Existence of the $(\alpha,\beta)$-Ricci-Yamabe flow on closed manifolds

On a smooth closed Riemannian manifold, we show short time existence of smooth solutions to the $(\alpha,\beta)$-Ricci-Yamabe flow, which is a natural generalization of the Ricci flow and the Yamabe flow. We also establish some long time existence theorems for the closed $(\alpha,\beta)$-Ricci-Yamabe flow by estimating its curvatures.

Using the Nash-Moser inverse function theorem, Hamilton [11] proved short time existence for smooth solutions to the Ricci flow ∂ ∂t g = −2Ric on a closed manifold for the first time.Shortly after that, DeTurck [7] subsequently simplified the short time existence proof by modifying the flow and showing that the Ricci flow could be replaced by an equivalent PDE which is strictly parabolic. 1 More generally, Shi [35] proved short time existence for the Ricci flow on a complete noncompact Riemannian manifold if the initial Riemannian curvature is bounded.
In 1995, Hamilton [12] provided a long time existence criterion that a closed Ricci flow defined on a maximal interval [0, T ) satisfies either T = ∞ or the maximum of the norm of the Riemannian curvature blows up at the finite time T .About a decade later, Šešum [34] proved that if the Ricci curvature is uniformly bounded along the closed Ricci flow defined on a finite time interval, then the Riemannian curvature stays uniformly bounded along the flow by using Perelman's noncollapsing theorem (see [30]) and Hamilton's compactness theorem (see [13]).Šešum's result still holds on the noncompact setting that the uniformly bounded Ricci curvature implies the uniformly bounded Riemannian curvature along the Ricci flow on a complete noncompact manifold with bounded Riemannian curvature at the initial time (see the work of Ma-Cheng [28], Kotschwar-Munteanu-Wang [19], and Hsu [17]).
There is a well-known conjecture on a closed Ricci flow g(•, t) 0 ≤ t < T < ∞ that a uniform bound for the scalar curvature is enough to extend Ricci flow over time T .This conjecture is achieved in dimension 3 by Hamilton-Ivey's pinching estimate (see [14,16]).Zhang [40] and Enders-Müller-Topping [9] partially settled this conjecture for Kähler-Ricci flow and Type I Ricci flow, respectively.Wang [37] proved that the closed Ricci flow g(•, t) 0 ≤ t < T < ∞ can be smoothly extended past T if the Ricci curvature tensor is uniformly lower bounded and the scalar curvature satisfies some space-time integral bounds.This result was generalized to the complete case by Di Matteo [29] with additional conditions in 2021.
Ma and Cheng [28] derived a smooth extension result for a closed Ricci flow with finite L n+2 2 norms of the scalar curvature and Weyl tensor, and showed the Riemannian curvature of a complete Ricci flow stays uniformly bounded along the flow if the sectional curvature is bounded at the initial time and the scalar curvature and Weyl tensor are uniformly bounded.For a closed Ricci flow g(•, t) defined on a maximal interval [0, T ), Cao [5] proved either the scalar curvature blows up at the finite time T or where W = {W ijkl } is the Weyl tensor of g(•, t) defined by R(g ik g jl − g il g jk ).
The Yamabe flow, which is an intrinsic geometric flow on a Riemannian manifold, is defined by ∂ ∂t g = −Rg, while the volume preserving normalized Yamabe flow is where r is the average scalar curvature.Given an initial metric on a compact locally conformally flat manifold with positive Ricci curvature, Ye [39] proved that the solution g(t) to the normalized Yamabe flow (1.2) exists for all time and converges in C ∞ norm to a conformal metric of constant scalar curvature.In 2005, Brendle [3] proved the flow (1.2) on a closed manifold exists for all time and converges to a metric with constant scalar curvature if the dimension n of the underlying manifold is 3 ≤ n ≤ 5 or the initial metric is locally conformally flat.Catino, Cremaschi, Djadli, Mantegazza and Mazzieri [4] proved short time existence and curvature estimates for the Ricci-Bourguignon flow which is a family of geometric flows introduced by Bourguignon [2] in 1981 and can be viewed as a special case of (1.1) by taking α = 2 and β = −2ρ.
Liang and Zhu [26] proved that the norm of the Weyl tensor of any smooth solution to the Ricci-Bourguignon flow can be explicitly estimated in terms of its initial value on a given ball and a local uniform bound on the Ricci tensor.As an application, they [26] concluded that the Riemannian curvature is uniformly bounded along the Ricci-Bourguignon flow defined on a finite time interval with uniformly bounded Ricci tensor.More recently, Qiu and Zhu [32] showed the short time existence of the Ricci-Bourguignon flow on a compact Riemannian manifold with constant mean curvature on the boundary if the initial metric has constant mean curvature and satisfies some compatibility conditions.
In this paper, we consider short and long time existence of the (α, β)-Ricci-Yamabe flow (1.1) on a smooth closed Riemannian manifold.The first main result is a short time existence theorem of a smooth solution g(x, t) to (1.1).
There exists a long time existence criteria for smooth solutions to (1.1) that is a generalization of Hamilton's long time existence theorem for the Ricci flow (see [12]).
Similary, we also have a smooth extension result for a smooth solution g(x, t) to (1.1) with conditions on the Ricci curvature and the scalar curvature.
then this flow can be extended smoothly over time T .
Moreover, we derive a long time existence theorem of the (α, β)-Ricci-Yamabe flow (1.1) in terms of its scalar curvature and Weyl tensor, which extends a previous result of Cao [5].
Theorem 1.4.Let g(x, t), 0 ≤ t < T < +∞, be a maximal smooth solution to the (α, β)-Ricci-Yamabe flow (1.1) with α > 0 and β > − α n−1 on an n-dimensional This paper is arranged as follows.In Section 2, we establish the short time existence of a smooth solution g(x, t) to the (α, β)-Ricci-Yamabe flow (1.1).We mainly derive evolution equations of curvature operators in Section 3. In Section 4, we estimate every ordered covariant derivative of the Riemannian curvature step by step and give a proof of Theorem 1.2.In Section 5, we show how the Riemannian curvature of g(x, t) can be locally controlled by the Riemannian curvature of the initial metric, the Ricci curvature and the second-order derivative of the scalar curvature.Moreover, we finish the proof of Theorem 1.3.In the last section, we prove Theorem 1.4 that relates the long time existence of the (α, β)-Ricci-Yamabe flow to the behavior of the scalar curvature and Weyl tensor.

Short time existence
In this section, we prove the short time existence of the (α, β)-Ricci-Yamabe flow (1.1) by DeTurck's trick which DeTurk [7,8] established to show that the Ricci flow is equivalent to an initial value problem for a strictly parabolic linear second ordered partial differential equation.
Proof of Theorem 1.1:For each h ∈ Γ(S 2 M ), the differentials of the Ricci curvature and the scalar curvature at normal coordinates of g 0 in the direction of h (see Theorem 1.174 in [1], [4] or [36]) are , respectively, where we omit the lower ordered terms.Then the linearization (without lower ordered terms) of the second ordered nonlinear partial differential operator Let X : Γ(S 2 M ) → Γ(T M ) be the vector field defined by Therefore, the linearization (without lower ordered terms) of the Lie derivative L X g 0 of g 0 in the direction of X is Following DeTuck's trick ( [7,8]), we only need to check the operator D(P −L X ) g0 is strongly elliptic.

Curvature evolution equations
In this section, we calculate evolution equations for the Riemannian curvature Rm, Ricci curvature Ric and the scalar curvature R of a smooth solution g(x, t) to the (α, β)-Ricci-Yamabe flow (1.1) as Hamilton [11] did for the Ricci flow in 1980s.Moreover, we show some curvature conditions that are preserved along the (α, β)-Ricci-Yamabe flow (1.1).
The notations we used are in accordance with that in Hamilton [11].The Riemannian metric is g ij (without the subscripts x and t in the components) and its inverse is g ij .The Levi-Civita connection is given by the Christoffel symbols

The tensor introduced by Hamilton [11]
B ijkl = g pr g qs R piqj R rksl satisfies the symmetries B ijkl = B jilk = B klij .In fact, the evolution equations of the Christoffel symbols and the scalar curvature can be found in [4].For completeness, we give a detail proof in this section.
The following two formulas, which are independent of any evolution equation, given in Hamilton [11] are needed in the proof of curvature evolution equations.Lemma 3.1 (Hamilton [11]).For any metric g ij the curvature tensor R ijkl satisfies the identity while the tensor B ijkl satisfies the identity Theorem 3.2.The Riemannian curvature satisfies the evolution equation Proof.We calculate the formulas in normal coordinates at any fixed (x, t).Using (1.1), we have Hence, where we used (1.1) and (3.5) in the second equality, and (3.1) in the second.This proves (3.3).
Then we derive evolution equations for the Ricci curvature and the scalar curvature, respectively.Theorem 3.3.The Ricci curvature satisfies the evolution equation while the scalar curvature satisfies where By direct computation, we have where we used (3.3) and (3.9) in the third equality, (3.2) in the fourth.Similarly, where we used (3.9) in the second equality and (3.7) in the third.This completes the proof.
As applications of (3.8), we show how the lower bound of the scalar curvature can be preserved along the (α, β)-Ricci-Yamabe flow (1.1).
Proof.It follows from (3.8) that Since n ≥ 2 and β(n − 1) + α > 0, the minimum of R on M satisfies ) for any t ∈ [0, T ).This proves the first result.
If a > 0, we can rewrite (3.11) to Integrating (3.12) from 0 to t for any 0 < t < T and using the fact of R min (0) ≥ a gives that Therefore, the arbitrary of t implies the maximal existence time T satisfies Now we consider the case of R g(•,0) ≥ 0. by applying the strong maximum principle to (3.10), we know that R g(•,t) > 0 or R g(•,t) ≡ 0 for any 0 < t < T .In the latter case, (3.8) implies that Ric g(•,t) ≡ 0.
This completes the proof.
There is a result for the stationary solution to (1.1) when it exists.Proof.Let g ∞ be the stationary solution to (1.1).Then its Ricci curvature and scalar curvature satisfy By Schur's lemma (see e.g.[31]), (M, g ∞ ) must be Einstein.Assume that R ∞ = 0, then Therefore, We conclude (M, g ∞ ) is Ricci flat.

Derivative estimates for the Riemannian curvarure and long time existence I
In this section, we show that every ordered covariant derivative of the Riamnnian curvature of smooth solutions to (1.1) is bounded by only assuming the Riemannian curvature is uniformly bounded.As an application, we prove a long time existence theorem (Theorem 1.2) for (1.1).
As in [35,24,19], we use the standard * -notation that A * B represents some linear combination of contractions of the tensor product A ⊗ B of any time-dependent tensor fields A and B by using the metric g(x, t).
Hence, (3.3) and (3.4) can be rewritten as and respectively.More generally, a time-dependent tensor field A satisfies and 3) Then we prove the evolution equation for kth-order covariant derivative of the Riemannian curvature.Proposition 4.1.For any nonnegative integer k, we have along the (α, β)-Ricci-Yamabe flow (1.1).
Proof.We prove this result by induction.The case of k = 0 holds since (3.3) and (3.8), respectively.Suppose that (4.4) and (4.5) hold for k = m > 0, we need to show that they hold for k = m + 1. Substituting A by ∇ m Rm in (4.2) and (4.3), and using the inductive hypothesis, we obtain that which yields (4.4) for k = m + 1.
Similarly, we have Lemma 4.2.For any nonnegative integer k, we have along the (α, β)-Ricci-Yamabe flow (1.1) with α > 0 and β > − α n−1 , where C k is a positive constant depends only on n and k.
Proof.Calculating directly, we have where we used (1.1) in the first inequality and (4.4) in the second.This proves (4.6).Similarly, by using (1.1), (4.5) and Cauchy's inequality, we can obtain that which yields (4.7).
Then we show that all the L 2 -norms of derivatives of the Riemannian curvature and the scalar curvature are bounded if the Riemannian curvature is uniformly bounded.The arguments are motivated by curvature estimates for the Ricci flow by Shi [35] and a geometric flow over Kähler manifold by Li, Yuan and Zhang [24].
The following analytic result is necessary.
There exists a constant C(n, k, p, q) such that for all 0 ≤ j ≤ k and all tensors T on M × [0, a/K] for some positive constants a and K, then for each positive integer k, there exists a nonnegative constant B k that depends only on k, n, α, β, a, K and V ol g(0) (M ) so that Here for any time-dependent L p function A on M with 1 ≤ p < ∞.
Proof.We derive (4.8) step by step.First of all, we consider the case of k = 1.Define with A 0 is a positive constant to be determined later.We need to compute du0 dt .It follows from (1.1) that i.e., Integrating (4.10) from 0 to t, we arrive at and Hence, where we used (4.9), (4.12) and Stokes's formula, and where we used (4.9) and (4.13) and Stokes's formula.Moreover, it follows from (3.8) that Combining with (4.9) and using Stokes's formula, we have By Cauchy's inequality and the fact of t ≤ a/K, we obtain that and Choose then the first term of RHS of (4.20) vanishes.For an arbitary 0 < t ≤ a/K, integrating (4.20) from 0 to t, we conclude that u 0 ≤ B 0 t for some constant B 0 depends only on n, α, β, a, K and V ol g(0) (M ).Moreover, Next, we estimate ∇Rm 2 2 + ∇ 2 R 2 2 based on (4.22).Define , where A 0 is defined in (4.21) and A 1 is a positive constant to be determined later. Then In the following, we deal with RHS of (4.25).By Cauchy's inequality, we have ) and where we used Hölder's inequality in the first inequality, Proposition 4.3 by taking j = 1, k = 2, p = ∞ and q = 2 in the second inequality and Cauchy's inequality in the last.Plugging (4.26) to (4.30) into (4.25) and rearranging, we can obtain Choose Applying (4.22) to (4.31) and using t ≤ a/K and (4.11) again, we conclude that For an arbitrary 0 < t ≤ a/K, integrating (4.32) from 0 to t gives u 1 ≤ B 1 t for some constant B 1 depends only on n, α, β, a, K and V ol g(0) (M ).Moreover, Define Using (4.22), (4.33) and reasoning similar to (4.25), (4.31), we can estimate du2 dt and then obtain that B 2 t 2 for some constant B 2 depends only on n, α, β, a, K and V ol g(0) (M ).
Repeating this procedure step by step, we conclude (4.8) for each positive integer k.
In the rest of this section, we prove derivative estimates for the Riemannian curvature under the flow (1.1).⌉ that is the smallest integer larger than n(k+1)

2
. By taking p = ∞ and q = 2 in Proposition 4.3 and then using the fact from Theorem 4.4 of By Sobolev embedding theorem (see e.g.Theorem 11.1.1 in [18]), we know that sup From Theorem 4.5, we know that all covariant derivatives of the Riemannian curvature stay uniformly bounded along this flow.Hence, g(•, t) converges smoothly to a complete limit metric g(•, T ).Then the flow (1.1) can be restarted from the new initial metric g(•, T ) by the short-time existence result Theorem 1.1, which contradicts that the finite T is a maximal existence time.
This proves this theorem.

Local curvature estimates and long time existence II
Inspired by the works of Kotschwat-Munteanu-Wang [19] in the study of Ricci flow and Li-Yuan [23] in the study of κ-LYZ flow, we prove that upper bounds for the norm of the Riemmanian curvature on an given geodesic ball of a smooth solution to the (α, β)-Ricci-Yamabe flow (1.1) can be locally explicitly estimated in terms of its local L ∞ -norm on such a ball at the initial time and a uniform local bound for the Ricci curvature and the second-order derivative of the scalar curvature.
Proceeding as in [19,23], we prove a local curvature estimate without loss of generality by assuming that M is a smooth complete manifold with is a open geodesic ball under the initial metric for x 0 ∈ M and positive constants L and ρ.
Choose the cutoff function which is Lipschitz with support B g0 (x 0 , ρ/ √ L).Denote a universal positive constant that depends only on n and p by c in this subsection.
In the following, we calculate some necessary differential inequalities.
Proposition 5.1.For any p ≥ 3, we have where we used (3.9) in the first equality and (3.7) in the second.Moreover, we used Cauchy's inequality and the fact of |Ric| ≤ L in the last inequality.By rearranging, we have (5.5) Similarly, we can derive from (3.3) that where we used (1.1), (3.5) and (3.9).
Then, for any p ≥ 3, we have Proof.By direct computation, we have where we used (5.10) and (5.11) in the first equality, (5.7) in the second equality, Cauchy's inequality in the second and third inequalities and Proposition 5.1 in the last.
Now we deal with the second term of RHS of (5.15).
where we used (5.6) in the first inequality, (5.10) in the second inequality and Cauchy's inequality in the last.We conclude (5.14) by combining (5.15) and (5.16) and rearranging.
In the rest of this section, we denote a universal constant depending only on α, β, L and n by C. Then we have a L p estimate for the Riemannian curvature.
(5.18) Hence, for all 0 ≤ t ≤ T ′ , and (5.20) Moreover, where we used (5.20) in the first inequality, and Young's inequality and (5.19) in the last.Using Young's inequality again, we can get Applying (5.21) and (5.22) to (5.14), we obtain This implies d dt e −CLt U (t) (5.23) For any τ ∈ (0, T ′ ], integrating (5.23) from 0 to τ yields On the other hand, for any constant θ > 1, φ(x) (5.25)By definition, we have where we used |Ric| ≤ L and |Ric| ≤ √ n|Rm| in the first inequality and Young's inequality in the second.
We conclude (5.17) from (5.27) by the arbitrary of τ . Define to be the average of a smooth function h ∈ M × [0, T ′ ] on some geodesic ball B with respect to g(0).It is clear from (5.18) that for all t ∈ (0, T ′ ].Now we are ready to prove a local curvature estimate by applying De Giorgi-Nash-Moser iteration method (see e.g.[20]).The proof follows from the work of Kotschwar, Munteanu and Wang [19] with minor modifications.Theorem 5.4.For any x 0 ∈ M , there exist constants a 1 and a 2 depending only on n so that sup where Λ 0 := sup |Rm|(x, 0).
Finally, we give a proof of Theorem 1.3.
Proof of Theorem 1.3: Since the Ricci curvature is uniformly bounded, it follows from Theorem 5.4 and the compactness of M that the Riemannian curvature of g(t) Since the Weyl tensor is trace-less and Then we derive a inequality that are essential in the curvature pinching estimate.

. 36 ) 5 . 2 :
where we used(4.11)and(4.35)and Kt ≤ a in the second inequality.This completes the proof.Now we can prove Theorem 1.2 with the aid of Theorem 4.Proof of Theorem 1.We prove the result by contraction.If T < ∞ is a maximal existence time for a smooth solution g(x, t) to the (α, β)-Ricci-Yamabe flow (1.1) and supM×[0,T ) |Rm| < ∞.