The Backward behavior of the Ricci and Cross Curvature Flows on SL(2,R)
aa r X i v : . [ m a t h . DG ] J a n THE BACKWARD BEHAVIOR OF THE RICCI AND CROSSCURVATURE FLOWS ON SL (2 , R ) XIAODONG CAO ∗ , JOHN GUCKENHEIMER ♮ , AND LAURENT SALOFF-COSTE ♭ Abstract.
This paper is concerned with properties of maximal solutions of theRicci and cross curvature flows on locally homogeneous three-manifolds of typeSL ( R ). We prove that, generically, a maximal solution originates at a sub-Riemanniangeometry of Heisenberg type. This solves a problem left open in earlier work bytwo of the authors. Introduction
Homogeneous evolution equations.
On a closed 3-dimensional Riemannianmanifold (
M, g ), let Rc be the Ricci tensor and R be the scalar curvature. Thecelebrated Ricci flow [Ham82] starting from a metric g is the solution of(1.1) (cid:26) ∂∂t g = − Rcg (0) = g . Another tensor, the cross curvature tensor, call it h , is used in [CH04] to define thecross curvature flow (XCF) on 3-manifolds with either negative sectional curvatureor positive sectional curvature. In the case of negative sectional curvature, the flow(XCF) starting from a metric g is the solution of(1.2) (cid:26) ∂∂t g = − hg (0) = g . Assume that computations are done in an orthonormal frame where the Ricci tensoris diagonal. Then the cross curvature tensor is diagonal and if the principal sectionalcurvatures are k , k , k ( k i = K jkjk , circularly) and the Ricci and cross curvaturetensors are given by(1.3) R ii = k j + k l (circularly in i, j, l ) , and(1.4) h ii = k j k l . Date : Dec. 10, 2009.2000
Mathematics Subject Classification.
Primary 53C44. ∗ Research partially supported by NSF grant DMS 0904432. ♮ Research partially supported by DOE grant DE-FG02-93ER25164. ♭ Research partially supported by NSF grant DMS 0603866.
A very special case arises when the 3-manifold (
M, g ) is locally homogeneous. Inthis case, both flows reduce to ODE systems. As a consequence, the cross curvatureflow can be defined even if the sectional curvatures do not have a definite sign, as thisis the case for most of the homogeneous 3-manifolds.We say a Riemannian manifold (
M, g ) is locally homogenous if, for every two points x ∈ M and y ∈ M , there exist neighborhoods U of x and V of y , and an isometry ϕ from ( U, g | U ) to ( V, g | V ) with ϕ ( x ) = y . We say that ( M, g ) is homogeneous ifthe isometry group acts transitively on M , i.e., U = V = M for all x and y . Bya result of Singer [Sin60] , the universal cover of a locally homogeneous manifold ishomogeneous.For a closed 3-dimensional Riemannian manifold ( M, g ) that is locally homoge-neous, there are 9 possibilities for the universal cover. They can be labelled by theminimal isometry group that acts transitively:(a) H (3) ( H ( n ) denotes the isometry group of hyperbolic n -space); SO(3) × R ; H (2) × R ;(b) R ; SU(2); SL(2 , R ); Heisenberg; E (1 ,
1) = Sol (the group of isometry of planewith a flat Lorentz metric); E (2) (the group of isometries of the Euclideanplane). This is called the Bianchi case in [IJ92].The crucial difference between cases (a) and (b) above is that, in case (b), the universalcover of the corresponding closed 3-manifold is (essentially) the minimal transitivegroup of isometries itself (with the caveat that both SL(2 , R ) and E (2) should bereplaced by their universal cover) whereas in case (a) this minimal group is of higherdimension. Case (b) corresponds exactly to the classification of 3-dimensional simplyconnected unimodular Lie groups (non-unimodular Lie groups cannot cover a closedmanifold). Any ( M, g ) in case (b) is of form M = G/H , where G = f M is the universalcover, H is a co-compact discrete subgroup of G , and the metric g descends from aleft-invariant metric e g on G .The cases of H (3), SO(3) × R , H (2) × R and R all lead to well-understood andessentially trivial behaviors for both the Ricci and cross curvature flows.The forward behavior of the Ricci flow on locally homogeneous closed 3-manifoldswas first analyzed by Isenberg and Jackson [IJ92]. The forward and backward be-haviors of the cross curvature flows are treated in [CNSC08, CSC08] whereas thebackward behavior of the Ricci flow is studied in [CSC09]. Related works include[Gli08, KM01, Lot07]. In [CSC09, CSC08], the following interesting asymptotic be-havior of these Ricci and cross curvature flows in the backward direction was observed:Let g t be a maximal solution defined on ( − T b , T f ) and passing through a generic g at t = 0. Then either T b = ∞ and g ( t ) = e λt g , or T b < ∞ and there is a positivefunction r ( t ) such that r ( t ) g ( t ) converges to a sub-Riemannian metric of Heisen-berg type, see [Mon02, CSC08]. More precisely, in [CSC09, CSC08], this result wasproved for all locally homogeneous closed 3-manifolds, except those of type SL(2 , R ).Indeed, the structure of the corresponding ODE systems turns out to be somewhatmore complicated in the SL(2 , R ) case. he Backward behavior of the Ricci and Cross Curvature Flows on SL(2 , R ) 3 The aim of this paper is to prove the result described above in the case of locallyhomogeneous 3-manifolds of type SL(2 , R ), i.e., ^ SL(2 , R ). This will finish the proofof the following statement announced in [CSC08, CSC09]. Theorem 1.1.
Let ( M, g ) be a complete locally homogeneous -manifold (compactor not), corresponding to the case (b) discussed above. Let g ( t ) , t ∈ ( − T b , T f ) , be themaximal solution of either the normalized Ricci flow (2.1) or the cross curvature flow (1.2) passing through g at t = 0 . Let d ( t ) be the corresponding distance function on M . Assume that g is generic among all locally homogeneous metrics on M . Then • either M is of type R , T b = ∞ and g ( t ) = g , • or T b < ∞ and there exists a function r ( t ) : ( − T b , → (0 , ∞ ) such that, as t tends to − T b , the metric spaces ( M, r ( t ) d ( t )) converge uniformly to a sub-Riemannian metric space ( M, d ( T b )) whose tangent cone at any m ∈ M is theHeisenberg group H equipped with its natural sub-Riemannian metric. Each of the manifolds M considered in the above Theorem is of the type M = G/H , where G is a simply connected unimodular 3-dimensional Lie group and H is a discrete subgroup of G . By a locally homogeneous metric on M , we mean ametric that descends from an invariant metric on G . A generic locally homogeneousmetric on M is a metric that descends from a generic invariant metric on G . Notethat homogeneous metrics on G can be smoothly parametrized by an open set Ω in afinite dimensional vector space and generic can be taken to mean “for an open densesubset of Ω”.By definition, the uniform convergence of metric spaces ( M, d t ) to ( M, d ) meansthe uniform convergence over compact sets of ( x, y ) → d t ( x, y ) to ( x, y ) → d ( x, y ),see [CSC08, CSC09] for more details.1.2. The Ricci and cross curvature flows on homogeneous -manifolds. As-sume that g is a 3-dimensional real Lie unimodular algebra equipped with an orientedEuclidean structure. According to J. Milnor [Mil76] there exists a (positively oriented)orthonormal basis ( e , e , e ) and reals λ , λ , λ such that the bracket operation ofthe Lie algebra has the form[ e i , e j ] = λ k e k (circularly in i, j, k ) . Milnor shows that such a basis diagonalizes the Ricci tensor and thus also the crosscurvature tensor. If f i = a j a k e i with nonzero a i , a j , a k ∈ R , then [ f i , f j ] = λ k a k f k (circularly in i, j, k ). Using a choice of orientation, we may assume that at most oneof the λ i is negative and then, the Lie algebra structure is entirely determined bythe signs (in {− , , +1 } ) of λ , λ , λ . For instance, + , + , + corresponds to SU(2)whereas + , + , − corresponds to SL(2 , R ).In each case, let ǫ = ( ǫ , ǫ , ǫ ) ∈ {− , , +1 } be the corresponding choice of signs.Then, given ǫ and an Euclidean metric g on the corresponding Lie algebra, we canchoose a basis f , f , f (with f i collinear to e i above) such that(1.5) [ f i , f j ] = 2 ǫ k f k (circularly in i, j, k ) . Xiaodong Cao, John Guckenheimer, Laurent Saloff-Coste
We call ( f i ) a Milnor frame for g . The metric, the Ricci tensor and the cross cur-vature tensor are diagonalized in this basis and this property is obviously maintainedthroughout either the Ricci flow or cross curvature flow. If we let ( f i ) be the dualframe of ( f i ) , the metric g has the form(1.6) g = A f ⊗ f + B f ⊗ f + C f ⊗ f . Assuming existence of the flow g ( t ) starting from g , under either the Ricci flow orthe cross curvature flow (positive or negative), the original frame ( f i ) stays a Milnorframe for g ( t ) along the flow and g ( t ) has the form(1.7) g ( t ) = A ( t ) f ⊗ f + B ( t ) f ⊗ f + C ( t ) f ⊗ f . It follows that these flows reduce to ODEs in (
A, B, C ). Given a flow, the explicitform of the ODE depends on the underlying Lie algebra structure. With the help ofthe curvature computations done by Milnor in [Mil76], one can find the explicit formof the equations for each Lie algebra structure. The Ricci flow case was treated in[IJ92]. The computations of the ODEs corresponding to the cross curvature flow arepresented in [CNSC08, CSC08].1.3.
Invariant metrics on SL (2 , R ) . This paper is devoted to study of the Ricciand cross curvature flows on 3-dimensional Riemannian manifolds that are covered by ^ SL(2 , R ). Since it makes no differences, we focus on SL(2 , R ). Given a left-invariantmetric g on SL(2 , R ), we fix a Milnor frame { f i } such that[ f , f ] = − f , [ f , f ] = 2 f , [ f , f ] = 2 f and g = A f ⊗ f + B f ⊗ f + C f ⊗ f . The sectional curvatures are K ( f ∧ f ) = 1 ABC ( − A + B + C − BC − AC − AB ) ,K ( f ∧ f ) = 1 ABC ( − B + A + C + 2 BC + 2 AC − AB ) ,K ( f ∧ f ) = 1 ABC ( − C + A + B + 2 BC − AC + 2 AB ) . Recall that the Lie algebra sl(2 , R ) of SL(2 , R ) can be realized as the space of twoby two real matrices with trace 0. A basis of this space is W = (cid:18) −
11 0 (cid:19) , H = (cid:18) − (cid:19) , V = (cid:18) (cid:19) . These satisfy [
H, V ] = − W, [ W, H ] = 2 V, [ V, W ] = 2 H. This means that (
W, V, H ) can be taken as a concrete representation of the aboveMilnor basis ( f , f , f ). In particular, f corresponds to rotation in SL(2 , R ). Notefurther that exchanging f , f and replacing f by − f produce another Milnor basis.This explains the B, C symmetry of the formulas above. he Backward behavior of the Ricci and Cross Curvature Flows on SL(2 , R ) 5 Normalizations.
Let g ( t ) , t ∈ I , be a maximal solution of(1.8) (cid:26) ∂∂t g = − vg (0) = g , where v denotes either the Ricci tensor Rc or the cross curvature tensor h . Byrenormalization of g ( t ), we mean a family e g ( e t ) , e t ∈ e I, obtained by a change of scalein space and a change of time, that is e g ( e t ) = ψ ( t ) g ( t ) , e t = φ ( t )where φ is chosen appropriately. The choices of φ are different for the two flowsbecause of their different structures. For the Ricci flow, take φ ( t ) = Z t ψ ( s ) ds. In the case of the cross curvature flow, take φ ( t ) = Z t ψ ( s ) ds. Now, set e ψ ( e t ) = ψ ( t ). Then we have ∂ e g∂ e t = − e v + (cid:18) dd e t ln e ψ (cid:19) e g, where e v is either the Ricci or the cross curvature tensor of e g .On compact 3-manifolds, it is customary to take ddt ln ψ = v , where v = R tr ( v ) dµ R dµ is the average of the trace of either the Ricci or the cross curvature tensor. In bothcases, this choice implies that the volume of the metric e g is constant. Obviously,studying any of the normalized versions is equivalent to studying the original flow.Notice that the finiteness of T b or T f is not preserved under different normalizationof flows. 2. The Ricci Flow on SL (2 , R )2.1. The ODE system.
Mostly for historical reasons, we will consider the normal-ized Ricci flow(2.1) ∂g∂t = − Rc + 23 Rg, g (0) = g , where g is a left-invariant metric on SL(2 , R ). Let g ( t ), t ∈ ( − T b , T f ) be the maximalsolution of the normalized Ricci flow through g . In a Milnor frame { f i } for g , wewrite (see (1.7)) g = Af ⊗ f + Bf ⊗ f + Cf ⊗ f . Xiaodong Cao, John Guckenheimer, Laurent Saloff-Coste
Under (2.1),
ABC = A B C is constant, and we set A B C ≡
4. For this normalizedRicci flow,
A, B, C satisfy the equations(2.2) dAdt = 23 [ − A (2 A + B + C ) + A ( B − C ) ] ,dBdt = 23 [ − B (2 B + A − C ) + B ( A + C ) ] ,dCdt = 23 [ − C (2 C + A − B ) + C ( A + B ) ] . Asymptotic results.
Because of natural symmetries, we can assume withoutloss of generality that B ≥ C . Then B ≥ C as long as a solution exists. Throughoutthis section, we assume that B ≥ C . Theorem 2.1 (Ricci flow, forward direction, [IJ92]) . The forward time T f satisfies T f = ∞ . As t tends to ∞ , B − C tends to exponentially fast and B ( t ) ∼ (2 / t, C ( t ) ∼ (2 / t and A ( t ) ∼ t − . In the backward direction, the following was proved in [CSC09].
Theorem 2.2 (Ricci flow, backward direction) . We have T b ∈ (0 , ∞ ) , i.e., the max-imal backward existence time is finite. Moreover, (1) If there is a time t < such that A ( t ) ≥ B ( t ) then, as t tends to − T b , A ( t ) ∼ η ( t + T b ) − / , B ( t ) ∼ η ( t + T b ) / , C ( t ) ∼ η ( t + T b ) / with η = √ / and constants η i ∈ (0 , ∞ ) , i = 2 , . (2) If there is a time t < such that A ≤ B − C then, as t tends to − T b , A ( t ) ∼ η ( t + T b ) / , B ( t ) ∼ η ( t + T b ) − / , C ( t ) ∼ η ( t + T b ) / with η = √ / , and constants η i ∈ (0 , ∞ ) , i = 1 , . (3) If for all time t < , B − C < A < B then, as t tends to − T b , A ( t ) ∼ B ( t ) ∼ √
64 ( t + T b ) − / , C ( t ) ∼
323 ( t + T b ) . As far as the normalized Ricci flow is concerned, the goal of his paper is to showthat the third case in the theorem above can only occur when the initial condition( A , B , C ) belongs to a two dimensional hypersurface. In particular, it does notoccur for a generic initial metric g on SL(2 , R ). Theorem 2.3.
Let Q = { ( a, b, c ) ∈ R : a > , b > c > } . There is an opendense subset Q of Q such that, for any maximal solution g ( t ) , t ∈ ( − T b , T f ) , of thenormalized Ricci flow with initial condition ( A (0) , B (0) , C (0) ∈ Q , as t tends to − T b , (1) either A ( t ) ∼ ( √ / t + T b ) − / , B ( t ) ∼ η ( t + T b ) / , C ( t ) ∼ η ( t + T b ) / (2) or A ( t ) ∼ η ( t + T b ) / , B ( t ) ∼ ( √ / t + T b ) − / , C ( t ) ∼ η ( t + T b ) / he Backward behavior of the Ricci and Cross Curvature Flows on SL(2 , R ) 7 In fact, let Q (resp. Q ) be the set of initial conditions such that case (1) (resp.case (2) ) occurs. Then there exists a smooth embedded hypersurface S ⊂ Q such that Q , Q are the two connected components of Q \ S . Moreover, for initial conditionon S , the behavior is given by case (3) of Theorem 2.2 . In order to prove this result, it suffices to study case (3) of Theorem 2.2. This isdone in the next section by reducing the system (2.2) to a 2-dimensional system.
Remark 2.1.
The study below shows that, when the initial condition varies, all valueslarger than of the ratio η /η are attained in case (1). Similarly, as the initialcondition varies, all positive values of the ratio η /η are attained in case (2). The two-dimensional ODE system for the Ricci flow.
For convenience,we introduce the backward normalized Ricci flow, for which the ODE is(2.3) dAdt = −
23 [ − A (2 A + B + C ) + A ( B − C ) ] ,dBdt = −
23 [ − B (2 B + A − C ) + B ( A + C ) ] ,dCdt = −
23 [ − C (2 C + A − B ) + C ( A + B ) ] . By Theorem 2.2, the maximal forward solution of this system is defined on [0 , T b )with T b < ∞ .We start with the obvious observation that if ( A, B, C ) is a solution, then t ( λA ( λ t ) , λB ( λ t ) , λC ( λ t ))is also a solution. By Theorem 2.2, there are solutions with initial values in Q suchthat B/A tends to ∞ and others such that B/A tends to 0. Let Q be the set ofinitial values in Q such that B/A tends to 0 and Q be the set of those for which B/A tends to ∞ . Let S be the complement of Q ∪ Q in Q . Again, by Theorem2.2, for initial solution in S , B/A tends to 1.The sets Q , Q , S must be homogeneous cones, i.e., are preserved under dilations.Hence, they are determined by their projectivization on the plane A = 1. So we set b = B/A, c = C/A and compute(2.4) (cid:26) db/dt = 2 A b (1 + b )( b − c − dc/dt = − A c (1 + c )( b − c + 1) . This means that, up to a monotone time change, the stereographic projection of anyflow line of (2.3) on the plane A = 1 is a flow line of the planar ODE system(2.5) (cid:26) db/dt = b (1 + b )( b − c − dc/dt = − c (1 + c )( b − c + 1) . Set Ω = { b > c > } . By Theorem 2.2, any integral curve of (2.5) tends in the forwardtime direction to either (0 , ∞ , c ∞ ), 0 < c ∞ < ∞ , or (1 , db/dt, dc/dt ) = (0 , , , (0 ,
0) (notice that they
Xiaodong Cao, John Guckenheimer, Laurent Saloff-Coste are in ∂ Ω but not in Ω). To investigate the nature of these equilibrium points, wecompute the Jacobian of the right-hand side of the (2.5) which is (cid:18) b − bc − c − − b − b − c − c c − bc − b − (cid:19) . In particular, at (0 , − ,
0) is attractive. At (1 , (cid:18) − − (cid:19) . This point is a hyperbolicsaddle point (the eigenvalues are 2 and − ,
0) must stay in the region { b − c < < b } . In that region, b and c are decreasing functions of time t and dcdb is positive. Using this observation and thestable manifold theorem, we obtain a smooth increasing function φ : [1 , ∞ ) → [0 , ∞ )whose graph γ = { ( b, c ) : c = φ ( b ) } is the stable manifold at (1 ,
0) in Ω. By Theorem2.1 and (2.5), γ is asymptotic to c = b at infinity, and φ ′ (1) = 2. In particular, Ω \ γ has two components Ω and Ω where (0 , ∈ Ω . Further any initial condition inΩ whose integral curve tends to (1 ,
0) must be on γ . It is now clear that the cases(1), (2) and (3) in Theorem 2.2 correspond respectively to initial conditions in Ω ,Ω and γ . This proves Theorem 2.3 with Q i the positive cone with base Ω i and S the positive cone with base γ .Figure (1) shows the curve γ and some flow lines of (2.5). It is easy to see from(2.5) that the flow lines to the right of γ have horizontal asymptotes { c = c ∞ } andthat all positive values of c ∞ appear. This proves Remark 2.1 in case (2) of Theorem2.3. The proof in case (1) is similar, but a different choice of coordinates must bemade. c Figure 1.
The flow line diagram of system (2.5) he Backward behavior of the Ricci and Cross Curvature Flows on SL(2 , R ) 9 The cross curvature flow on SL (2 , R )3.1. The ODE system.
We now consider the cross curvature flow (1.2) where g isa left-invariant metric on SL(2 , R ). Let g ( t ), t ∈ ( − T b , T f ), be the maximal solutionof the cross curvature flow through g . Writing g = Af ⊗ f + Bf ⊗ f + Cf ⊗ f , we obtain the system(3.1) dAdt = − AF F ( ABC ) ,dBdt = − BF F ( ABC ) ,dCdt = − CF F ( ABC ) , where F = − A + B + C − BC − AC − AB,F = − B + A + C + 2 BC + 2 AC − AB,F = − C + A + B + 2 BC − AC + 2 AB.
Asymptotic Results.
Without loss of generality we assume throughout thissection that B ≥ C . Then B ≥ C as long as a solution exists. The behavior of theflow in the forward direction can be summarized as follows. See [CNSC08]. Theorem 3.1 ((XCF) Forward direction) . If B = C then T f = ∞ and there existsa constant A ∞ ∈ (0 , ∞ ) such that, B ( t ) = C ( t ) ∼ (24 A ∞ t ) / and A ( t ) ∼ A ∞ as t → ∞ . If B > C , T f is finite and there exists a constant E ∈ (0 , ∞ ) such that, A ( t ) ∼ B ( t ) ∼ E ( T f − t ) − / and C ( t ) ∼ T f − t ) / as t → T f . In the backward direction, the following was proved in [CSC08].
Theorem 3.2 ((XCF) Backward direction) . We have T b ∈ (0 , ∞ ) , i.e., the maximalbackward existence time is finite. Moreover, (1) If there is a time t < such that A ( t ) ≥ B ( t ) − C ( t ) then, as t tends to − T b , A ( t ) ∼ η ( t + T b ) − / , B ( t ) ∼ η ( t + T b ) / , C ( t ) ∼ η ( t + T b ) / for some constants η i ∈ (0 , ∞ ) , i = 1 , , . (2) If there is a time t < such that A < (cid:0) √ B − BC + C − B − C (cid:1) then,as t tends to − T b , A ( t ) ∼ η ( t + T b ) / , B ( t ) ∼ η ( t + T b ) − / , C ( t ) ∼ η ( t + T b ) / for some constants η i ∈ (0 , ∞ ) , i = 1 , , . (3) If for all time t < , (cid:0) √ B − BC + C − B − C (cid:1) ≤ A < B − C then, as t tends to − T b , A ( t ) ∼ η − ( t + T b ) , B ( t ) ∼ η + 4 p t + T b ) , C ( t ) ∼ η − p t + T b ) for some η ∈ (0 , ∞ ) . As far as the cross curvature flow is concerned, the goal of this paper is to showthat the third case in the theorem above can only occur when the initial condition( A , B , C ) belongs to a two dimensional hypersurface. In particular, it does notoccur for a generic initial metric g on SL(2 , R ). Theorem 3.3.
Let Q = { ( a, b, c ) ∈ R : a > , b > c > } . There is an open densesubset Q of Q such that, for any maximal solution g ( t ) , t ∈ ( − T b , T f ) , of the crosscurvature flow with initial condition ( A (0) , B (0) , C (0)) ∈ Q , as t tends to − T b , (1) either A ( t ) ∼ η ( t + T b ) − / , B ( t ) ∼ η ( t + T b ) / , C ( t ) ∼ η ( t + T b ) / (2) or A ( t ) ∼ η ( t + T b ) / , B ( t ) ∼ η ( t + T b ) − / , C ( t ) ∼ η ( t + T b ) / .In fact, let Q (resp. Q ) be the set of initial conditions such case (1) (resp. case (2) )occurs. Then there exists a smooth embedded hypersurface S ⊂ Q such that Q , Q are the two connected components of Q \ S . Moreover, for initial condition on S ,the behavior is given by case (3) of Theorem 3.2 . In order to prove this result, it suffices to study case (3) of Theorem 3.2. In thatcase, it is proved in [CSC08] that A , B and C are monotone ( A, B non-decreasing, C non-increasing) on ( − T b , A/C, B/C ). This leads to a two-dimensional ODE system whoseorbit structure can be analyzed.
Remark 3.1.
The analysis below shows that, in case (1) of Theorem 3.3 and whenthe initial condition varies, all the values larger than of the ratio η /η are attained.Similarly, in case (2), all the values of the ratio η /η are attained. The two-dimensional ODE system for the cross curvature flow.
Forconvenience, we introduce the backward cross curvature flow, for which the ODE is(3.2) dAdt = 2 AF F ( ABC ) ,dBdt = 2 BF F ( ABC ) ,dCdt = 2 CF F ( ABC ) , where { F i } are defined as before. By Theorem 3.2, the maximal forward solution ofthis system is defined on [0 , T b ) with T b < ∞ .Note that if ( A, B, C ) is a solution, then t ( λA ( t/λ ) , λB ( t/λ ) , λC ( t/λ )) he Backward behavior of the Ricci and Cross Curvature Flows on SL(2 , R ) 11 is also a solution. By Theorem 3.2, there are solutions with initial values in Q suchthat A/B tends to ∞ and others such that ( A/B, C/B ) tends to (0 , Q bethe set of initial values in Q such that A/B tends to ∞ and Q be the set of thosefor which ( A/B, C/B ) tends to (0 , S be the complement of Q ∪ Q in Q .Again, by Theorem 3.2, for initial solution in S , ( A/B, C/B ) tends to (0 , Q , Q , S must be homogeneous cones, i.e., are preserved under dilations.Hence, they are determined by their projectivization on the plane B = 1. So we set a = A/B, c = C/B and compute(3.3) ( da/dt = Bac ) a ( a + 1)( a + c − φ dc/dt = − Bac ) c (1 − c )( a + c + 1) φ , where φ = − a + 1 + c − c − ac − a,φ = − c + a + 1 + 2 c − ac + 2 a. This means that, up to a monotone time change, the stereographic projection ofany flow line of (3.2) on the plane B = 1 is a flow line of the planar ODE system(3.4) (cid:26) da/dt = a ( a + 1)( a + c − φ dc/dt = − c (1 − c )( a + c + 1) φ . Set Ω = { a > , > c > } . By Theorem 3.2, any integral curve of (3.3) tendsin the forward time direction to either (0 , ∞ , c ∞ ), 0 < c ∞ < ∞ , or (0 , da/dt, dc/dt ) = (0 , , , (1 , , (0 , (cid:18) Y Y Y Y (cid:19) , where Y = (3 a + 2 ac + c − φ + 2 a ( a + 1)( a + c − a − c + 1) ,Y = a ( a + 1)[ φ + 2( a + c − − c + 1 − a )] ,Y = − c (1 − c )[ φ − a + c + 1)(3 a + c + 1)] ,Y = (3 c + 2 ac − a − φ − c (1 − c )( a + c + 1)( c − − a ) . In particular, at (0 , − ,
0) is attractive. Its basin of attraction corresponds to region Q definedabove. At (1 , (cid:18) (cid:19) . This point is a repelling fixed point. Itreflects the behavior of the forward Ricci flow described in Theorem 3.1( B > C ). At(0 , (cid:18) (cid:19) . This is the equilibrium point of interest to us and amore detailed analysis is required to determine trajectories that tend toward it. Thisis done with coordinate transformations that blow-up the equilibrium. Blow-up transformations introduce coordinates in which the blown up equilibriumbecomes a circle or projective line representing directions through the equilibrium.Blow-up transformations reduce the analysis of flows near degenerate equilibria toflows with less degenerate equilibria. The blown up system allows us to analyzethe trajectories that are asymptotic to the equilibrium. In particular, trajectoriesapproaching the equilibrium of the original system from different directions yielddifferent equilibria in the blown up system.Translating the equilibrium point to the origin by setting e = c −
1, the equations(3.3) become(3.5) (cid:26) da/dt = − a ( a + 1)( a + e )(3 e + 4 e + 2 ae − a ) ,de/dt = e ( e + 1)( a + e + 2)( e − ae − a − a ) . The leading order terms of equations (3.5) have degrees 3 and 2: the next coordinatetransformation a = u , e = v of the region a ≥ (cid:26) du/dt = − u ( u + 1)( v + u )(3 v + 4 v + 2 u v − u ) ,dv/dt = v ( v + 1)( v + 2 + u )( v − u v − u − u ) . To blow up the origin, the equations (3.6) are transformed to polar coordinates( u, v ) = ( r cos( θ ) , r sin( θ )) and then rescaled by a common factor of r , yielding thevector field X defined by(3.7) dr/dt = 1 / r [ − r (cos ( θ )) (sin ( θ )) − r (cos ( θ )) (sin ( θ )) − r (cos ( θ )) (sin ( θ )) − r (cos ( θ )) sin ( θ ) − r (cos ( θ )) sin ( θ )+ r (cos ( θ )) − r (cos ( θ )) (sin ( θ )) −
20 (cos ( θ )) (sin ( θ )) − r (cos ( θ )) sin ( θ ) + r (cos ( θ )) + 2 r (sin ( θ )) − r (sin ( θ )) (cos ( θ )) − r (cos ( θ )) (sin ( θ )) − r (sin ( θ )) (cos ( θ )) + 6 r (sin ( θ )) − r (sin ( θ )) (cos ( θ )) + 4 (sin ( θ )) ] ,dθ/dt = − / θ ) sin ( θ ) [ − r (sin ( θ )) − r (sin ( θ )) (cos ( θ )) +9 r (cos ( θ )) (sin ( θ )) + 5 r (sin ( θ )) (cos ( θ )) − r (sin ( θ )) +28 r (cos ( θ )) sin ( θ ) + 25 r (cos ( θ )) sin ( θ ) + 5 r sin ( θ ) (cos ( θ )) − θ )) + 16 (cos ( θ )) + 20 r (cos ( θ )) +7 r (cos ( θ )) + r (cos ( θ )) ] . In these equations, the origin of equations (3.6) is blown up to the invariant circle r = 0, and the complement of the origin becomes the cylinder r >
0. Trajectoriesthat tend to the origin in equation (3.6) yield trajectories that tend to an equilibriumpoint of equations (3.7) on the circle r = 0. Now the zeros of dθ/dt on the circle r = 0 are equilibria of the rescaled equations obtained from (3.7). They are locatedat points where cos ( θ ) = 0 , /
3. The equilibria determine the directions in whichtrajectories of (3.6) can approach or leave the origin. These directions correspond todifferent approach to the equilibrium (0 ,
1) of (3.4). In the ( a, c ) coordinates, thedirections θ = ± π/ ,
1) along curves tangent to the he Backward behavior of the Ricci and Cross Curvature Flows on SL(2 , R ) 13 c -axis with tangency degree greater than 2. The direction θ = 0 corresponds toapproaching (0 ,
1) along curves tangent to c = 1. The directions ± θ with cos θ =1 / ,
1) along curves asymptotic to the parabola a = ( c − . Observe that this is consistent with Theorem 3.2(3). Our goal is to showthat this can only happen along a particular curve.Since the circle r = 0 is invariant, the Jacobians at the equilibria discussed above aretriangular. The stability of each equilibrium is determined by the signs of r ( dr/dt )and ∂ ( dθ/dt ) /∂θ when these are non-zero. The equilibria with cos ( θ ) = 0 have r ( dr/dt ) = 2 and ∂ ( dθ/dt ) /∂θ = −
4, so the point is a saddle with an unstablemanifold in the region r >
0. Equilibria with cos ( θ ) = 1 / r ( dr/dt ) = − / ∂ ( dθ/dt ) /∂θ = 16 /
3, so these equilibria are also saddles but with stable manifoldsin the region r >
0. After change of coordinates, only one of these stable manifolds,call it γ , belongs to the region { c < } . This curve γ provides the only way toapproach (0 ,
1) which is consistent with case (3) of Theorem 3.2. The equilibria withcos ( θ ) = 1 have r ( dr/dt ) = 0 and ∂ ( dθ/dt ) /∂θ = −
8, so further analysis is requiredto determine the properties of nearby trajectories. However, because of Theorem 3.2,it is clear that any solution of (3.4) approaching the line { c = 1 } has a → ∞ , hencecannot approach (1 , θ = 0, dθ/dt = 0 and dr/dt = ( r + r ) /
2. Therefore, the r axisis invariant and weakly unstable. The trajectory along this axis approaches (0 ,
0) as t → −∞ . To prove that no other trajectories in r > t → ±∞ ,we consider the vector field Y defined by subtracting r cos ( θ ) / r cos ( θ ) / dr/dt in X :(3.8) dr/dt = 1 / θ ) r [ − r (cos ( θ )) − r (cos ( θ )) − r (cos ( θ )) sin ( θ ) −
20 sin ( θ ) (cos ( θ )) − r (cos ( θ )) (sin ( θ )) − r (cos ( θ )) − r (cos ( θ )) sin ( θ ) − r (cos ( θ )) (sin ( θ )) − r (cos ( θ )) (sin ( θ )) − r (cos ( θ )) (sin ( θ )) − r (cos ( θ )) (sin ( θ )) − r (cos ( θ )) (sin ( θ )) + 6 r (sin ( θ )) +2 r (sin ( θ )) + 4 (sin ( θ )) ] ,dθ/dt = − / θ ) sin ( θ ) [ − r (sin ( θ )) − r (sin ( θ )) (cos ( θ )) +9 r (cos ( θ )) (sin ( θ )) + 5 r (sin ( θ )) (cos ( θ )) − r (sin ( θ )) +28 r (cos ( θ )) sin ( θ ) + 25 r (cos ( θ )) sin ( θ )+5 r sin ( θ ) (cos ( θ )) − θ )) + 16 (cos ( θ )) + 20 r (cos ( θ )) +7 r (cos ( θ )) + r (cos ( θ )) ] . The vector field Y is transverse to the vector field X in the interior of the firstquadrant: dθ/dt < X and Y and the r component of X is larger thanthe r component of Y . Therefore, trajectories of X cross the trajectories of Y frombelow to above as they move left in the ( θ, r ) plane. The vector field Y has a commonfactor of sin( θ ) in its two equations. When Y is rescaled by dividing by this factor,the result is a vector field that does not vanish in a neighborhood of the origin. Since the θ axis is invariant for Y , the Y trajectory γ starting at ( θ, r ) , r > r axis at a point with r >
0. The X trajectory starting at ( θ, r ) lies above γ , soit does not approach the origin. This proves that the only trajectories of equations(3.7) asymptotic to the origin lie on the r and θ axes.In conclusion, the above analysis shows that the regions Q , Q are separated bythe 2-dimensional cone S determined by the curve γ in the ( a, c )-plane. This provesTheorem 3.3. Figure 2 describes the flow lines of (3.4). The part of interest to usis the part below the line { c = 1 } , which corresponds to { B > C } . The part abovethe line { c = 1 } corresponds to the case { B < C } , where the role of B and C areexchanged. The most important component of this diagram are the flow lines thatare forward asymptotic to (0 , S in Theorem(3.3). The flow lines in the upper-left corner have vertical asymptotes { a = a ∞ } with all positive values of a ∞ appearing. Similarly, the flow lines on the right havehorizontal asymptotes { c = c ∞ } with all positive values of c ∞ appearing. Thesefacts can easily be derived from the system (3.4). This proves the part of Remark3.1 dealing with case (1) of Theorem 3.3. The other case is similar using differentcoordinates. c Figure 2.
The flow line diagram of system (3.4)
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Department of Mathematics, Cornell University, Ithaca, NY 14853-4201
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