in X has the normal form given by (1.1), then the metric ψ ∗ (h+ ) has the normal form given by ψ ∗ (h+ ) =

d(ψ ∗ (ρ))2 + ψ ∗ (gρ ) (ψ ∗ (ρ))2

(2.10)

where ψ ∗ (ρ) is a defining function on M satisfying ψ ∗ (ρ) > 0

in X,

ψ ∗ (ρ) = 0

on M,

d(ψ ∗ (ρ)) ̸= 0

on M

since ψ is a diffeomorphism on X. Note that ψ ∗ (gρ ) is a one-parametric family of metric on M satisfying ψ ∗ (g0 ) = ψ ∗ ( g ).

(2.11)

Now suppose u is a solution of the generalized eigenvalue problem n  n  − ∆h+ u − +γ − γ u = 0 in X 2 2 where u = F ρn−s + Gρs ,

F, G ∈ C ∞ (X),

(2.12)

F |ρ=0 = 1.

(2.13)

 n Γ (γ) Γ (γ) S + γ (1) = 22γ G|M . g Γ (−γ) 2 Γ (−γ)

(2.14)

Then by definition g Pγ (1) = 22γ

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P.T. Ho / Nonlinear Analysis 139 (2016) 211–217

It follows from (2.12) and (2.13) that ψ ∗ (u) is a solution of the generalized eigenvalue problem n  n  −∆ψ∗ (h+ ) ψ ∗ (u) − +γ − γ ψ ∗ (u) = 0 in X 2 2 ∗ and ψ (u) has the form ψ ∗ (u) = ψ ∗ (F )(ψ ∗ (ρ))n−s + ψ ∗ (G)(ψ ∗ (ρ))s where ψ ∗ (F ), ψ ∗ (G) ∈ C ∞ (X),

ψ ∗ (F )|ψ∗ (ρ)=0 = 1

since ψ is a diffeomorphism on X. Therefore, by definition, we have n   g  ∗ Γ (γ) ∗ Γ (γ) g) Sψ∗ (ρ) + γ (1) = 22γ ψ (G)|M = ψ ∗ Pγ (1) (1) = 22γ Pγψ ( Γ (−γ) 2 Γ (−γ) by (2.10), (2.11) and (2.14). This proves the assertion.



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