Symmetries of geometric flows 0 1 0 Xu Chao 2 ∗ Department of Mathematics n a Zhejiang University, Hangzhou, China J 9 January 11, 2010 ] T G . h Abstract t a m By applying the theory of group-invariantsolutions we investigate the symme- [ tries of Ricci flow and hyperbolic geometric flow both on Riemann surfaces. The warped products on n+1 of both flows are also studied. 1 S v 4 9 3 §1 Introduction 1 . 1 The Ricci flow is the geometric evolution equation in which one starts with a smooth 0 0 Riemannian manifold ( n,g ) and evolves its metric by the equation 0 1 M : v ∂ i g = 2Rc, (1.1) X ∂t − r a where Rc denotes the Ricci tensor of the metric g. The Ricci flow has been exhaus- tively studied and successfully applied to solve the famous Poincare´’s Conjecture [2]. Recently, Kong and Liu [9] introduced the hyperbolic geometric flow which is the hy- perbolic version of Ricci flow ∂2 g = 2Rc, (1.2) ∂t2 − which shows different behavior with the original Ricci flow. On Riemann surfaces ( 2,g), equations (1.1) and (1.2) can be simplified to scalar M equations ∗e-mail address: [email protected] 1 Symmetries of geometric flows 2 u = lnu, (1.3) t △ u = lnu, (1.4) tt △ where function u(x,y,t) is the conformal factor of g: g = u(x,y,t)δ ij ij Later we will use the theory of group-invariant solutions to investigate (1.3) and (1.4). As we will see, the sets of symmetries of the two equations are quite large and we expect to find large classes of exact solutions to both flows on Riemann surfaces and the symmetries dependent on the solution of two-dimensional Laplace equation. The technique here we use to investigate the symmetries and exact solutions of the equations is the theory of group-invariant solutions for differential equations which applies Lie group, Lie algebra and adjoint representation to differential equations. For most cases, there is a one-to-one correspondence between different symmetries of an equation and the conjugate classes of subgroups of its one-parameter transformation group. So finally through the classification of subalgebras of the Lie algebra of the transformation group, we are able to classify all the symmetries of the equations. We will introduce this technique briefly later in this paper. For more details, see [11]. We will also study warped products on n+1 or SO(n + 1)-invariant metrics on S n+1 of both flows on the set ( 1,1) n: S − ×S g = ϕ2(x,t)dx2+ψ2(x,t)g , can where g denotes the canonical metric on n. This metric under Ricci flow was can S studied in [1]. Analyzing its asymptotic behavior leads to significant information about thenechpinchwhichisimportantinthesurgeryofRicciflow. Byachangeofcoordinate x s(x) = ϕ(x)dx, Z0 the evolutions of ϕ(s,t) and ψ(s,t) under Ricci flow and hyperbolic geometric flow are the followings respectively: ϕ = nψssϕ t ψ (1.5) ( ψt = ψss−(n−1)1−ψψs2 Symmetries of geometric flows 3 under Ricci flow, and ϕ = nψssϕ ϕ2t tt ψ − ϕ (1.6) ( ψtt = ψss−(n−1)1−ψψs2 − ψψt2 under hyperbolic geometric flow. In contrast to (1.3) and (1.4), equations (1.5) and (1.6) have few symmetries especially in higher dimensions. This paper is organized as follows: We would begin with the theory of group- invariant solutions for differential equations in Section 2. In Section 3, we will study the symmetries and exact solutions of Ricci flow on surfaces. In Section 4, we investi- gate hyperbolic geometric flow on surfaces. In section 5, the warped product of n+1 S on both flows are studied. In Section 6, we give some further discussions. Finally, in section 7, we derive the evolutions of warped products on both flow. Acknowledgement. The author thanks the Center of Mathematical Sciences at Zhejiang University where he wrote this paper during the summer of 2009. §2 Theory of group-invariant solutions for differential equations In this section, we briefly introduce the theory of group-invariant solutions for differ- ential equations. The following main definitions and theorems are cited from [11]. First we introduce the jet space. Given u = lnu, t △ let w = lnu, so eww w w = 0. (2.1) t xx yy − − We regardw andits derivatives asvariables in (2.1), so(2.1) can beregardedas defined on X U(2) = (x,y,t;w;w ,w ,w ;w ,w ,w ,w ,w ,w ) , x y t xx xy xt yy yt tt × { } where X = (x,y,t) is the space of independent variables (x,y,t). { } Symmetries of geometric flows 4 In general, we denote an n-th order differential equation of w with independent variables x =(x1,...,xp) by (x,w(n)) = 0. △ Thus can be regarded as a smooth map form the jet space X U(n) to R △ × :X U(n) R, △ × → and the differential equation tells where the given map vanishes on X U(n), thus △ × determines a subvariety S = (x,w(n)) : (x,w(n)) = 0 X U(n) △ { △ } ⊂ × of the total jet space. Definition 2.1. Let S be a system of differential equations. A symmetry group of the system S is a local group of transformations G acting on an open subset M of the space of independent and dependent variables for the system with the property that whenever u = f(x) is a solution of S, and whenever g f is defined for g G, then · ∈ u = g f(x) is also a solution of the system. · Theorem 2.2. Let M be an open subset of X U(n) and suppose (x,w(n)) = 0 × △ is an n-th order equation defined over M, with corresponding subvariety S M. △ ⊂ Suppose G is a local group of transformations acting on M which leaves S invariant, △ meaning that whenever (x,w(n)) S , we have g (x,w(n)) S for all g G such ∈ △ · ∈ △ ∈ that this is defined. Then G is a symmetry group of the equation in the sense of Defi- nition 2.1. Next we introduce the prolongation of vector fields correspondingto one-parameter transformation group acting on M X U = (x,w) . We only state its formula here ⊂ × { } for our use. The interested reader can see [11]. Theorem 2.3. Let Symmetries of geometric flows 5 p ∂ ∂ v = ξi(x,w) +φ(x,w) ∂xi ∂w i=1 X be a vector field defined on an open subset M X U. The n-th prolongation of v is ⊂ × the vector field ∂ pr(n)v = v+ φJ(x,w(n)) ∂w J J X defined on the corresponding jet space M(n) X U(n), the summation being over all ⊂ × (unordered) multi-indices J = (j ,...,j ), with 1 j p, 1 k n. The coefficient 1 k k ≤ ≤ ≤ ≤ functions φJ of pr(n)v are given by the following formula: p p φJ(x,w(n)) = D (φ ξiw )+ ξiw , J i J,i − i=1 i=1 X X where D is the total derivative operator, and w = ∂w/∂xi, w = ∂w /∂xi. i J,i J We state two important definitions which play important role in the theory. Definition 2.4. Let (x,w(n)) = 0, △ be a differential equation. The equation is said to be of maximal rank if the Jacobian matrix ∂ ∂ (x,w(n)) = ( △, △ ) J△ ∂xi ∂wJ of with respect to all the variables (x,w(n)) is of rank 1 whenever (x,w(n)) = 0. △ △ For example, consider (2.1), the corresponding Jacobian matrix is (x,y,t;w;w ,w ,w ;w ,w ,w ,w ,w ,w )= (0,0,0;eww ;0,0,ew; 1,0,0, 1,0,0), x y t xx xy xt yy yt tt t J△ − − which is of rank 1 whenever (x,y,t,w(2)) = 0. So (2.1) is of maximal rank. △ Symmetries of geometric flows 6 Definition 2.5. An n-th order differential equation (x,w(n)) = 0 is locally solv- △ able at the point (x ,w(n)) S = (x,w(n)) : (x,w(n)) =0 0 0 ∈ △ { △ } ifthere existsasmooth solutionu= f(x)oftheequation, definedforxinaneighborhood of x , which has the prescribed initial condition w(n) = pr(n)f(x ), where pr(n)f(x ) 0 0 0 0 means f and all its derivatives up to order n at point x . The equation is locally 0 solvable if it is locally solvable at every point of S . A differential equation is nonde- generate ifat everypoint (x ,w(n)) S itisboth△locally solvable and of maximal rank. 0 0 ∈ △ The main theorem we will use is the following: Theorem 2.6. Let (x,w(n)) = 0 be a nondegenerate differential equation. A △ connected local group of transformations G acting on an open subset M X U is a ⊂ × symmetry group of the equation if and only if pr(n)v[ (x,w(n))] = 0, whenever (x,w(n))= 0, △ △ for every infinitesimal generator v of G. Wecalculatesomeprolongationformulasherethatwewilluselater. OnM X U, ⊂ × given a vector ∂ ∂ ∂ ∂ v = ξ(x,y,t,w) +η(x,y,t,w) +τ(x,y,t,w) +φ(x,y,t,w) , 1 ∂x ∂y ∂t ∂w its second order prolongation is ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ pr(2)v = v +φx +φy +φt +φxx +φxy +φxt +φyy +φyt +φtt . 1 1 ∂w ∂w ∂w ∂w ∂w ∂w ∂w ∂w ∂w x y t xx xy xt yy yt tt We will use the followings: φt = φ ξ w η w +(φ τ )w ξ w w η w w τ w2, t t x t y w t t w x t w y t w t − − − − − − φx = φ +(φ ξ )w η w τ w ξ w2 η w w τ w w , x w x x x y x t w x w x y w x t − − − − − − Symmetries of geometric flows 7 φy = φ ξ w +(φ η )w τ w ξ w w η w2 τ w w , y y x w y y y t w x y w y w y t − − − − − − φtt = φ +(2φ τ )w η w ξ w +(φ 2τ )w2 2η w w tt tw tt t tt y tt x ww tw t tw y t − − − − − 2ξ w w τ w3 η w2w ξ w2w +(φ 2τ )w tw x t ww t ww t y ww t x w t tt − − − − − 2ξ w wη w 3τ w w η w w ξ w w t xt t yt w t tt w y tt w x tt − − − − − 2η w w 2ξ w w , w t yt w t xt − − φxx = φ +(2φ ξ )w η w τ w +(φ 2ξ )w2 2η w w xx xw xx x xx y xx t ww xw x xw x y − − − − − 2τ w w ξ w3 η w2w τ w2w +(φ 2ξ )w xw x t ww x ww x y ww x t w x xx − − − − − 2τ w 2η w 3ξ w w η w w τ w w x xt x xy w x xx w y xx w t xx − − − − − 2η w w 2τ w w , w x xy w x xt − − φyy = φ +(2φ η )w ξ w τ w +(φ 2η )w2 2ξ w w yy yw yy y yy x yy t ww yw y yw x y − − − − − 2τ w w η w3 ξ w2w τ w2w +(φ 2η )w yw y t ww y ww y x ww y t w y yy − − − − − 2τ w 2ξ w 3η w w y ξ w w y τ w w y yt y xy w y y w x y w t yy − − − − − 2ξ w w 2τ w w . w y xy w y yt − − Next, given ∂ ∂ ∂ v = ξ(s,t,ψ) +τ(s,t,ψ) +φ(s,t,ψ) , 2 ∂s ∂t ∂ψ its second order prolongation is ∂ ∂ ∂ ∂ ∂ pr(2)v = v +φs +φt +φss +φst +φtt . 2 2 ∂ψ ∂ψ ∂ψ ∂ψ ∂ψ s t ss st tt We will use the followings: φt = φ ξ ψ +(φ τ )ψ ξ ψ ψ τ ψ2, t t s ψ t t ψ s t ψ t − − − − φs = φ +(φ ξ )ψ τ ψ ξ ψ2 τ ψ ψ , s ψ s s s t ψ s ψ s t − − − − Symmetries of geometric flows 8 φtt = φ +(2φ τ )ψ ξ ψ +(φ 2τ )ψ2 2ξ ψ ψ tt tψ tt t tt s ψψ tψ t tψ s t − − − − τ ψ3 ξ ψ ψ2+(φ 2τ )ψ 2ξ ψ 3τ ψ ψ ψψ t ψψ s t ψ t tt t st ψ t tt − − − − − ξ ψ ψ 2ξ ψ ψ , ψ s tt ψ t st − − φss = φ +(2φ ξ )ψ τ ψ +(φ 2ξ )ψ2 2τ ψ ψ ss sψ ss s ss t ψψ sψ s sψ s t − − − − ξ ψ3 τ ψ2ψ +(φ 2ξ )ψ 2τ ψ 3ξ ψ ψ ψψ s ψψ s t ψ s ss s st ψ s ss − − − − − τ ψ ψ 2τ ψ ψ . ψ t ss ψ s st − − When we solve out for example ξ(x,y,t,w), η(x,y,t,w), τ(x,y,t,w), φ(x,y,t,w) for v , we get some vectors 1 v , ,v . 1 k ··· These vectors generate the Lie algebra of the transformation group G. To classify its subalgebras, we need to calculate the structure constants [v ,v ] = Cl v , i j ij l and the adjoint representations Ad(exp(εvi))vj = ∞n=0 εnn!(advi)n(vj) = v ε[v ,v ]+ ε2[v ,[v ,v ]] . Pj− i j 2 i i j −··· Aftertheclassification ofsubalgebrasandsubgroupsofG, wegetanoptimalsystem for the equation. By constructing invariants from v in these subalgebras, we can sim- plify the equation to ODE or lower order PDE, thus we expect to find exact symmetric solutions to the original equation. These will be investigated in detail for our equations in the following sections. Symmetries of geometric flows 9 §3 Ricci flow on Riemann surfaces Onasurface, allof theinformationaboutcurvatureis contained inthescalar curvature function R. The Ricci curvature is given by 1 R = Rg , ij ij 2 and the Ricci flow equation can be simplified to ∂ g = Rg . ij ij ∂t − The metric for a surface can always be written (at least locally) in the following form g = u(x,y,t)δ , ij ij where u(x,y,t) > 0. Therefore, we have lnu R = △ . − u Thus ∂ lnu u = △ u, ∂t u · namely, u lnu= 0. (3.1) t −△ Denote w = lnu, thus (x,y,t,w(2)) = w w eww = 0 (3.2) xx yy t △ − − we will use the techniques developed in section 2 to analyze (3.2). First note that the Jacobian matrix of (3.2) is (x,y,t;w;w ,w ,w ;w ,w ,w ,w ,w ,w )= (0,0,0; eww ;0,0, ew;1,0,0,1,0,0), x y t xx xy xt yy yt tt t J△ − − Symmetries of geometric flows 10 which is obviously of rank 1 in S and (3.2) is obviously locally solvable. So we can △ apply Theorem 2.6. Given a vector ∂ ∂ ∂ ∂ v = ξ(x,y,t,w) +η(x,y,t,w) +τ(x,y,t,w) +φ(x,y,t,w) , ∂x ∂y ∂t ∂w we have pr(2)v( (x,y,t,w(2))) = φeww φtew +φxx+φyy. t △ − − Weapplytheformulasforφt,φxx,φyy derivedintheabovesection,sincepr(2)v( (x,y,t,w(2))) = △ 0 whenever (3.2) holds, we use w = e w(w +w ) to cancel w and set coefficients t − xx yy t of every monomial zero. For example the coefficient of w w is 2τ . So τ = 0, i.e. y yt w w − τ = τ(x,y,t). See the following table for all the coefficients. monomial coefficient monomial coefficient e ww : τ τ = 0 e ww : τ τ = 0 − xx xx yy − yy xx yy − − − − e ww w : 2τ = 0 e ww w : 2τ = 0 − x xx xw − x yy xw − − e ww2w : τ = 0 e ww2w : τ = 0 − x xx ww − x yy ww − − e ww w : τ = 0 e ww w : 2τ = 0 − y xx yw − y yy yw − − e ww2w : τ = 0 e ww2w : τ = 0 − y xx ww − y yy ww − − ew : φ = 0 eww : ξ = 0 t x t − eww : η = 0 1 : φ +φ = 0 y t xx yy w : φ= τ 2ξ = 0 w : φ+τ 2η = 0 xx t x yy t y − − − − w w : 2ξ = 0 w w : 2η = 0 x xx w y yy w − − w : 2φ ξ ξ = 0 w : 2φ η η = 0 x xw xx yy y yw xx yy − − − − w2 : φ 2ξ = 0 w w : 2η 2ξ = 0 x ww xw x y xw yw − − − w3 : ξ = 0 w2w : η = 0 x ww x y ww − − w : τ = 0 w : 2η 2ξ = 0 xt x xy x y − − − w w : 2η = 0 w w : 2τ = 0 x xy w x xt w − − w2 : φ 2η = 0 w3 : η = 0 y ww yw y ww − − w2w : ξ = 0 w : 2τ = 0 y x ww yt y − − w w : 2ξ = 0 w w : 2τ = 0 y xy w y yt w − − Thus we finally get