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Basic structural equations for almost Ricci-harmonic solitons and applications

This paper studies gradient almost Ricci-harmonic soliton with respect to a fixed metric. We rely on analytic techniques to estabilish some basic elliptic and integral equations for the structure of almost Ricci-harmonic soliton which generalizes that of Ricci-hamonic solitons on one hand and that of almost Ricci soliton on the other hand. The consequences of these formulas in relation to classification and rigidity results for gradient almost Ricci-harmonic solitons are also discussed.


Introduction and the setting
Let (M, g) be a complete Riemannian manifold. (M, g) is said to have Ricci soliton structure if it satisfies the soliton equation, for any smooth vector field X and some constant σ ∈ R, where Rc is the Ricci tensor of M, g is the metric and L X is the Lie derivative in the direction of X. In this case, the triple (g, X, σ) is called Ricci soliton and said to be shrinking, steady or expanding if σ > 0, σ = 0 or σ < 0, respectively. If X is a gradient of some smooth function f on M, that is, X = ∇f , Ricci soliton equation (1.1) is then written as Rc + Hessf = σg, (1.2) where Hessf is the Hessian of function f . Then the triple (g, f, σ) is called gradient Ricci soliton and f is called the potential function. If either X or ∇f is a Killing vector field or f is a constant, we say that the soliton is trivial and the underlying metric is Einstein. Compact Ricci solitons are special solutions to Hamilton's Ricci flow [20] and they occur as the flow's singularity models or blow-up limits [8,26]. We refer the reader to [8,12,13] for background on Ricci solitons and their connection to the Ricci flow. Ricci soitons are also very useful in theoretical physics as they have connection with renormalization group flow, Einstein field and string theories [10,18]. A generalization of Ricci solitons is the Ricci-harmonic solitons which serve as special solutions of the Ricci-harmonic flow.
1.1. The Ricci-harmonic flow. Let (M, g) and (N, h) be compact Riemannian manifolds of dimensions m and n respectively. Let a smooth map u : M → N be a critical point of the energy integral E(u) = M |∇u| 2 g dµ g , where N is isometrically embedded in R d , d ≥ n, by the Nash embedding theorem. These critical points are called harmonic maps between M and N, which are generalization of harmonic functions such as identity and constant maps. The configuration (g(x, t), u(x, t)), t ∈ [0, T ) of a one parameter family of Riemannian metrics g(x, t) and a family of smooth maps u(x, t) is defined to be Ricci-harmonic flow if it satisfies the coupled system of nonlinear parabolic equations Here α(t) ≡ α > 0 is a time-dependent coupling constant and τ g u is the intrinsic Laplacian of u which denotes the tension field of map u. The system (1.3) was first studied by List [23] in a special case (N, h) = (R, dr 2 ), with motivation coming from general relativity. When u is a constant function, List's flow becomes the well known Hamilton's Ricci flow. List's flow was generalized by Müller [24] to the general case N ֒→ R d , for sufficiently large d as we see in system (1.3). Precisely, the system couples together the Hamilton's Ricci flow [20] and Eells and Sampson's heat flow for harmonic maps [14]. The Ricci-harmonic flow is closer to the Ricci flow in behaviours, though less singular. For a detail introduction to the study of the Ricci flow see the book by Chow and Knopf [12] or Chow, Lu and Ni [13]. The note [15] will provide necessary background on harmonic map theory. The short-time existence, uniqueness and obstruction to long-time existence of Ricci-harmonic flow are discussed in [24], see also [22]. In recent time, several researchers have extended fundamental results such as Perelman's entropy formulas, no breather theorems, noncollapsing theorems, volume growth, gradient estimates and Harnack inequalities in the study of Ricci flow to the setting of Ricci-harmonic flow, for example, see [1,2,3,19,23,25,29,30,32,34]. where α > 0 is a positive constant depending on m and σ ∈ R. The function f is called the potential. The gradient Ricci-harmonic soliton is said to be shrinking, steady or expanding depending on whether σ > 0, σ = 0 or σ < 0. We say that the soliton is trivial if f is set to be constant. Here are some examples as they appear in certain contexts: (1) If u is a constant map, then (1.4) becomes which is the equation of gradient Ricci soliton. Rc − α∇u ⊗ ∇u = σg τ g u = 0, which is usually called harmonic-Einstein and generalizes the notion of Einstein manifold. For generalization and some extension of results on gradient Ricci solitons see [7,11,16,21,28,33]. The paper [8] (see also the references therein) provides details on the geometry of Ricci solitons. Recently, a particular generalization of gradient Ricci soliton, called gradient almost Ricci soliton, was proposed in [28]. The concept appears to be natural and meaningful, though, almost Ricci soliton are not necessarily gradient. See [28,4,5,6,17] for detail discussion on existence, rigidity, triviality, classification and so on for gradient almost Ricci soliton. In this paper however, we are considering gradient almost Ricci-harmonic solitons, in which case, σ appearing in (1.4) is not necessarily a constant but a smooth function on M. In this setting we adopt similar terminology, where we say gradient almost Ricci-harmonic soliton is shrinking, steady or expanding depending on whether σ is positive, null or negative on M. The gradient almost Ricci-harmonic soliton is said to be trivial if f is a constant.
The organization of the paper is as follows: In Section 2, we give detail of notation adopted in the paper and statements of results to be proved later. Sections 3 discusses basic equations which will be used to prove some classification and rigidity results for almost Ricci harmonic solitons. The last section is devoted to the proofs of main results.

2.1.
Notation. In the sequel, Ricci harmonic soliton will be abbreviated as (RH) α soliton. We write (0, 2)-tensor Sc := Rc − α∇u ⊗ ∇u, its components as S ij := R ij − α∇ i u∇ j u in local coordinates while its metric trace g ij S ij is denoted by S := R − α|∇u| 2 , where Rc, R ij and R are the Ricci curvature tensor, its components and scalar curvature, respectively. div is the divergence operator, i.e., (divSc) k = g ij ∇ i S jk . For a smooth map u(x) : (M, g) → (N, h), where g = g ij dx i dx j and h = h νγ du ν du γ , the energy density of u(x) at a point x ∈ M is |∇u| 2 g = g ij h νγ ∂ i u ν ∂ j u γ , i.e., the trace g u * h and ∇u ⊗ ∇u = u * h is the pullback of the metric tensor h on N via the map u. The tension field of map u is denoted by τ g u = trace g ∇du, which is a section of the bundle u * (T u N), where ∇ is the connection on the vector bundle T * M ⊗ u * (T N) induced by the Levi-Civita connections on M and N.
For a smooth function f on M, we write |∇f | 2 = g ij ∇ i f ∇ j f and ∆ is the Laplace Beltrami operator, while the volume measure on M is denoted by dM = |g|dx i , i = 1, 2, ..., m, where |g| is the determinant of g.

2.2.
Statement of results. The main results of this paper which concern classification and rigidity of the gradient almost (RH) α solitons are stated in this section. Basic formulas which will help in proving them are dealt with in Section 3. (iv) ∇f is a conformal vector field. Here dM is the volume measure of M.
We notice also that if u is a nonconstant map, one can still deduce the last result. See the discussion in Remark 4.4.

Basic equations for almost Ricci-harmonic solitons
Firstly, we shall prove some basic equations which will be applied to derive some classification and rigidity results. The equations are extension to almost (RH) α solitons of the classical ones derived for almost Ricci solitons. We include the proofs for completeness sake and since some new terms come in.  The proofs of the above equations follow from standard computation which is usually performed with the method of the moving frame in a local orthonormal frame.
Note that the first equation in (1.4) implies Taking covariant derivative of (3.6) we have since g ij,k = 0 (covariant derivative of g) by metric compatibility of Levi-Civita connection. Tracing with respect to j and k we have Recall the Ricci identity and the first formula in the proposition Now, putting (3.9) and (3.10) into (3.8), we have Tracing (3.7) with respect to i and j, we have Substituting (3.12) into (3.11) we have In particular, we have from (3.13) ∇S, ∇f = 2(m − 1) ∇σ, ∇f + 2Rc(∇f, ∇f ) − 2ατ g (u) ∇u, ∇f = 2(m − 1) ∇σ, ∇f + 2Sc(∇f, ∇f ), (3.14) where we have used τ g (u) = ∇u, ∇f and Rc = Sc + α∇u ⊗ ∇u. This proves the second formula in the proposition. The third equation in the proposition can be seen as a direct consequence of (3.13) or (3.14). But we prove it this way: Taking divergence of the first equation in (1.4) yields divRc − αdiv(∇u ⊗ ∇u) + div Hessf = div(σg). The fourth equation is another consequence of (3.13). It has been previously derived in [4,17] by different methods. Equation (3.13) implies The fourth equation is proved.
Inserting ∆f = mσ − S into the well known Bochner formula we have Using the second equation of the proposition yields which implies the fifth equation. Therefore the proof of the proposition is complete.
Lastly, we derive some integral equations which will be used to derive an application which generalizes results obtained in [4] for almost Ricci solitons.

Proofs of results
The aim of this section is to use some of the equations obtained in the last section to classify gradient almost (RH) α solitons and determine when they are rigid as stated in Section 2. The first result to prove is Theorem 2.1. Since M is compact, there exists a point q ∈ M, where S attains its minimum. By the maximum principle, at this point ∇S(q) = 0 and ∆S(q) ≥ 0. Then the last inequality implies The first case is when σ = 0, that is, the case of steady gradient almost (RH) α soliton, we have S min = 0 by (4.2). In the case of shrinking gradient almost (RH) α soliton σ > 0 with 0 < σ ≤ σ * , by (4.2) we have 0 ≤ S min ≤ mσ * . In the case of expanding gradient almost (RH) α soliton σ < 0 with σ * ≤ σ < 0, we have mσ * ≤ S min < 0 and S ≥ S min ≥ mσ * .
Remark 4.1. If one follows the steps in the proof of Theorem 0.4 of [28], it can be further shown that under a pairwise control on σ, the curvature function S of gradient almost (RH) α soliton is bounded from below and the lower bound can be estimated from both below and above.
We note that when S attains its minimum, by the maximum principle the inequality implies that if S is nonconstant, then it must be everywhere positive and then σ is positive, which says that gradient almost (RH) α soliton is shrinking. This implies the following proposition. This implie that ∆S min ≤ 0 since S min = 0 ( for the steady case) and S min ≥ mσ * (for the expanding case). Then S ≥ S min ≥ 0 is nonnegative superharmonic and it is therefore constant.
The above proposition forces ∆f = mσ−S to be ∆f ≥ 0 since S = mσ * ≤ mσ. Thus f is subharmonic and the compactness of M implies that f is constant and the soliton is trivial. Hence, the following corollary.  Note that the second on the right hand side of (4.5) is nonnegative and by assumption the first term is nonpositive. Then we obtain Hessf = 0 which implies ∇f is a killing vector field and that f is a constant by compactness of M.  and we conclude that (M, g) is Einstein, since every Einstein manifold belongs to the class of Riemannian manifolds (M, g) realizing the relation R ij = ag ij + b∇ i v∇ j v, where a and b are certain nonzero constants (see [9]).