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CHAI Zhao YANG ZhiHua ZHANG RuiYan [email protected] [email protected] [email protected] SCIENCE CHINA Mathematics

. ARTICLES . October 2013 Vol. 56 No. 10: 2007–2013 doi: 10.1007/s11425-012-4549-x

Canonical solitons associated with generalized Ricci flows

CHEN BingLong & GU HuiLing∗

Department of Mathematics, Sun Yat-sen University, Guangzhou 510275,China Email: [email protected], [email protected]

Received May 7, 2012; accepted July 18, 2012; published online December 7, 2012

Abstract We construct the canonical solitons, in terms of Cabezas-Rivas and Topping, associated with some generalized Ricci flows.

Keywords canonical soliton, generalized Ricci flow, harmonic map heat flow, differential form heat flow

MSC(2010) 53C25, 53C44

Citation: Chen B L, Gu H L. Canonical solitons associated with generalized Ricci flows. Sci China Math, 2013, 56: 2007–2013, doi: 10.1007/s11425-012-4549-x

1 Introduction

The optimal transportation has been recently introduced to the theory of Ricci flow (see Lott [5], Mccann and Topping [6] and Topping [11]). The leit-motif is to investigate the Wasserstein distances of two diffusions by taking certain L-distance functions derived from Perelman’s L-length as the cost functions. A precursor of these results is a theorem on stationary manifolds (see Corollary 1.4(viii) in [10]), which says that the Wasserstein distance of two diffusions is decreasing on a background manifold with a time- independent metric of nonnegative Ricci curvature. Of particular interest is the link, revealed by Cabezas- Rivas and Topping [1], between L-Wasserstein distance and the so-called canonical solitons associated with a given Ricci flow; more precisely, they found that these solitons are the appropriate background, in terms of [6], to understand the L-Wasserstein distance. The advantage of this construction is that it is finite-dimensional, and Perelman’s construction is potentially infinite-dimensional; the drawback is that only approximate solitons are obtained, instead of a Ricci flat metric. The goal of this note is to generalize the above canonical soliton construction to other super solutions of the Ricci flow. Apparently we cannot expect the whole construction to hold for very general super solutions of the Ricci flow, so in this paper we only concern two natural cases which have geometric or physical significance. They are generalized Ricci flows coupled with harmonic map heat flow or differential form heat flow. The latter contains the “stringy” Ricci flow (see [8]) as a special case. The application of these generalized Ricci flow to geometry needs further study. In the following, we explain these generalized Ricci flows separately. Case I. Generalized Ricci flow coupled with harmonic map heat flow.

∗Corresponding author

c Science China Press and Springer-Verlag Berlin Heidelberg 2012 math.scichina.com www.springerlink.com 2008 Chen B L et al. Sci China Math October 2013 Vol. 56 No. 10

Let (M,g)and(N ,gN ) be two Riemannian manifolds and ϕ : M→N a smooth map. We evolve (g,ϕ) by the following system of equations, ∂g ∂ϕ ij = −2R + ∇ ϕα∇ ϕβgN , = ϕ, (1.1) ∂t ij i j αβ ∂t where ϕ is the harmonic map Laplacian given by   2 α α β γ ∂ ϕ ∂ϕ α ∂ϕ ∂ϕ ϕα = gij (x, t) − Γk +ΓN . ∂xi∂xj ij ∂xk βγ ∂xi ∂xj

This flow was studied in [7]. It shows that many results of Ricci flow can be parallel generalized to this flow, for example, the monotonicity of (generalized Perelman’s) W-entropy holds true and local noncollapsing result follows similarly to that in the Ricci flow case. Case II. Generalized Ricci flow coupled with differential form heat flow. M Let ( ,g)beann-dimensional closed Riemannian manifold with a Riemannian metric g, let H = n p p=1 H be a closed (dH = 0) (graded) differential form. We evolve (g,H) by the following equations:

n ∂gij p p i2j2 ipjp ∂H = −2Rij + H ··· H ··· g ···g , = −HodgeH, (1.2) ∂t ii2 ip jj2 jp ∂t p=1

where HodgeH =(dδ + δd)H is the (positive) Hodge Laplacian (with respect to evolving metric) on differential forms. It is easy to see that H maintains closed and represents the same De-Rham cohomology class. Let H˚ be a fixed representative, we know H˚ + dB satisfies the second equation of (1.2), provided that B fulfills the equation ∂B = −δ(H˚ + dB), (1.3) ∂t where δ is the L2 adjoint, with respect to the metric given by (1.2), of the exterior derivative d. Hence the equation (1.2) is equivalent to

∂g n ij = −2R + Hp Hp gi2j2 ···gipjp , ∂t ij ii2 ···ip jj2 ···jp p=1 p−1 (1.4) ∂Bi2···ip = gkl∇ Hp ,p=1, 2,...,n, ∂t k li2···ip H = H˚ + dB.

We remark that when H consists of three-form, (1.2) becomes the 1-loop RG flow for (g,B)in[8]. Recall that solitons are a sort of special solutions whose geometric shape is maintained during the evolution. The soliton equations of (1.1) and (1.2) are

1 α β N 1 1 Rij − ∇iϕ ∇j ϕ g + LX gij = εgij , 2 αβ 2 2 (1.5) LX ϕ = ϕ,

and n 1 i2j2 ipjp 1 1 R − H ··· H ··· g ···g + L g = εg , ij 2 ii2 ip jj2 jp 2 X ij 2 ij p=0 (1.6)

LX H = −HodgeH,

respectively, where LX is the Lie derivative with respect to a vector field X on M and ε is a constant (e.g. −1, 0, +1 corresponding to expanding, steady or shrinking solitons respectively). Chen B L et al. Sci China Math October 2013 Vol. 56 No. 10 2009

We remark that

iX H = −δH (1.7)

implies the second equation of (1.6) by using the homotopy formula LX = diX +iX d on differential forms. It turns out that we are able to construct all three types of canonical solitons for (1.1) (see The- orems 1.2 and 2.2), whereas only canonical steady soliton could be constructed for (1.2) (see Theorem 1.1) if H contains differential forms of degree  2. In what follows, for an arbitrarily fixed big N>0, we write A  B for two tensors A and B (depending on N) on the said manifold if for a (any) fixed metric g (independent of N) on the said manifold, any ∈ ∪{ } |∇l − |  →∞  l N 0 , we have g(N(A B)) g Const. as N . If A B, we say that A equals B up to 1 errors of order N . Theorem 1.1. Assume that (g,B) satisfies (1.4). Fix a big N>0, and a time T. Let τ = T − t run in an interval I, M˜ = M×I be the spacetime manifold equipped with the Riemannian metric g˜ and closed differential form H˜ defined by

n 2p − 1 g˜ = g , g˜ =0, g˜ = N + R − |Hp|2, H˜ = H + d B, ij ij i0 00 2p 0 M˜ p=1

where we use 0 to indicate the τ coordinate, and dM˜ B is the exterior derivative of B (regarded as a differential form on M) on M˜ .Then(˜g,H˜ ) on M˜ satisfies the approximate steady soliton equation

n 1 a2b2 apbp R˜ − H˜ ··· H˜ ··· g˜ ···g˜ +Hess (−Nτ)  0, (1.8) ab 2 aa2 ap bb2 bp g˜ ab p=0 ˜ − ˜ i∇˜ (−Nτ)H δH, (1.9)

for a, b =0, 1, 2,...,n. Theorem 1.2. Assume that (g,ϕ) satisfies (1.1). Fix a big N>0, andatimeT.

Let τ = T − t run in an interval I contained in R+, Mˆ = M×I be the spacetime manifold equipped with (ˆg,ϕˆ), a Riemannian metric gˆ and a map ϕˆ : Mˆ →N, defined by   gij N n 1 1 2 gˆij = , gˆi0 =0, gˆ00 = − + R − |∇ϕ| , τ 2τ 3 2τ 2 τ 2 (1.10) ϕˆ(x, τ)=ϕ(x, t).

Then (ˆg,ϕˆ) satisfies the approximate shrinking soliton equations:   ˆ − 1∇ˆ α∇ˆ β N N  1 Rab aϕˆ bϕˆ gαβ +Hessgˆ gˆab, 2 2τ ab 2 (1.11)

L ˆ N ϕˆ  ˆ ϕ,ˆ ∇( 2τ ) for a, b =0, 1, 2,...,n.

2 Construction of canonical solitons 2.1 Proof of Theorem 1.1

To prove Theorem 1.1, we need some preliminary calculations. Let hp = Hp Hp and h =  jk jj2 ···jp kj2 ···jp jk n p ∂ − p=1 hjk. The first equation of (1.2) becomes ∂tgjk = 2Rjk + hjk. By direct computations as in [3], we have   ∂ − R =2(B − B − B + B ) ∂t ijkl ijkl ijlk iljk ikjl

− (RipRpjkl + RjpRipkl + RkpRijpl + RlpRijkp) 2010 Chen B L et al. Sci China Math October 2013 Vol. 56 No. 10

1 + (−∇ ∇ h + ∇ ∇ h + ∇ ∇ h −∇ ∇ h ) 2 i k jl i l jk j k il j l ik 1 + (Rijplhpk + Rijkphpl), (2.1)   2 ∂ − R =2|Ric|2 − R ,h −(trh)+∇ ∇ h . (2.2) ∂t ij ij i j ij ∇ ∇ ∇ p To compute the term i j hij , we need to compute j hij . p ∇ p p ∇ p Note that dH =0, we have j Hi1i2···ip = l=1 il Hi1i2···j···ip . This implies 1 ∇ Hp Hp = ∇ |Hp|2, j ii2···ip ji2 ···ip 2p i 1 ∇ hp = ∇ |Hp|2 + Hp ∇ Hp (2.3) j ij 2p i ii2···ip j ji1 ···ip and 1 1 ∇ ∇ hp = |δHp|2 − R ,hp + |Hp|2 − |∇Hp|2 i j ij ij ij p p p − 1 p p   + Ri1i2i i H ··· H   ··· . (2.4) 2 1 2 i1i2 ip i1i2i3 ip Hence   ∂ n 1 − R =2|Ric|2 − 2 R ,h + |δHp|2 − |∇Hp|2 ∂t ij ij p p=1   n n 1 p − 1 p p − | p|2   + 1 H + Ri1i2i i H ··· H   ··· . (2.5) p 2 1 2 i1i2 ip i1i2i3 ip p=1 p=1 Write the equation (1.4) in local coordinates

p ∂   Hp = Hp + R Hi1···r···s···ip − R Hp , (2.6) ∂t i1···ip i1···ip ik ilrs ilk i1···k···ip l=1 which gives   ∂ p 2 p 2 p − |H | = −2|∇H | + p(p − 1)R   H ··· H   ··· − p h ,h . (2.7) ∂t i1i2i1i2 i1i2 ip i1i2i3 ip ij ij  n p p ProofofTheorem1.1. Letg ˜ be assumed as in Theorem 1.1, and let hjk = p=1 Hjj2 ···jp Hkj2 ···jp .We first calculate the Christoffel symbols ofg ˜,   1 1 n 2p − 1 Γ˜i =Γi , Γ˜i = Ri − hi , Γ˜i = − ∇i R − |Hp|2 , jk jk j0 j 2 j 00 2 2p p=1     n − ˜0 −1 1 ˜0 1 −1 2p 1 p 2 Γ =˜g − Rjk + hjk , Γ = g˜ ∇j R − |H | , (2.8) jk 00 2 j0 2 00 2p p=1   1 ∂ n 2p − 1 Γ˜0 = g˜−1 R − |Hp|2 . 00 2 00 ∂τ 2p p=1 Then we calculate the Ricci curvature and the Hessian of the function −Nτ:

R˜jk  Rjk,   n − n 1 2p 1 p 2 1 p kl j2l2 jplp R˜ ∇ − R + |H | − H ∇ H ··· g g ···g , j0 j 2 4p 2 jj2 ···jp k ll2 lp p=1 p=1 (2.9)       n −  2 ˜ ∂ 1 1 2p 1 p 2  1  R00 − R − trh −  R − |H | − Rij − hij  , ∂τ 2 2 2p 2 p=1 Chen B L et al. Sci China Math October 2013 Vol. 56 No. 10 2011

and 1 Hess(−Nτ)jk −Rjk + hjk,  2  1 n 2p − 1 Hess(−Nτ)  ∇ R − |Hp|2 , j0 2 j 2p (2.10) p=1   1 ∂ n 2p − 1 Hess(−Nτ)  R − |Hp|2 . 00 2 ∂τ 2p p=1

By combining (2.5), (2.7) and the second equation of (1.2), we have 1 R˜ +Hess (−Nτ)  h , jk g˜ jk 2 jk  1 ∂ − R˜ +Hess (−Nτ)  Hp Bp 1 , j0 g˜ j0 2 ji2 ···ip ∂τ i2···ip p (2.11)   n−1  2 ˜ 1  ∂ p  R00 +Hessg˜(−Nτ)00   B ···  . 2 ∂τ i1i2 ip p=0 g On the other hand, we have n n ˜  ˜ ˜ a2b2 ··· apbp  i2j2 ··· ipjp hjk Hja2 ···ap Hkb2···bp g˜ g˜ Hji2···ip Hkj2 ···jp g g = hjk, p=1 p=1 n n ∂ ˜  ˜ ˜ a2b2 ··· apbp  i2j2 ··· ipjp hj0 Hja2···ap H0b2···bp g˜ g˜ Hji2···ip Bj2···jp g g , (2.12) ∂τ p=1 p=1   n n  ∂ 2 ˜  ˜ ˜ a2b2 ··· apbp   p  h00 H0a2···ap H0b2···bp g˜ g˜  B ···  , ∂τ i1i2 ip p=1 p=1 g

−1 1 sinceg ˜00 = O( N ). Combining (2.11) and (2.12), the Ricci curvature equation (1.8) follows. Now we show (1.9). According to whether 0 appears in the indices and using (2.8), the equation (1.9) follows from the calculations:

 cd kl − (δH˜ )∗ =˜g ∇˜ cH˜d∗  g ∇kHl∗,

 cd ∂ (i H˜ )∗ =˜g ∇˜ (−Nτ)H˜ ∗  B∗. ∇˜ (−Nτ) c d ∂t The proof of Theorem 1.1 is completed.

2.2 Proof of Theorem 1.2

∇ α∇ β N Set the new hij = iϕ j φ gαβ and assume that (g,φ) satisfies the equation (1.1). By direct calcula- tions, we have ∂ h = h − (R h + R h )+2RN ∇ ϕα∇ ϕβ∇ ϕγ ∇ ϕδ ∂t ij ij il jl jl il αβγδ i p j p − ∇ ∇ α∇ ∇ β N 2 i lϕ j lϕ gαβ, (2.13) ∂ trh = trh −|h |2 +2RN ∇ ϕα∇ ϕβ∇ ϕγ∇ ϕδ. (2.14) ∂t ij αβγδ i p i p Since 1 ∇ h = ∇ trh + ∇ ϕαφβgN j ij 2 i i αβ and

∇ ∇  | |2 −|∇2 |2 − N ∇ α∇ β∇ γ ∇ δ i j hij = trh + ϕ ϕ Rij ,hij + Rαβγδ iϕ pϕ iϕ pϕ , (2.15) 2012 Chen B L et al. Sci China Math October 2013 Vol. 56 No. 10

we have ∂ R = R +2|Ric|2 + |ϕ|2 −|∇2ϕ|2 − 2 R ,h ∂t ij ij N ∇ α∇ β∇ γ∇ δ + Rαβγδ iϕ pϕ iϕ pϕ (2.16)

and       2 ∂ 1  1  2 − R − trh =2Rij − hij  + |ϕ| . (2.17) ∂t 2 2 We list the relevant calculations of Theorem 1.2 in the following Proposition 2.1 and leave the verifi- cations to the readers. To finish the proof of Theorem 1.2, we still need to use the equation (2.17). Proposition 2.1. Let gˆ be assumed as in Theorem 1.2.Then

Rˆ  R , jk jk  1 1 1 Rˆ ∇ − R + trh − ∇ h , (2.18) j0 j 2 2 2 l lj          2 ˆ ∂ 1 1 1 1 1  1  R00 − R − trh −  R − trh − R − trh − Rij − hij  , ∂τ 2 2 2 2τ 2 2   N 1 g Hess −R + h + jk , gˆ 2τ jk 2 jk 2τ  jk   N 1 1 Hess  ∇ R − trh , (2.19) gˆ 2τ 2 j 2  j0     N  N 1 − 1 1 ∂ − 1 − n Hessgˆ 3 + R trh + R trh 2 , 2τ 00 4τ τ 2 2 ∂τ 2 4τ and ˆ ϕˆα  τϕα,   α (2.20) α  ˆ α ˆ N ab ∂ϕ L∇ˆ N ϕˆ = ∇aϕˆ ∇b gˆ −τ . ( 2τ ) 2τ ∂τ In conclusion, we include for completeness the construction of steady and expanding solitons of gener- alized Ricci flow equation (1.1). Theorem 2.2. Assume that (g,ϕ) satisfies (1.1). Fix a big N>0, andatimeT. (i) Let τ = T − t run in an interval I, M¯ = M×I be the spacetime manifold equipped with (¯g, ϕ¯), a Riemannian metric g¯ and a map ϕ¯ : M¯ →N, defined by

1 2 g¯ij = gij , g¯i0 =0, g¯00 = N + R − |∇ϕ| , 2 (2.21) ϕ¯(x, τ)=ϕ(x, t).

Then (¯g,ϕ¯) on M¯ satisfies the approximate steady soliton equation:

1 α β N R¯ab − ∇¯ aϕ¯ ∇¯ bϕ¯ g +Hessg¯(−Nτ)ab  0, 2 αβ (2.22) L  ¯ ∇¯ (−Nτ)ϕ¯ ϕ¯ for a, b =0, 1, 2,...,n.

(ii) Let t run in an interval I contained in R+, Mˇ = M×I be the spacetime manifold equipped with (ˇg,ϕˇ), a Riemannian metric gˇ and a map ϕˇ : Mˇ →N, defined by   gij N n 1 1 2 gˇij = , gˇi0 =0, gˇ00 = + + R − |∇ϕ| , t 2t3 2t2 t 2 (2.23) ϕˇ(x, t)=ϕ(x, t). Chen B L et al. Sci China Math October 2013 Vol. 56 No. 10 2013

Then (ˇg,ϕˇ) satisfies the approximate expanding soliton equation:   ˇ − 1∇ˇ α∇ˇ β N − N −1 Rab aϕˇ bϕˇ gαβ +Hessgˇ gˇab, 2 2t ab 2 (2.24)

L ˇ N ϕˇ  ˇ ϕˇ ∇(− 2t ) for a, b =0, 1, 2,...,n.

Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos. 11025107, 10831008 and 10901165), the Fundamental Research Funds for Central Universities (Grant No. 2010- 34000-3162643), High Level Talent Project in High Schools in Guangdong Province (Grant No. 34000-5221001), the Fundamental Research Funds for the Central Universities (Grant No. 101gpy25) and China Post-doctoral Science Foundation (Grant No. 201003382). The authors are grateful to Prof. Xi-Ping Zhu for his constant encouragement.

References

1 Cabezas-Rivas E, Topping P. Canonical shrinking solitons associated to a Ricci flow. Calc Var Partial Differential Equations, 2012, 43: 173–184 2 Cabezas-Rivas E, Topping P. Canonical expanding soliton and Harnack ineqalities for Ricci flow. Trans Amer Math Soc, 2012, 364: 3001–3021 3 Hamilton R. Three manifolds with positive Ricci curvature. J Differ Geom, 1982, 17: 255–306 4 Li P, Yau S T. On the parabolic kernel of the Schr¨odinger operator. Acta Math, 1986, 156: 153–201 5 Lott J. Optimal transport and Perelman’s reduce volume. Calc Var Partial Differential Equations, 2009, 36: 49–84 6 Mccann R J, Topping P. Ricci flow, entropy and optimal transportation. Amer J Math, 2010, 132: 711–730 7M¨uller R. Ricci flow coupled with harmonic map flow. Ann Sci Ec Norm Super, 2012, 45: 101–142 8 Oliynyk T, Suneeta V, Woolgar E. A gradient flow for nonlinear sigma models. Nuclear Phys B, 2006, 739: 441–458 9 Perelman G. The entropy formula for the Ricci flow and its geometric applications. ArXiv.org/math.DG/0211159v1, 2002 10 Von Renesse M K, Sturm K T. Transport inequalities, gradient estimates, entropy and Ricci curvature. Comm Pure Appl Math, 2005, 58: 923–940, 153–201 11 Topping P. L-optimal transportation for the Ricci flow. J Reine Angew Math, 2009, 636: 93–122

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An inequality of the holomorphic invariant forms ...... 1965 LU QiKeng Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation ...... 1969 SHENG WeiJie, LI WanTong & WANG ZhiCheng

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Disjoint long cycles in a graph ...... 1983 WANG Hong q On invariant function spaces ( ) ...... 1999 HQP R CHEN HuaiHui & AULASKARI Rauno Canonical solitons associated with generalized Ricci flows ...... 2007 CHEN BingLong & GU HuiLing ...... Totally real conformal minimal tori in the hyperquadric Q 2 2015 ZHONG Xu, WANG Jun & JIAO XiaoXiang On a certain type of nonlinear differential equations admitting transcendental meromorphic solutions ...... 2025 ZHANG Xia & LIAO LiangWen Multiplicity results for the unstirred chemostat model with general response functions ...... 2035 NIE Hua & WU JianHua Global smoothing for the periodic Benjamin equation in low-regularity spaces ...... 2051 SHI ShaoGuang & LI JunFeng Exponential decay of expansive constants ...... 2063 SUN Peng Robust estimation for partially linear models with large-dimensional covariates ...... 2069 ZHU LiPing, LI RunZe & CUI HengJian The generalized Bouleau-Yor identity for a sub-fractional Brownian motion ...... 2089 YAN LiTan, HE Kun & CHEN Chao Some properties of g-convex functions ...... 2117 LI XiaoJuan On strong Markov property for Fleming-Viot processes ...... 2123 LI QinFeng, MA ChunHua & XIANG KaiNan Successful couplings for diffusion processes with state-dependent switching ...... 2135 XI FuBao & SHAO JingHai On convergence of the inexact Rayleigh quotient iteration with the Lanczos method used for solving linear systems ...... 2145 JIA ZhongXiao Positivity and boundedness preserving schemes for the fractional reaction-diffusion equation ...... 2161 YU YanYan, DENG WeiHua & WU YuJiang A proximal point algorithm revisit on the alternating direction method of multipliers ...... 2179 CAI XingJu, GU GuoYong, HE BingSheng & YUAN XiaoMing

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