EIGENVALUES AND ENTROPYS UNDER THE HARMONIC-RICCI FLOW 0 1 YI LI 0 2 Abstract. In this paper, the author discuss the eigenvalues and en- v tropys under the harmonic-Ricci flow, which is the Ricci flow coupled o N with theharmonicmapflow. Wegiveanalternativeproofofresultsfor compactsteadyandexpandingharmonic-Riccibreathers. Inthesecond 8 part,wederivesomemonotonicityformulasforeigenvaluesofLaplacian under the harmonic-Ricci flow. Finally, we obtain the first variation of ] the shrinkerand expandingentropysof theharmonic-Ricci flow. G D . h t a Contents m [ 1. Introduction 1 2. Notation and commuting identities 8 1 v 3. Harmonic-Ricci flow and the evolution equations 9 7 4. Entropys for harmonic-Ricci flow 10 9 5. Compact steady harmonic-Ricci breathers 13 6 6. Compact expanding harmonic-Ricci breathers 14 1 . 7. Eigenvalues of the Laplacian under the harmonic-Ricci flow 19 1 1 8. Eigenvalues of the Laplacian-type under the harmonic-Ricci flow 25 0 9. Another formula for dλ(f(t)) 30 dt 1 10. The first variation of expander and shrinker entropys 35 : v References 43 i X r a 1. Introduction After successfully applying the Ricci flow to topological and geometric problems, people study some analogues flows, including the harmonic-Ricci flow[9, 11], connection Ricci flow[14], Ricci-Yang-Mills flow[13, 16, 17], and renormalization group flows[6, 8, 12, 15], etc. In this note, we study the eigenvalue problems of the harmonic-Ricci flow which is the following cou- pled system ∂ (1.1) g(x,t) = −2Ric +4du(x,t)⊗du(x,t), g(x,t) ∂t ∂ (1.2) u(x,t) = ∆ u(x,t). g(x,t) ∂t 1 2 YILI For convenience, we introduce a new symmetric 2-tensor S whose g(t),u(t) components S are defined by ij S := R −2∂ u∂ u. ij ij i j Its trace is S := gijS = R −2 g(t)∇u(t) 2 . g(t),u(t) ij g(t) g(t) Suppose that M is a Riemannian manifold. For any Riemannian metric (cid:12) (cid:12) g and any smooth functions u,f, we have(cid:12) a numbe(cid:12)r of functionals F(g,u,f) = R +|g∇f|2 −2|g∇u|2 e−fdV , g g g g ZM (cid:16) (cid:17) E(g,u,f) = R −2|g∇u|2 e−fdV , g g g ZM (cid:16) (cid:17) F (g,u,f) = kR +|g∇f|2 −2k|g∇u|2 e−fdV . k g g g g ZM (cid:16) (cid:17) List[9]andMu¨ller[11]showedthat,asinthecaseofPerelman’sF-functional, under the following evolution equation ∂ g(t) = −2Ric +4du(t)⊗du(t), g(t) ∂t ∂ (1.3) u(t) = ∆ u(t), g(t) ∂y ∂ 2 2 f(t) = −∆ f(t)−R + g(t)∇f(t) +2 g(t)∇u(t) ∂t g(t) g(t) g(t) g(t) (cid:12) (cid:12) (cid:12) (cid:12) the evolution equation for F-functional(cid:12)is (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) d 2 F(g(t),u(t),f(t)) = 2 S +g(t)∇2f(t) e−f(t)dV g(t),u(t) g(t) dt ZM (cid:12) (cid:12)g(t) (1.4) +4 (cid:12)(cid:12) ∆ u(t)−hdu(t),d(cid:12)(cid:12)f(t)i 2 e−f(t)dV g(t) g(t) g(t) g(t) ZM (cid:12) (cid:12) that is nonnegative. Based on (1.4), we derive (cid:12) (cid:12) Theorem 1.1. Under the evolution equation (1.3), one has d E(g(t),u(t),f(t)) = 2 S 2 e−f(t)dV dt g(t),u(t) g(t) g(t) ZM (cid:12) (cid:12) (1.5) +4 (cid:12) ∆ u(cid:12)(t) 2 e−f(t)dV , g(t) g(t) g(t) ZM d (cid:12) (cid:12) F (g(t),u(t),f(t)) = 2(k−1(cid:12)) S (cid:12) 2 e−f(t)dV dt k g(t),u(t) g(t) g(t) ZM (cid:12) (cid:12) 2 (1.6) +2 S (cid:12) +g(t)(cid:12)∇2f(t) e−f(t)dV g(t),u(t) g(t) ZM (cid:12) (cid:12)g(t) +4(k−(cid:12)(cid:12)1) ∆ u(t) 2 e(cid:12)(cid:12)−f(t)dV g(t) g(t) g(t) ZM (cid:12) (cid:12) +4 ∆ u(cid:12)(t)−hdu((cid:12)t),df(t)i 2 e−f(t)dV ,. g(t) g(t) g(t) g(t) ZM (cid:12) (cid:12) (cid:12) (cid:12) EIGENVALUES AND ENTROPYS UNDER THE HARMONIC-RICCI FLOW 3 As a corollary we give a new proof of the following Corollary 1.2. There is no compact steady harmonic-Ricci breather other than (M,g(t)) is Ricci-flat and u(t) is constant. When we deal with the expanding harmonic-Ricci breather, we need the following two functionals n L (g,u,τ,f) = τ2 R + +∆ f −2|g∇u|2 e−fdV , + g 2τ g g g ZM (cid:16) (cid:17) n L (g,u,τ,f) = τ2 k R + +∆ f −2k|g∇u|2 e−fdV . +,k g 2τ g g g ZM h (cid:16) (cid:17) i Under the following evolution equation ∂ g(t) = −2Ric +4du(t)⊗du(t), g(t) ∂t ∂ u(t) = ∆ u(t), ∂t g(t) ∂ 2 2 f(t) = −∆ f(t)+ g(t)∇f(t) −R +2 g(t)∇u(t) , g(t) g(t) ∂t g(t) g(t) (cid:12) (cid:12) (cid:12) (cid:12) d (cid:12) (cid:12) (cid:12) (cid:12) τ(t) = 1, (cid:12) (cid:12) (cid:12) (cid:12) dt we have Theorem 1.3. Under the above evolution equation, one has d (1.7) L (g(t),u(t),τ(t),f(t)) + dt 2 1 = 2τ(t)2 S +g(t)∇2f(t)+ g(t) e−f(t)dV g(t),u(t) g(t) 2τ(t) ZM (cid:12) (cid:12)g(t) (cid:12) (cid:12) +4τ(t)2 (cid:12)(cid:12) ∆ u(t)−hdu(t),df(t)i 2 (cid:12)(cid:12)e−f(t)dV , g(t) g(t) g(t) g(t) ZM d (cid:12) (cid:12) (1.8) L (g(t),(cid:12)u(t),τ(t),f(t)) (cid:12) +,k dt 2 1 = 2τ(t)2 S +g(t)∇2f(t)+ g(t) e−f(t)dV g(t),u(t) g(t) 2τ(t) ZM (cid:12) (cid:12)g(t) (cid:12) (cid:12) (cid:12) 1 2 (cid:12) +2(k−1)τ(cid:12)(t)2 S + g(t) (cid:12)e−f(t)dV g(t),u(t) 2τ(t) g(t) ZM (cid:12) (cid:12)g(t) (cid:12) (cid:12) +4τ(t)2 ∆ u(cid:12)(cid:12)(t)−hdu(t),df(t)i (cid:12)(cid:12)2 e−f(t)dV g(t) g(t) g(t) g(t) ZM (cid:12) (cid:12) +4(k−1)τ((cid:12)t)2 ∆ u(t) 2 e−f(t)dV(cid:12) . g(t) g(t) g(t) ZM (cid:12) (cid:12) As a corollary, we obtain a(cid:12)new proof(cid:12)of the following 4 YILI Corollary 1.4. There is no expanding harmonic-Ricci breather on compact Riemannian manifolds other than M is an Einstein manifold and u(t) is constant. The second part of this paper focuses on the eigenvalue of the Laplacian operator under the harmonic-Ricci flow. Suppose that λ(t) is an eigenvalue of the Laplacian ∆ . We prove g(t) Theorem 1.5. If (g(t),u(t)) is a solution of the harmonic-Ricci flow on a compact Riemannian manifold M and λ(t) denotes the eigenvalue of the Laplacian ∆ with eigenfunction f(t), then g(t) d λ(t)· f(t)2dV = λ(t) S f(t)2dV g(t) g(t),u(t) g(t) dt ZM ZM 2 (1.9) − S g(t)∇f dV g(t),u(t) g(t) ZM (cid:12) (cid:12)g(t) (cid:12) (cid:12) +2 hS (cid:12),df(t)(cid:12)⊗df(t)i dV . g(t),u(t) g(t) g(t) ZM The above equation (1.9) is a general formula to describe the evolution of λ(t) under the harmonic-Ricci flow. Under a curvature assumption, we can derive some monotonicity formulas for the eigenvalue λ(t). Set (1.10) S (0) := minS (x) min g(t),u(t) x∈M the minimum of S over M at the time 0. g(t),u(t) Theorem 1.6. Let (g(t),u(t))t∈[0,T] be a solution of the harmonic-Ricci flow on a compact Riemannian manifold M and λ(t) denote the eigenvalue of the Laplacian ∆ . Suppose that S −αS g(t) ≥ 0 along the g(t) g(t),u(t) g(t),u(t) harmonic-Ricci flow for some α≥ 1. 2 (1) If S (0) ≥ 0, then λ(t) is nondecreasing along the harmonic-Ricci min flow for any t ∈ [0,T]. (2) If S (0) > 0, then the quantity min nα 2 1− S (0)t λ(t) min n (cid:18) (cid:19) is nondecreasing along the harmonic-Ricci flow for T ≤ n . 2Smin(0) (3) If S (0) < 0, then the quantity min nα 2 1− S (0)t λ(t) min n (cid:18) (cid:19) is nondecreasing along the harmonic-Ricci flow for any t ∈[0,T]. Corollary 1.7. Let (g(t),u(t)) be a solution of the harmonic-Ricci t∈[0,T] flow on a compact Riemannian surface Σ and λ(t) denote the eigenvalue of the Laplacian ∆ . g(t) EIGENVALUES AND ENTROPYS UNDER THE HARMONIC-RICCI FLOW 5 (1) Suppose that Ric ≤ ǫdu(t)⊗du(t) where g(t) 1−α 1 ǫ ≤ 4 , α > . 1−2α 2 (1-1) If S (0) ≥ 0, then λ(t) is nondecreasing along the harmonic- min Ricci flow for any t ∈ [0,T]. (1-2) If S (0) > 0, then the quantity min (1−S (0)t)2αλ(t) min is nondecreasing along the harmonic-Ricci flow for T ≤ 1 . Smin(0) (3) If S (0) < 0, then the quantity min (1−S (0)t)2αλ(t) min is nondecreasing along the harmonic-Ricci flow for any t ∈ [0,T]. (2) Suppose that 2 g(t)∇u(t) g(t) ≥ 2du(t)⊗du(t). g(t) (cid:12) (cid:12) (1-1) If S ((cid:12)0) ≥ 0, th(cid:12)en λ(t) is nondecreasing along the harmonic- min (cid:12) (cid:12) Ricci flow for any t ∈ [0,T]. (1-2) If S (0) > 0, then the quantity min (1−S (0)t)λ(t) min is nondecreasing along the harmonic-Ricci flow for T ≤ 1 . Smin(0) (3) If S (0) < 0, then the quantity min (1−S (0)t)λ(t) min is nondecreasing along the harmonic-Ricci flow for any t ∈ [0,T]. When we restrict to the Ricci flow, we obtain Corollary 1.8. Let (g(t)) be a solution of the Ricci flow on a compact t∈[0,T] Riemannian surface Σ and λ(t) denote the eigenvalue of the Laplacian ∆ . g(t) (1) If R (0) ≥ 0, then λ(t) is nondecreasing along the Ricci flow for min any t ∈ [0,T]. (2) If R (0) > 0, then the quantity (1−R (0)t)λ(t) is nondecreasing min min along the Ricci flow for T ≤ 1 . Rmin(0) (3) If R (0) < 0, then the quantity (1−R (0)t)λ(t) is nondecreasing min min along the Ricci flow for any t ∈[0,T]. Remark 1.9. Let (g(t)) be a solution of the Ricci flow on a compact t∈[0,T] Riemannian surface Σ with nonnegative scalar curvsture and λ(t) denote the eigenvalue of the Laplaican ∆ . Then λ(t) is nondecreasing along the g(t) Ricci flow for any t ∈ [0,T]. 6 YILI Since (1.11) µ(g,u) := inf F(g,u,f) f ∈ C∞(M), e−fdV = 1 g (cid:26) (cid:12) ZM (cid:27) (cid:12) (cid:12) is the smallest eigenvalue of the operator ∆ := −4∆ +R −2|g∇u|2, we g,u g g g can consider the evolution equation for this eigenvalue under the harmonic- Ricci flow. To the operator ∆ we associate a functional g,u (1.12) λ (f):= f ·∆ f ·dV . g,u g,u g ZM When f is an eigenfunction of the the operator ∆ with the eigenvalue λ g,u and normalized by f2dV = 1, we obtain X g R λ (f)= λ. g,u So, we can suffice to study the evolution equation for d λ (f) under the dy g,u harmonic-Ricci flow. Theorem 1.10. Suppose that (g(t),u(t)) is a solution of the harmonic- Ricci flow on a compact Riemannian manifold M and f(t) is an eigenvalue of ∆ , i.e., ∆ f(t) = λ(t)f(t)(where λ(t) is only a function of g(t),u(t) g(t),u(t) time t), with the normalized condition f(t)2dV = 1. Then we have M g(t) R d d λ(t) = λ (f(t)) = 2 S ,df(t)⊗df(t) dV dt dt g,u g(t),u(t) g(t) g(t) ZM (cid:10) (cid:11) (1.13) + f(t)2 S 2 +2 ∆ u(t) 2 dV . g(t),u(t) g(t) g(t) g(t) g(t) ZM h(cid:12) (cid:12) (cid:12) (cid:12) i (cid:12) (cid:12) (cid:12) (cid:12) In [9], List proved the nonnegativity of the operator S is preserved g(t),u(t) by the harmonic-Ricci flow, hence Corollary 1.11. If Ric −2du(0) ⊗du(0) ≥ 0, then the eigenvalues of g(0) the operator ∆ are nondecreasing under the harmonic-Ricci flow. g(t),u(t) Remark 1.12. If we choose u(t)≡ 0, then we obtain X. Cao’s result [3]. There is another expression of dλ(t). dt Theorem 1.13. Suppose that (g(t),u(t)) is a solution of the harmonic- Ricci flow on a compact Riemannian manifold M and f(t) is an eigenvalue of ∆ , i.e., ∆ f(t) = λ(t)f(t)(where λ(t) is only a function of g(t),u(t) g(t),u(t) EIGENVALUES AND ENTROPYS UNDER THE HARMONIC-RICCI FLOW 7 time t), with the normalized condition f(t)2dV = 1. Then we have M g(t) d d R λ(t) = λ (f(t)) g,u dt dt 1 2 = S +g(t)∇2ϕ(t) e−ϕ(t)dV g(t),u(t) g(t) 2ZM (cid:12) (cid:12)g(t) +1 (cid:12)(cid:12) S 2 e−ϕ(t)d(cid:12)(cid:12)V + hdu(t),dϕ(t)i 2e−ϕ(t)dV 4 g(t),u(t) g(t) g(t) g(t) g(t) ZM ZM (cid:12) (cid:12) 2 (cid:12) (cid:12) (1.14)+2 (cid:12)g(t)∇2u(t(cid:12)) e−ϕ(t)dV (cid:12) (cid:12) g(t) ZM (cid:12) (cid:12)g(t) +1 (cid:12)(cid:12)S +(cid:12)(cid:12) 4du(t)⊗du(t) 2 e−ϕ(t)dV 4 g(t),u(t) g(t) g(t) ZM (cid:12) 2 (cid:12) − ∆(cid:12) g(t)∇u(t) e−ϕ(t(cid:12))dV g(t) g(t) ZM (cid:18)(cid:12) (cid:12)g(t)(cid:19) (cid:12) (cid:12) where f(t)2 = e−ϕ(t(cid:12)). (cid:12) Remark 1.14. When u≡ 0, (1.14) reduces to J. Li’s formula [7]. SupposethatM isaclosed manifoldof dimensionn. For anyRiemannian metric g, any smooth functions u,f, and any positive number τ, we define e−f (1.15) W±(g,u,f,τ) := τ Sg +|g∇f|2g ∓f ±n (4πτ)n/2dVg. ZM h (cid:16) (cid:17) i Set e−f ∞ µ±(g,u,τ) := inf W±(g,u,f,τ) f ∈ C (M), dVg = 1 , (4πτ)n/2 (cid:26) (cid:12) ZM (cid:27) ν±(g,u) := inf{µ±(g,u,τ)|τ >(cid:12)0}. (cid:12) The first variation of ν±(g(s),u(s)) is Theorem 1.15. Suppose that (M,g) is a compact Riemannian manifold and u a smooth function on M. Let h be any symmetric covariant 2-tensor on M and set g(s) := g + sh. Let v be any smooth function on M and u(s) := u + sv. If ν±(g(s),u(s)) = W±(g(s),u(s),f±(s),τ±(s)) for some smooth functions f±(s) with e−f±(s)dV/(4πτ±(s))n/2 = 1 and constants M τ±(s) > 0, then R d (1.16) ν±(g(s),u(s)) ds s=0 = −τ(cid:12)(cid:12)(cid:12)±ZM (cid:18)hh,Sg,uig + h,g∇2f g ± 2τ1±trgh(cid:19) (4πeτ−±f±)n/2dVg (cid:10) (cid:11) e−f± +4τ± v(∆gu−hdu,df±ig) dVg, ZM (4πτ±)n/2 8 YILI where f± := f±(0) and τ± := τ±(0). In particular, the critical points of ν±(·,·) satisfy 1 Sg,u+g∇2f ± 2τ±g = 0, ∆gu = hdu,df±ig. Consequently, if W±(g,u,f,τ) and ν±(g,u) achieve their minimums, then (M,g)isagradient expanding andshrinkerharmonic-Ricci solitonaccording to the sign. Corollary 1.16. Suppose that (M,g) is a compact Riemannian manifold and u a smooth function on M. Let h be any symmetric covariant 2-tensor on M and set g(s) := g + sh. Let v be any smooth function on M and u(s) := u + sv. If ν±(g(s),u(s)) = W±(g(s),u(s),f±(s),τ±(s)) for some smooth function f±(s) with e−f±(s)dV/(4πτ±(s))n/2 = 1 and a constant M τ±(s) > 0, and (g,u) is a critical point of ν±(·,·), then R 1 Ricg = ∓ g, f± ≡ constant, u≡ constant. 2τ± Thus, if W±(g,u,·,·) achieve their minimum and (g,u) is a critical point of ν±(·,·), then (M,g) is an Einstein manifold and u is a constant function. Remark 1.17. In the situation of Corollary 1.16, by normalization, we my choose f± = n and u = 0. 2 Acknowledgements. Part of work was done when the author visited Center of Mathematical Science at Zhejiang University. The author would like to thank Professor Kefeng Liu, who teaches the author mathematics. Furthermore, I also thank Professor Hongwei Xu and other staffs in Center of Mathematical Science. 2. Notation and commuting identities LetM beaclosed(i.e., compactandwithoutboundary)Riemannianman- ifoldofdimensionn. ForanyvectorbundleE overM,wedenotebyΓ(M,E) the space of smooth sections of E. Set ⊙2(M) := {v = (v ) ∈Γ(M,T∗M ⊗T∗M)|v = v }, ij ij ji ⊙2(M) := {g = (g )∈ ⊙2(M)|g > 0}. + ij ij Thus, ⊙2(M) is the space of all symmetric covariant 2-tensors on M while ⊙2(M) the space of all Riemannian metrics on M. The space of all smooth + ∞ functions on M is denoted by C (M). For agiven Riemannianmetricg ∈ ⊙2(M), thecorrespondingLevi-civita + connection gΓ = (gΓk) is given by ij 1 (2.1) gΓk = gkl(∂ g +∂ g −∂ g ) ij 2 i jl j il l ij EIGENVALUES AND ENTROPYS UNDER THE HARMONIC-RICCI FLOW 9 where ∂ := ∂ for a local coordinate system {x1,··· ,xn}. The Riemann i ∂xi tensor Rm = (gRk ) is determined by g ijl (2.2) gRk = ∂ gΓk −∂ gΓk +gΓk gΓp −gΓk gΓp. ijl i jl j il ip jl jp il The Ricci curvature Ric = (gR ) is g ij (2.3) gR = gkl ·gRl . ij kij The scalar curvature R of the metric g now is given by g (2.4) R = gij ·gR . g ij For any tensor A= (Ak1···kq) the covariant derivative of A is j1···jp p q g∇ Ak1···kq = ∂ Ak1···kq − gΓm Ak1···kq + gΓksAk1···m···kq. i j1···jp i j1···jp ijr j1···m···jp im j1···jp r=1 s=1 X X Next we recall the Ricci identity: q p g∇ g∇ Al1···lq −g∇ g∇ Al1···lq = gRlr Al1···m···lq− gRm Al1···lq . i j k1···kp j i k1···kp ijm k1···kp ijks l1···m···kp r=1 s=1 X X ∞ In particular, for any smooth function f ∈ C (M) we have g∇ g∇ f = g∇ g∇ f. i j j i The Bianchi identities are (2.5) 0 = gR +gR +gR , ijkl iklj iljk (2.6) 0 = g∇ gR +g∇ gR +g∇ gR q ijkl i jqkl j qikl and the contracted Bianchi identities are (2.7) 0 = 2g∇jgR −g∇ gR, ij i (2.8) 0 = g∇ R −g∇ gR +g∇lgR . i jk j ik lkij 3. Harmonic-Ricci flow and the evolution equations Motivated by static Einstein vacuum equation, List[9] introduced the harmonic-Ricci flow(Originally, it is called the Ricci flow coupled with the harmonic map flow.). Such a flow is similar to the Ricci flow and is the following coupled system ∂ (3.1) g(x,t) = −2Ric +4du(x,t)⊗du(x,t), g(x,t) ∂t ∂ (3.2) u(x,t) = ∆ u(x,t) g(x,t) ∂t for a family of Riemannian metrics g(x,t)(or written as g(t)) and a family of smooth functions u(x,t)(or written as u(t)). Locally, we have ∂ ∂ (3.3) g = −2R +4∂ u·∂ u, u= ∆ u. ij ij i j g ∂t ∂t 10 YILI Introduce a new symmetric tensor field S = (S ) ∈ ⊙2(M) by g(t),u(t) ij (3.4) S := R −2∂ u·∂ u. ij ij i j Then its trace S is equal to g(t),u(t) 2 (3.5) S = gijS = R −2 g(t)∇u(t) . g(t),u(t) ij g(t) g(t) (cid:12) (cid:12) The evolution equation for R is (cid:12) (cid:12) g(t) (cid:12) (cid:12) ∂ R = ∆ R +2|Ric |2 ∂t g(t) g(t) g(t) g(t) g(t) 2 (3.6) +4 ∆ u(t) 2 −4 g(t)∇2u(t) −8 Ric ,du(t)⊗du(t) . g(t) g(t) g(t) g(t) g(t) (cid:12) (cid:12) Also, we have the(cid:12)(cid:12) evolution(cid:12)(cid:12) equati(cid:12)(cid:12)on for g(t)∇(cid:12)(cid:12) u 2 :(cid:10) (cid:11) g(t) (3.7) (cid:12) (cid:12) ∂ 2 2 (cid:12) (cid:12) 2 4 g(t)∇u(t) = ∆ g(t)∇u(t) −2 g(t)∇2u(t) −4 g(t)∇u(t) , g(t) ∂t g(t) g(t) g(t) g(t) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) and(cid:12)the evolu(cid:12)tion equatio(cid:12)n for S (cid:12) : (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) g(t(cid:12)),u(t) (cid:12) (cid:12) (cid:12) (cid:12) ∂ 2 2 (3.8) S = ∆ S +2 S +4 ∆ u(t) . ∂t g(t),u(t) g(t) g(t),u(t) g(t),u(t) g(t) g(t) g(t) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 4. Entropys for harmonic-Ricci flow Motivated by Perelman’s entropy, List [9] introduced the similar func- tional for the harmonic-Ricci flow: ⊙2(M)×C∞(M)×C∞(M) −→ R, (g,u,f) 7−→ F(g,u,f) + where (4.1) F(g,u,f) := R +|g∇f|2−2|g∇u|2 e−fdV . g g g g ZM (cid:16) (cid:17) He also showed that if (g(t),u(t),f(t)) satisfies the following system ∂ g(t) = −2Ric +4du(t)⊗du(t)−2g(t)∇2f(t), g(t) ∂t ∂ (4.2) u(t) = ∆ u(t)−hdu(t),df(t)i , ∂t g(t) g(t) ∂ 2 f(t) = −∆ f(t)−R +2 g(t)∇u(t) , g(t) g(t) ∂t g(t) (cid:12) (cid:12) then the evolution of the entropy is given by (cid:12) (cid:12) (cid:12) (cid:12) d 2 F(g(t),u(t),f(t)) = 2 S +g(t)∇2f(t) g(t),u(t) dt ZM (cid:18)(cid:12) (cid:12)g(t) (4.3) +2 ∆ (cid:12) u(t)−hdu(t),df(t)i(cid:12) 2 e−f(t)dV ≥ 0. g(cid:12)(t) (cid:12)g(t) g(t) g(t) (cid:12) (cid:12) (cid:17) (cid:12) (cid:12)