On the h-almost Yamabe Soliton

Fanqi Zeng

School of Mathematics and Statistics,Xinyang Normal University,Xinyang 464000,China.

Abstract.We introduce the concept h-almost Yamabe soliton which extends naturally the almost Yamabe soliton by Barbosa-Ribeiro and obtain some rigidity results concerning h-almost Yamabe solitons.Some condition for a compact h-almost Yamabe soliton to be a gradient soliton is also obtained.Finally,we give some characterizations for a special class of gradient h-almost Yamabe solitons.

Key words:Yamabe flow,h-almost Yamabe soliton,scalar curvature.

1 Introduction

The Yamabe flow was introduced by Hamilton at the same time as the Ricci flow as an attempt to solve the Yamabe problem on manifolds of positive conformal Yamabe invariant.Formally,the Yamabe flow deforms a given manifold by evolving its metric according to

where R(t)denotes the scalar curvature of the metric g(t).The Yamabe flow and the Ricci flow are equivalent in dimension n=2,but they are essentially different in higher dimensions[1].

A family of metrics g(t)=σ(t)g(0)solving(1.1),where σ(t)is a positive smooth function and ψt:M→M is a one-parameter family of diffeomorphisms of M,is said to be a self-similar solution of the Yamabe flow.Yamabe solitons are self-similar solutions of the Yamabe flow.A Riemannian manifold(Mn,g)is a Yamabe soliton if it admits a vector field X such that

where LXdenotes the Lie derivative in the direction of the vector field X and ρ is a real number.For ρ =0 the Yamabe soliton is steady,for ρ <0 is expanding,and for ρ >0 is shrinking.It has been known(see[2])that a compact Yamabe soliton has constant scalar curvature,thus trivial.For more details on Yamabe soliton we refer the reader to[3–5].

Barbosa and Ribeiro introduced the almost Yamabe soliton in[6]as follows.A Riemannian manifold(Mn,g)is an almost Yamabe soliton if there exists a complete vector field X and a smooth soliton function ρ on(Mn,g)satisfying

From the definition,if ρ is constant,almost Yamabe solitons are Yamabe solitons.

Recently,Gomes,Wang and Xia[7]have introduced the definition of the h-almost Ricci soliton.Such a soliton is a generalization of an almost Ricci soliton given by the authors in[8].An h-almost Ricci soliton is a complete Riemannian manifold(Mn,g)with a vector field X∈X(M),a soliton function λ:M→R and a function h:M→R which are smooth and satisfy the equation

In particular,the results contained in[7,9]indicates that h-almost Ricci solitons should reveal a reasonably broad generalization of the fruitful concept of the classical Ricci soliton,almost Ricci soliton and quasi Einstein manifolds[10–14].For further details about h-almost Ricci soliton,see[7,9].

Therefore,it is very interesting to consider a generalization of the almost Yamabe solitons.

Definition 1.1.We say that a Riemannian manifold(Mn,g)is an h-almost Yamabe soliton if there exists a complete vector field X,a smooth soliton function ρ:M→R and a signal function h:M→R which satisfy the equation:

where R denotes the scalar curvature of Mn.The function h is said to have a defined signal if either h>0 or h<0 on M.

In the following, we denote the h-almost Yamabe soliton satisfying(1.5)by(Mn,g,X,h,ρ).When ρ is constant the structure is said to be an h-Yamabe soliton.Moreover,if X=∇u,we call the equation

a gradient h-almost Yamabe soliton,where∇2u denotes the Hessian of u.An h-almost Yamabe soliton is said to be shrinking,steady or expanding if it admits a soliton field for which,respectively,ρ>0,ρ=0 or ρ<0.Otherwise,it will be called indefinite.

Moreover,when either the vector field X is trivial or the potential u is constant,an h-almost Yamabe soliton will be called trivial,otherwise it will be a nontrivial h-almost Yamabe soliton.Let us point out that the traditional Yamabe soliton is a 1-Yamabe soliton with constant ρ.Moreover,1-almost Yamabe soliton is just the almost Yamabe soliton.Some interesting examples of 1-almost Yamabe soliton are given in[6].

Inspired by[15],Leandro and Pina[16]have introduced the definition of generalized quasi Yamabe gradient soliton,more precisely,we say that a complete Riemannian manifold(Mn,g)with n≥3 is a generalized quasi Yamabe gradient soliton,if there exists a constant ρ and two functions f and µ defined on M such that

where df is the dual 1-form of∇f.When µ=,where m is not zero,the above generalized quasi Yamabe gradient soliton is called a quasi Yamabe gradient soliton(see[17])which will be denoted by(Mn,g,f).It has been proved in[17]that locally conformally flat quasi Yamabe gradient solitons with positive sectional curvature are rotationally symmetric.Moreover,they proved that a compact quasi Yamabe gradient soliton has constant scalar curvature.Wang[18]gave several estimates for the scalar curvature and the potential function of the quasi Yamabe gradient solitons.In[19,20],they defined the almost quasi Yamabe solitons

where ρ is a function.They also obtained some conditions for an almost quasi Yamabe soliton to be trivial and some necessary and sufficient conditions for a compact almost quasi Yamabe soliton to be a gradient soliton.

Now we consider a nonconstant function u=.We have

Consequently,Equation(1.8)can be rewritten as

which implies that all almost quasi Yamabe solitons are a gradient(−)-almost Yamabe soliton.

From the equivalence between Eqs.(1.8)and(1.10),we have the following example.

Example 1.1.Let(Rn,geuc)be the Euclidean space.For a nonzero real number m and a positive constant β,the functions

parameterize(Rn,geuc)with a nontrivial gradient(−)-almost Yamabe soliton.Indeed,since Rnhas null scalar curvature and∇2||x||2=2geuc,we conclude our statement by straightforward computation from Eq.(1.10).

In this paper,We treat the Yamabe soliton equation,the almost Yamabe soliton equation and almost quasi Yamabe soliton equation in a unified way,and it is interested to investigate whether the h-almost Yamabe solitons share a similar property.For what follows we assume that(Mn,g)has dimension n≥3 and that h has defined signal.

In the following we show that,with natural conditions and non-positive Ricci curvature,any complete h-almost Yamabe soliton is trivial.

Theorem 1.1.Suppose(Mn,g,X,h,ρ)is an h-almost Yamabe soliton with non-positive Ricci curvature such that

Then(Mn,g,X,h,ρ)is a trivial h-almost Yamabe soliton.Here d(x,x0)denotes the distance from x to x0with respect to the Riemannian metric g.

We remark that we do not assume in Theorem 1.1 that the h-almost Yamabe soliton(Mn,g)is either gradient or compact.Note also that the corresponding theorem for the Yamabe soliton was proved by[21].

The next result gives some integral conditions which force h-almost Yamabe soliton to be trival.Its proof is motivated by the corresponding result for almost Yamabe solitons proven in Barbosa-Ribeiro[6].

Theorem 1.2.Let(Mn,g,X,h,ρ)be an h-almost Yamabe soliton.Then(Mn,g,X,h,ρ)is trivial if one of the following conditions holds:

We remark that(1.5)implies that the associated vector field X is a conformal vector field with conformal factor h−1(R−ρ).Therefore,Theorems 1.1 and 1.2 give some conditions that X is a Killing vector field,i.e.,R=ρ.

On the other hand,a conformal vector field X is called closed when it satisfies the following condition:

for all Y∈X(M).Obviously,a gradient conformal vector field X=∇u is closed.When we have a compact h-almost Yamabe soliton,we can use the Hodge-de Rham decomposition theorem to write

where Y ∈X(M)satisfies div Y=0,whereas φ is a smooth function on(Mn,g).The following theorem gives some conditions for a compact h-almost Yamabe soliton to be a gradient soliton.

Theorem 1.3.Let(Mn,g,X,h,ρ)be a compact h-almost Yamabe soliton.The following statements hold:

1.If it is also a gradient h-almost Yamabe soliton with potential u,then,up to a constant,u agrees with the Hodge-de Rham potential φ.

2.The compact nontrivial h-almost Yamabe soliton of constant scalar curvature with closed conformal vector field X vanishing at some point of(Mn,g)admits a gradient structure.

3.The compact h-almost Yamabe soliton(Mn,g,X,h,ρ)is gradiant if and only if

We point out that Theorem 1.3 is an extension of the almost Yamabe soliton case and almost quasi-Yamabe soliton case proved recently in[6,20]to the h-almost Yamabe soliton case.

The last two theorems give strong restrictions to a special class of gradient h-almost Yamabe soliton.Its proof is motivated by[22]and[20].

Theorem 1.4.Let u be a positive function on(Mn,g)and m a nonzero constant.If(Mn,g,∇u,−,ρ)is a compact gradient(−)-almost Yamabe soliton with constant scalar curvature such that its first nonzero eigenvalue λ1satisfies

2.If(Mn,g)is compact and R is a constant,then(Mn,g)is isometric to a Euclidean sphere.

2 Preliminaries

3 Proof of Theorem 1.1

In view of(1.11),(3.1)and the assumption that(Mn,g)has non-positive Ricci curvature,we can conclude that the right hand side of(3.7)tends to zero as r→∞.So we can conclude that the left hand side of(3.7)tends to zero as r→∞,which implies(h−1(R−ρ))2=0 in(Mn,g).Hence Theorem 1.1 follows. □

4 Proof of Theorem 1.2

The following lemmas play crucial roles in this section:

Lemma 4.1([24]).Let T be a symmetric(0,2)-tensor on a Riemannian manifold(Mn,g)and ψ a smooth function on(Mn,g).Then we have

for all Z∈X(M).

Lemma 4.2([25]).For any conformal vector field X on a compact Riemannian manifold(Mn,g),the following identity holds

Since we are assuming that the right hand side is less than or equal to zero,we obtain∇X=0,therefore,LXg=0.Hence,(Mn,g)is a trivial h-almost Yamabe soliton.

(4)We use Lemma 4.3 to deduce from(2.1)that div X=0,therefore,R=ρ which implies that(Mn,g)must be a trivial h-almost Yamabe soliton.We complete the proof of the theorem.

5 Proof of Theorem 1.3

The following lemmas play crucial roles in this section:

Lemma 5.1([27]).Let X be a nontrivial conformal vector field on a compact Riemannian manifold(Mn,g)of constant scalar curvature R.Then R>0.

Lemma 5.2([28]).Let(Mn,g)be a connected compact Riemannian manifold with the constant curvature R>0.Then(Mn,g)is globally isometric to a sphere if(Mn,g)admits a closed conformal vector field X which vanishes at some point of(Mn,g).

(2)Since(Mn,g)is compact with constant scalar curvature R and X is a nontrivial closed conformal vector field vanishing at some point of(Mn,g),using Lemmas 5.1 and 5.2,we conclude that(Mn,g)is isometric to a Euclidean sphere Sn.Thus,(Mn,g)is an Einstein Riemannian manifold,i.e.,

Recall that for a compact h-almost Yamabe soliton(Mn,g,X,h,ρ)it follows that

6 Proof of Theorem 1.4

7 Proof of Theorem 1.5

We need the following key lemma:

Lemma 7.1([29]).Let u be an L2function,form or tensor that satisfies Δu=κu for some κ>0.Then u is identically zero.

(2)From(1.10),∇u defines a closed conformal vector field on(Mn,g).Assume that(Mn,g)is compact with constant scalar curvature R.Hence,from Lemma 5.1,we have that R is positive.Moreover,since(Mn,g)is compact,there exists a point p∈M such that∇u(p)=0.Therefore,we may use Lemma 5.2 to obtain that(Mn,g)is isometric to an Euclidean sphere Sn,which concludes the proof of the theorem.

Acknowledgments

The work is supported by the National Natural Science Foundation of China(Grant No 11971415),and Henan Province Science Foundation for Youths(No.212300410235),and the Key Scientific Research Program in Universities of Henan Province(No.21A110021)and Nanhu Scholars Program for Young Scholars of XYNU(No.2019).