Evolution of Yamabe constant along the Ricci–Bourguignon flow

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Yamabe problem which is more or less the generalization of uniformization of compact surfaces to higher dimensional smooth manifolds is well known in the literature since the 60s and remains an active research area to date. An important geometric quantity associated with this problem is an invariant quantity known as Yamabe constant or Yamabe invariant. Detail descriptions of Yamabe constant and Yamabe invariant are given in Sect. 2. Yamabe invariant of smooth manifolds carries several important geometric and topological consequences or information. For instance, Yamabe invariant is positive if and only if the underlying manifold admits a metric of positive scalar curvature. Due to Perelman's resolution of Poincaré conjecture, it follows that a simply connected manifold can have negative Yamabe invariant only if it is of 4-dimensional. Our interest at this time is to study the behaviors of Yamabe invariant of manifolds evolving by certain geometric flow, which will enable us to reach some useful conclusions.
The aim of this paper is to provide an evolution formula for the Yamabe constant, which is defined as (2.2), under the Ricci-Bourguignon flow of an n-dimensional closed Riemannian manifold for n ≥ 3. The evolution of subcritical Yamabe constant was studied by Chang and Lu [10] under the Ricci flow and they established a differential inequality of constant under some technical assumption. Later Chang-Lu's results were extended to the relative subcritical Yamabe constant under the Ricci flow with boundary under the condition that the mean curvature of the boundary vanishes by Botvinnik and Lu [5]. Danesvar Pip and Razavi [13] extended the same results to the case of Bernhard List's flow. See also [8] and [22] for similar results under Cotton flow and conformal Ricci flow, respectively. Also, in [23,24] have been investigated the evolution of some geometric constants along the geometric flows.
Motivated by the above works we are concerned with the evolution of the Yamabe constant under Ricci-Bourguignon flow, as an application, we show that under some conditions, the initial metric is an Einstein metric if and only if the Yamabe flow constant is nondecreasing along the Ricci-Bourguignon flow. The rest of this paper is, therefore, planned as follows: Sect. 2 gives some basics and preliminary results on the Yamabe constant vis-a-vis Yamabe problem. Section 3 is devoted to the main results and their applications. In Sect. 4, we give three examples of the evolution of Yamabe constant on Einstein metrics, Ricci-Bourguignon soliton, and 3-dimensional Heisenberg Lie group.

Preliminaries
Given a smooth manifold M of dimension n ≥ 3, we consider M to be the set of Riemannian metrics on M.
Recall that the normalized Einstein-Hilbert functional E : M → R is given by where R g and dμ are the scalar curvature and the volume element of metric g, respectively. It is well known that every compact surface has a conformal metric of constant Gaussian curvature. A generalization of this is Yamabe problem, which asks if any Riemannian metric g on a compact smooth manifold M n of dimension n ≥ 3 is conformal to a metric with constant scalar curvature. In 1960, Yamabe [27] attempted to solve this problem, but his proof contained some error, discovered in 1968 by Trudinger [26]. Trudinger [26], Aubin [2] and Schoen [25] solved the Yamabe problem with a rather restrictive assumption on the manifold M. They proved that a minimum value of E(g) is attained in each conformal class of metrics and that this minimum is achieved by a metric of constant scalar curvature. Note that any metric conformal to g can be written as g = e 2 f g, where f is smooth real-valued function on M. Now recall that the Yamabe constant of a smooth metric g on a closed manifold M is given by where ∇ is the Riemannian connection on M. A function u for which Y (g) get its infimum is called the Yamabe minimizer (see [11,20,22,27]).
In the next, we denote := −a + R, where a = 4 n−1 n−2 , is Laplace-Beltrami operator and R is the scalar curvature of M. Yamabe problem is reduced to saying thatg = u 4 n−2 g has constant scalar curvature Y if and only if u satisfies the Yamabe equation The metric u 4 n−2 g is called the Yamabe metric and has constant scalar curvature. It happens that the exponent q = p − 1 = (n + 2)/(n − 2) in (2.3) is precisely the critical value, below (subcritical) which the equation is easy to solve and above which may be delicate. The existence of solution to (2.3) follows from direct method in the calculus of variation (cf. [20]). It is also observed that equation (2.3) is the Euler-Lagrange equation for the functional E(g). Thus, for a positive smooth function u satisfyingg = u 4 n−2 g, we have infimums in (2.1) and (2.2) being equal, that is, This constant Y (M) is an invariant of the conformal classes of (M, g) which is usually called the Yamabe (or Yamabe invariant) constant. Aubin [2] showed that the Yamabe problem can be solved on any compact where sup is taken over all smooth metrics on M. Therefore, the Yamabe minimizer is u (cf. [20]). In the critical and supercritical, the existence of solutions becomes a delicate issue. A useful way to handle supercritical problems consists in reducing the problem to a more general elliptic critical or subcritical problem, either by considering rotational symmetries or by means of maps preserving Laplace operator, or a combination of both (cf. [12] for instance). Chang and Lu [10] assumed that Yamabe minimizer is C 1 -differentiable with respect to variable t and then they investigated the evolution of the subcritical Yamabe constant under the Ricci flow. They also showed that, if g(0) is a Yamabe metric at time t = 0 and R g n−1 is not a positive eigenvalue of the Laplacian g α for any Yamabe metric g α in the conformal class [g 0 ], then d dt | t=0 Y (g(t)) ≥ 0. Recently, Daneshvar Pip and Razavi in [13] studied the evolution of the Yamabe constant under Bernhard List's flow. The results in this paper generalize and extend the aforementioned results [10,13] as highlighted in the introduction.

Variation of Yamabe constant
In this section, we will find evolution formulae for Y (t) along the flow (1.1). First, we recall some evolution formulae for geometric structure along the Ricci-Bourguignon flow. Next, we will present a useful proposition about the variation of Yamabe constant under the flow (1.1). From [7], we have the following lemma.
As a consequence of Lemma 3.1 we obtain the following result.

Lemma 3.2 Let (M, g(t)), t ∈ [0, T ) be a solution to the flow (1.1) on a closed oriented Riemannian
). Then we have the following evolutions:

2)
where u t = ∂u ∂t . Proof By direct computation in local coordinates we have where we used Lemma 3.1 (1) and gives exactly (3.1). Next, by using again Lemma 3.1 and the twice-contracted second Bianchi identity 2∇ i R i j = ∇ j R we infer So, the proof of is complete.
In the sequel, we shall state our main result.
Now, multiplying (3.6) by u and upon integrating we use (3.4) to obtaiñ since M u p dμ = 1. Taking time derivative of (3.7) and using Lemmas 3.1 and 3.2 yields Integrating by parts we obtain Combining (3.8) and (3.12) yields the expected evolution formula.

Remark 3.4
The traceless Ricci tensor of Riemannian manifold (M n , g) is defined by S i j = R i j − R n g i j . So, we can write R i j = S i j + R n g i j and |R i j | 2 = |S i j | 2 + R 2 n . Substituting these into the formula (3.5) with assumptions of Proposition 3.3 we can rewrite the evolution ofỸ (t) along the Ricci-Bourguignon flow as follows: Taking p = 2n n−2 in (3.13), then we obtain d dtỸ (3.14) which contains Chang-Lu's results [10], Proposition 1 under the Ricci flow (i.e., when ρ = 0). We also observe that the evolution of Yamabe constant remains the same both under the Ricci flow (at point n+2 n−2 ) and Ricci-Bourguignon flow (at point 2n n−2 ). This can be seen from (3.14) which is equivalent to Chang-Lu's results [10], Proposition 1 under the Ricci flow without necessarily setting ρ = 0. Proof By Koiso's decomposition theorem (Corollary 2.9 in [19] or Theorem 4.44 in [4]), there exists a C 1family of smooth positive functions u(t) on [0, ) for some > 0 with constant u(0), which satisfies the assumption of Proposition 3.3 for p = 2n n−2 . Obviously,Ỹ (t) =Ỹ p (t) is the scalar curvature of u(t) 4 n−2 g(t). Since u(0) and R g 0 are constant, ∇u(0) = 0 and ∇ R g 0 = 0. Hence, we have

Corollary 3.5 Let g(t) be the solution of the Ricci-Bourguignon flow on closed n-dimensional
Since the right-hand side of (3.15) is nonnegative, if d dtỸ (t)| t=0 = 0 then the trace Ricci tensor S i j (g 0 ) vanishes identically. Consequently, g 0 is an Einstein metric.
Notice that,Ỹ (t) in Corollary 3.5 cannot be equal to the Yamabe constant Y (g(t)) even if g 0 satisfies 0)). If we suppose that u(t) 4 n−2 g(t) has unit volume and constant scalar curvature Y (g(t)), then we can conclude as follows, which says that infinitesimally the Ricci-Bourguignon flow will try to increase the Yamabe constant. In what follows, we consider n-dimensional Riemannian manifold M whose sigma invariant is realized by some metric, the assumption is a little different from that of Corollary 3.6.
the metric g 0 is diagonal and we denote by where {θ 1 , θ 2 , θ 3 } is the dual coframe to the Milnor frame {X 1 , X 2 , X 3 }. We assume that be a solution of Ricci-Bourguignon flow. According to [17] under the normalization A 0 B 0 C 0 = 1, we get The Ricci-Bourguignon flow equations are then Starting with the equation for A, these can be integrated directly to have Therefore, for ρ = −1 Eq. (3.5) yields Thus, the quantity is nonincreasing along the Ricci-Bourguignon flow. If a = −2(1 + ρ) then we get This shows that the quantity is nonincreasing under the Ricci-Bourguignon flow.

Conclusion
We have obtained the evolution formula for Yamabe constantỸ (t) under the Ricci-Bourguignon flow as (3.5) and (3.13) on closed n-dimensional Riemannian manifolds with initial metric g 0 . We assume that g 0 is a metric of constant scalar curvature and we conclude that d dt | t=0Ỹ (t) ≥ 0 and the equality holds if and only if g 0 is an Einstein metric, whenever one of the following condition holds: (1) Also, we give examples on Einstein manifold, Ricci-Bourguignon soliton, and Heisenberg group in support of our results.

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Conflict of interest
We declare that there is no conflict of interest between the authors.