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Mathematics CONTENTS Editorial Board Supported by NSFC Honorary Editor General ZHOU GuangZhao (Zhou Guang Zhao) Editor General ZHU ZuoYan Institute of Hydrobiology, CAS, China Editor-in-Chief YUAN YaXiang Academy of Mathematics and Systems Science, CAS, China Associate Editors-in-Chief CHEN YongChuan Tianjin University, China GE LiMing Academy of Mathematics and Systems Science, CAS, China SHAO QiMan The Chinese University of Hong Kong, China XI NanHua Academy of Mathematics and Systems Science, CAS, China ZHANG WeiPing Nankai University, China Members BAI ZhaoJun LI JiaYu WANG YueFei University of California, Davis, USA University of Science and Technology Academy of Mathematics and Systems of China, China Science, CAS, China CAO DaoMin Academy of Mathematics and Systems LIN FangHua WU SiJue Science, CAS, China New York University, USA University of Michigan, USA LIU JianYa CHEN XiaoJun Shandong University, China WU SiYe The Hong Kong Polytechnic University, The University of Hong Kong, China China LIU KeFeng University of California, Los Angeles, USA XIAO Jie CHEN ZhenQing Zhejiang University, China Tsinghua University, China University of Washington, USA LIU XiaoBo XIN ZhouPing CHEN ZhiMing Peking University, China The Chinese University of Hong Kong, Academy of Mathematics and Systems University of Notre Dame, USA China Science, CAS, China MA XiaoNan XU Fei CHENG ChongQing University of Denis Diderot-Paris 7, Capital Normal University, China Nanjing University, China France MA ZhiMing XU Feng DAI YuHong University of California, Riverside, USA Academy of Mathematics and Systems Academy of Mathematics and Systems Science, CAS, China Science, CAS, China XU JinChao MOK NgaiMing Pennsylvania State University, USA DONG ChongYing The University of Hong Kong, China University of California, Santa Cruz, USA PUIG Lluis XU XiaoPing CNRS, Institute of Mathematics of Jussieu, Academy of Mathematics and Systems DUAN HaiBao France Science, CAS, China Academy of Mathematics and Systems Science, CAS, China QIN HouRong YAN Catherine H. F. Nanjing University, China Texas A&M University, USA E WeiNan Princeton University, USA RINGEL Claus M. YANG DaChun Peking University, China University of Bielefeld, Germany Beijing Normal University, China SHANG ZaiJiu FAN JianQing YE XiangDong Princeton University, USA Academy of Mathematics and Systems Science, CAS, China University of Science and Technology of China, China FENG Qi SHEN ZhongMin Academy of Mathematics and Systems Indiana University-Purdue University YU XingXing Science, CAS, China Indianapolis, USA Georgia Institute of Technology, USA FU JiXiang SHU Chi-Wang ZHANG James J. Fudan University, China Brown University, USA University of Washington, USA GAO XiaoShan SIU Yum-Tong ZHANG JiPing Academy of Mathematics and Systems Harvard University, USA Peking University, China Science, CAS, China SUN LiuQuan ZHANG Ping GE GenNian Academy of Mathematics and Systems Academy of Mathematics and Systems Zhejiang University, China Science, CAS, China Science, CAS, China GUO XianPing SUN XiaoTao ZHANG PingWen Sun Yat-sen University, China Academy of Mathematics and Systems Peking University, China Science, CAS, China HE XuMing ZHANG ShouWu University of Michigan, USA TAN Lei Columbia University, USA University of Angers, France HONG JiaXing ZHANG Xu Fudan University, China TANG ZiZhou Sichuan University, China Beijing Normal University, China HSU Elton P. ZHOU XiangYu Northwestern University, USA TEBOULLE Marc Academy of Mathematics and Systems Tel Aviv University, Israel Science, CAS, China JI LiZhen University of Michigan, USA WANG FengYu ZHU XiPing Beijing Normal University, China Sun Yat-sen University, China JING Bing-Yi The Hong Kong University of Science WANG HanSheng ZONG ChuanMing and Technology, China Peking University, China Peking University, China E ditorial Staff 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.
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