FIRST EIGENVALUES OF GEOMETRIC OPERATOR UNDER THE RICCI-BOURGUIGNON FLOW

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1 J. Inones. ath. Soc. Vol. 24, No. 1 (2018), pp FIRST EIGENVALUES OF GEOETRIC OPERATOR UNDER THE RICCI-BOURGUIGNON FLOW Shahrou Azami Department of athematics, Faculty of Sciences Imam Khomeini International University, Qazvin, Iran azami@sci.ikiu.ac.ir Abstract. Let (, g(t)) be a compact Riemannian manifol an the metric g(t) evolve by the Ricci-Bourguignon flow. We fin the formula variation of the eigenvalues of geometric operator φ + cr uner the Ricci-Bourguignon flow, where φ is the Witten-Laplacian operator an R is the scalar curvature. In the final section, we show that some quantities epenent to the eigenvalues of the geometric operator are nonecreasing along the Ricci-Bourguignon flow on close manifols with nonnegative curvature. Key wors an Phrases: Laplace, Ricci-Bourguignon flow. Abstrak. isalkan (, g(t)) aalah manifol Riemann kompak an metrik g(t) berevolusi mengikuti aliran Ricci-Bourguignon. Kita mencari variasi formula nilainilai eigen ari operator geometrik φ + cr i bawah aliran Ricci-Bourguignon, engan φ menyatakan operator Laplace-Witten an R aalah kurvatur skalar. Di bagian akhir, kita menunjukkan bahwa beberapa besaran yang bergantung paa nilai-nilai eigen ari operator geometrik bersifat tak turun sepanjang aliran Ricci- Bourguignon paa manifol tutup engan kurvatur taknegatif. Kata kunci: Laplace, aliran Ricci-Bourguignon 1. Introuction Let (, g(t)) be a close Riemannian manifol. Stuying the eigenvalues of geometric operators is a very powerful tool for the unerstaning Riemannian manifols. Recently, there has been a lot of work on the eigenvalue problem uner 2000 athematics Subject Classification: Primary 58C40; Seconary 53C44. Receive: , revise: , accepte:

2 52 S. Azami geometric flow. In [9], Perelman shows that the functional F = (R + f 2 )e f ν is nonecreasing along the Ricci flow couple to a backwar heat-type equation, where R is the scalar curvature with respect to the metric g(t) an ν enote the volume form of the metric g = g(t). The nonecreasing of the functional F implies that the lowest eigenvalue of the operator 4 +R is nonecreasing along the Ricci flow. As an application, Perelman shown that there are no nontrivial steay or expaning breathers on compact manifols. Cao [2] extene the geometric operator 4 + R to the operator + R 2 on close Riemannian manifols, an showe that the eigenvalues of the operator + R 2 are nonecreasing along the Ricci flow with nonnegative curvature operator. Then, Li [8] an Cao [3] consiere the operator +cr an both them prove that the first eigenvalue of the operator +cr for c 1 4 is nonecreasing along the Ricci flow. Zeng an et al [12] stuie the monotonicity of eigenvalues of the operator + cr along the Ricci-bourguignon flow. Later Fang an Yang [7] stuie the evolution for the first eigenvalue of geometric operator φ + R 2 uner the Yamabe flow, where φ is the Witten-Laplacian operator, φ C 2 (), an constructe some monotonic quantities uner the Yamabe flow. Also, Wen an et al [10] investigate the evolution an monotonicity for eigenvalues of geometric operator φ + R 2 uner the Ricci flow. For the other recent research in this irection, see [5, 6, 11]. We consier an n-imensional close Riemannain manifol with a time epenent Riemannian metric g(t), where g = g(t) is evolving accoring to the Ricci- Bourguignon flow equation t g = 2Ric + 2ρRg = 2(Ric ρrg), g(0) = g 0 (1) where Ric is the Ricci tensor of the manifol, R is scalar curvature an ρ is a real constant. This family of geometric flows contains, as a special case, when ρ = 0, this flow is the Ricci flow. At the first time the Ricci-Bourguignon introuce by Bourguignon in [1] an then Catino an et al in [4] shown that if ρ < 1 2(n 1), then the evolution equation (1) has a unique solution for a position time interval on any smooth n-imensional close Riemannian with any initial metric g 0 an shown that some conitions on the curvature are preserve by the Ricci-Bourguignon flow. otivate by the above works, in this paper we will stuy the first eigenvalue of the geometric operator whose metric satisfying the Ricci-Bourguignon flow (1). 2. Preliminaries In this section, we will first the efinitions for the first eigenvalue of the geometric operator φ + cr (2) then we will fin the formula for the evolution of the first eigenvalue of the geometric operator (2) uner the Ricci-Bourguignon flow on a close manifol. Let (, g(t)) be a compact Riemannian manifol, an (, g(t)) be a smooth solution to the

3 First Eigenvalues of Geometric Operator 53 Ricci-Bourguignon flow (1) for t [0, T ). Let be the Levi-Civita connection on (, g(t)) an f : R be a smooth function on or f belong to the Sobolev space W 1,2 (). The Laplacian of f efine as f = iv( f) = g ij ( i j f Γ k ij k f). (3) Suppose that ν the Riemannian volume measure, an µ the weight volume measure on (, g(t)); i.e. µ = e φ(x) ν (4) where φ C 2 (). The Witten-Laplacian is efine by φ = φ. (5) which is a symmetric operator on L 2 (, µ) an satisfies the following integration by part formula: < u, v > µ = v φ u µ = u φ v µ u, v C (), The Witten-Laplacian is generalize of Laplacian operator, for example, when φ is a constant function, the Witten-Laplacian operator is just the Laplace-Belterami operator. In this paper we consier a generalize of the Witten-Laplacian operator as φ + cr where R is the scalar curvature. We say that λ(t) is an eigenvalue of the operator φ + cr at time t [0, T ), an f(x, t) the corresponing eigenfunction, whenever φ f(x, t) + crf(x, t) = λ(t)f(x, t). (6) Normalize eigenfuctions are efine as follow: f 2 µ = 1, (7) an assume that f(x, t) is a C 1 -family of smooth function on. ultiplying with f on both sies (6) an then by integration we get λ(t) = ( f φ f + crf 2 )µ (8) where efines the evolution of the first eigenvalue of the geometric operator (2) uner the variation of g(t) where the eigenfunction associate to λ(t) is normalize. In [10] Wen an et al shown that the following lemma. Lemma 2.1. ([10]) Suppose that λ(t) is an eigenvalue of the operator φ +cr, f is the eigenfunction of λ(t) at the time t, an the metric g(t) evolves by t g ij = v ij, where v ij is a symmetric two-tensor. Then we have t λ(t) = (v ij f ij v ij φ i f j + c R t f)µ + where V = tr(v), f ij = i j f, f i = i f, φ i = i φ. (v ij,i 1 2 V i)f j fµ (9) Now, we fin the evolution formula of eigenvalue λ(t) uner the Ricci-Bourguignon flow (1).

4 54 S. Azami Theorem 2.2. Let g(t), t [0, T ), be a solution of the Ricci-Bourguignon flow (1) on an n-imensional close manifol. Assume that λ(t) is the lowest eigenvalue of φ + cr an f = f(x, t) satisfies in (6) with (7). Then uner the Ricci- Bourguignon flow, we have λ(t) = 2A R 1 t 2 f f φ 2 µ A Rf 2 φµ +[ 3 2 A 1 + (n 2)ρ] R f 2 µ λ(2a 1 + nρ) Rf 2 µ (10) +2 R ij f i f j µ + 2c Ric 2 f 2 µ + c(2a 1 + (n 2)ρ) R 2 f 2 µ where A = c(1 2(n 1)ρ). Proof. In [4], G. Catino an et al shown that the evolution of scalar curvature uner the Ricci-Bourguignon flow is R t = (1 2(n 1)ρ) R + 2 Ric 2 2ρR 2. (11) Substiuting v ij = 2R ij + 2ρRg ij an (11) into the equality (9) we get t λ(t) = [ 2R ij f ij + 2ρR f + 2R ij φ i f j 2ρR φ. f] fµ (n 2)ρ i R f i fµ (12) [ +c (1 2(n 1)ρ) R f Ric 2 f 2 2ρR 2 f 2] µ. Integration by parts results that f 2 Rµ = 2 R f 2 µ + 2 Rf 2 φ µ + Rf φ fµ 2 Rf φ. fµ Rf 2 φ 2 µ, (13) an i R f i fµ = Rf φ fµ R f 2 µ. (14)

5 First Eigenvalues of Geometric Operator 55 Also, using integration by parts an 1 2 R = iv Ric we have R ij f ij fµ = (R ij fe φ ) j f i ν = iv(ric)fe φ f i ν + R ij f j e φ f i ν R ij fφ j f i e φ ν = 1 i Rff i e φ ν + R ij f j f i µ R ij fφ j f i µ 2 = 1 R(ff i e φ ) i ν + R ij f j f i µ R ij fφ j f i µ 2 = 1 R f 2 µ 1 Rf fµ + 1 Rff i φ i µ (15) R ij f j f i µ R ij fφ j f i µ = 1 R f 2 µ 1 Rf φ fµ + R ij f j f i µ R ij fφ j f i µ. 2 2 Inserting (13), (14) an (15) in (12), yiels t λ(t) = R f 2 µ Rf φ fµ + 2 R ij f i f j µ 2 R ij f i φ j fµ Rf φ. fµ +2ρ Rf fµ + 2 R ij f j φ i fµ 2ρ +(n 2)ρ Rf φ fµ + (n 2)ρ R f 2 µ +2c(1 2(n 1)ρ) R f 2 µ + 2c(1 2(n 1)ρ) Rf φ fµ 2c(1 2(n 1)ρ) Rf φ. fµ c(1 2(n 1)ρ) Rf 2 φµ +2c(1 2(n 1)ρ) Rf 2 φ 2 µ + 2c Ric 2 f 2 µ 2cρ R 2 f 2 µ,

6 56 S. Azami therefore λ(t) = [2A 1 + (n 2)ρ] R f 2 µ + [2A 1 + nρ] Rf φ fµ t +2 R ij f i f j µ + 2c Ric 2 f 2 µ 2cρ R 2 f 2 µ (16) 2A Rf φ. fµ A Rf 2 φµ + 2A Rf 2 φ 2 µ = 2A R 1 2 f f φ 2 µ A Rf 2 φµ +[ 3 2 A 1 + (n 2)ρ] R f 2 µ +(2A 1 + nρ) R( λf 2 + crf 2 )µ +2 R ij f i f j µ + 2c Ric 2 f 2 µ 2cρ R 2 f 2 µ. Here in the last equality we have use (6). In theorem (2.2) if φ is a constant function, we can get the evolution for the first eigenvalue of the geometric operator + cr uner the Ricci-Bourguignon flow (1), which stuie in [12]. Remark: In theorem (2.2), we assume that φ oes not epenent on the time t. If φ is epen to t then the eigenvalue λ(t) introuce in (6) an (7) satisfies λ(t) = 2A R 1 t 2 f f φ 2 µ A Rf 2 φµ + f i (φ t ) i fµ +[ 3 2 A 1 + (n 2)ρ] R f 2 µ λ(2a 1 + nρ) Rf 2 µ +2 R ij f i f j µ + 2c Ric 2 f 2 µ (17) +c(2a 1 + (n 2)ρ) R 2 f 2 µ where A = c(1 2(n 1)ρ). Because the evolution equation of eigenvalues will have an aitional term f i(φ t ) i fµ. In the following we show that some quantity epenent on the eigenvalue of geometric operator (6) are monotonic along the Ricci-Bourguignon flow. Not that the scalar curvature uner the Ricci-Bourguignon flow evolves by R t = (1 2(n 1)ρ) R + 2 Ric 2 2ρR 2, by Ric 2 R 2 we have R t (1 2(n 1)ρ) R + 2(1 ρ)r2.

7 First Eigenvalues of Geometric Operator 57 Let σ(t) be the solution to the ODE y = 2(1 ρ)y 2 with initial value α = max R(0). x By the maximum principle, we have ( R(t) σ(t) = 2(1 ρ)t + α) 1 1 (18) on [0, T ), where T 1 = min T, 2(1 ρ)α. Also, the inequality Ric 2 R2 n R t (1 2(n 1)ρ) R + 2( 1 n ρ)r2. result that we assume that γ(t) be the solution to the ODE y = 2( 1 n ρ)y2 with initial value β = min R(0). Then the maximum principle implies that x R(t) γ(t) = nβ n 2(1 nρ)βt on [0, T ). (19) Theorem 2.3. Let (, g(t)) be a solution of the Ricci-Bourguignon flow (1) for t [0, T ] on a close n-imensional manifol an ρ < 1 2(n 1) with nonnegative scalar curvature. Let the Ricci curvature operator be a nonnegative along the Ricci- Bourguignon flow an scalar curvature satisfies R 1 2(n 1)ρ φ, in [0, T ]. (20) 2A 1 + (n 2)ρ If λ(t) is the first eigenvalue of (2) then for c 2(1 (n 2)ρ) 3(1 2(n 1)ρ), the quantity e t 0 [ (2A+nρ)γ(τ)+σ(τ)]τ λ(t) is nonecreasing uner the Ricci-Bourguignon flow on [0, T ) where A = c(1 2(n 1)ρ) an σ(t) an γ(t) introuce in (18) an (19), respectively. Proof. Accoring to hypothesis of the theorem the Ricci curvature operator is nonnegative along the Ricci-Bourguignon flow an on the other han in [4], shown that the nonnegative of the scalar curvature is preserve along the Ricci-Bourguignon flow. Therefore (10) an (20) imply that for ρ < 1 2(1 (n 2)ρ) 2(n 1) an c 3(1 2(n 1)ρ) we get λ(t) λ(2a 1 + nρ) Rf 2 µ λ[(2a + nρ)γ(t) σ(t)]. (21) t in last inequality we use f 2 µ = 1. Hence the theorem follows from the last inequality. Theorem 2.4. Let g(t), t [0, T ) be a solution to the Ricci-Bourguignon flow (1) on a close Riemannian manifol n with nonnegative scalar curvature an the scalar curvature satisfies R 2(n 1) φ, in [0, T ). (22) 4c(n 1) n + 2

8 58 S. Azami Suppose that n 3 an the Ricci curvature satisfies where c > Ric 1 4c 4c 1 φ 2 (4c 1) 2 φ 2, in [0, T ) (23) n 2 2(n 1) an φ C () satisfies the heat equation φ t = φ. (24) Then for ρ 0 the quantity e t 0 ( ρnγ(τ)+4ρc(n 1)σ(τ))τ λ(t) is nonecreasing along the Ricci-Bourguignon flow, where σ(t) an γ(t) introuce in (18) an (19), respectively. Proof. From (12) an (17), we have [ λ(t) = 2ρ R f f + R fφ i f i + n 2 i R f i f t 2 we set I = an II = +c(n 1) R f 2 + cr 2 f 2] µ (25) [ + 2Rij f ij f + 2R ij φ i f j f + c R f 2 + 2c Ric 2 f 2 + f i (φ t ) i f ] µ [ R f f + R fφ i f i + n 2 ] i R f i f + c(n 1) R f 2 + cr 2 f 2 µ 2 [ 2Rij f ij f + 2R ij φ i f j f + c R f 2 + 2c Ric 2 f 2 + f i (φ t ) i f ] µ. Notice that, using (13) an (14) we can rewrite I as follow: I = c(n 1) R f f φ 2 µ + [ n 2 c + 2c2 (n 1) + c] R 2 f 2 µ +[ ( n 2 1) + c(n 1)] R f 2 µ [ n 2 + 2c(n 1)]λ R f 2 µ c(n 1) Rf 2 φµ, (26) on the other han, in [10], has been shown that II = 1 R ij + ψ ij 2 e ψ µ + 4c 1 Ric 2 e ψ µ (ψ ij φ ij ψ i( φ) i )e ψ µ 1 ψ i (φ t ) i e ψ µ (27) 2 where f 2 = e ψ for some smooth function ψ. Therefore if φ t = φ then we have 1 ψ i (φ t ) i e ψ µ = 1 ψ i ( φ) i e ψ µ. (28) 2 2

9 First Eigenvalues of Geometric Operator 59 Combining (25), (26), (27) an (28) we arrive at t λ(t) = 1 R ij + ψ ij 2 e ψ µ + 4c 1 Ric 2 e ψ µ + ψ ij φ ij e ψ µ 2 2 2ρc(n 1) R 1 2 ψ φ 2 e ψ µ 2ρ[ n 2 c + 2c2 (n 1) + c] R 2 e ψ µ (29) + ρ 2 [(n 2 1) c(n 1)] R ψ 2 e ψ µ +2ρ[ n 2 + 2c(n 1)]λ R e ψ µ + 2ρc(n 1) Re ψ φµ = 1 R ij + ψ ij + φ ij 2 e ψ µ 2 + 4c 1 ( R ij 1 2 4c 1 φ ij 2 4c (4c 1) 2 φ ij 2 )e ψ µ 2ρc(n 1) R 1 2 ψ φ 2 e ψ µ 2ρ[ n 2 c + 2c2 (n 1) + c] R 2 e ψ µ + ρ 2 [(n 2 1) c(n 1)] R ψ 2 e ψ µ +2ρ[ n 2 + 2c(n 1)]λ R e ψ µ + 2ρc(n 1) Re ψ φµ. The nonnegative scalar curvature is preserve along the Ricci-Bourguignon flow, then (22) an (23) for ρ 0 an c > n 2 2(n 1) imply that t λ(t) 2ρ[ n + 2c(n 1)]λ R e ψ µ [ ρnγ(t) + 4ρc(n 1)σ(t)] λ (30) 2 the last inequality complete the proof of theorem. If we assume that φ is constant function an ρ = 0 then (29) implies that for c 1 4 the eigenvalues is strictly increasing uner the Ricci-Bourguignon flow, where this result fune by Cao in [3]. References [1] Bourguignon, J.P., Ricci curvature an Einstein metrics, Global ifferential geometry an global analysis (Berlin,1979) Lecture nots in ath. vol. 838, Springer, Berlin, 1981, [2] Cao, X.D., Eigenvalues of ( + R ) on manifols with nonnegative curvature operator, 2 ath. Ann. 337:2(2007),

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