The Ricci Flow for Nilmanifolds

We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.'s for some of these flows and describe their qualitative properties. We also present some explicit solutions for the evolution of soliton metrics under the Ricci flow.


Introduction
The Ricci flow, defined by R. Hamilton ([Ham82]), is an important tool for understanding the topology and geometry of three-manifolds. It is key in Perelman's work ([Pera], [Perc], [Perb]), and has useful applications in other areas of geometry as well. Due to the difficulty of solving the partial differential equations involved, very few explicit examples of Ricci flow solutions are known.
For a homogeneous manifold, the Ricci flow can be presented as a set of O.D.E.'s rather than P.D.E.'s. In many cases, it is possible to solve these systems exactly or to make estimates that allow a description of the qualitative behavior of the system. The Ricci flows for the universal covers of compact homogeneous spaces of dimension three have been analyzed in [IJ92] and [KM01]. Solutions for certain metrics on the universal covers of compact homogeneous four-manifolds are described in [IJL06].
These examples provide insight into the behavior of the Ricci flow in general, exhibiting many of the phenomena of interest in variable curvature cases. If a manifold M admits a homogeneous metric, the set of homogeneous metrics on M is invariant under the Ricci flow on the space of all metrics on M . One hopes that this invariant set is an attractor for the Ricci flow. If that is true, a description of the Ricci flow for homogeneous metrics on M would yield an understanding of the long-term behavior of metrics in a large subset of the set of all metrics on M .
In this work, we continue the study of the Ricci flow for homogeneous spaces initiated in [IJ92], [KM01], and [IJL06]. We analyze the Ricci flow for the class of homogeneous spaces consisting of simply connected nilmanifolds of arbitrary dimension. A nilmanifold is a Riemannian manifold with universal cover (N, g), where N is a simply-connected nilpotent Lie group N endowed with a left-invariant metric g. Although no Einstein metrics exist on a nilpotent Lie group N unless N is abelian (Corollary 2, [Jen69]), many nilpotent groups admit soliton metrics. A soliton metric is a metric g that evolves under the Ricci flow by diffeomorphisms and rescaling; that is, g t = t · η * t g 0 , where η t is a one-parameter family of diffeomorphisms. If a nilpotent Lie group does admit a soliton metric, then it is unique up to scaling ( [Lau01]). However, not all nilpotent Lie groups admit soliton metrics. Using geometric invariant theory, it can be shown that under the Ricci flow, any left-invariant metric g on a nilpotent Lie group N approaches, modulo rescaling, a unique soliton metric g ′ on a nilpotent Lie group N ′ , and if N admits a soliton metric, that limiting nilmanifold is the soliton metric on N. ( [Jab]).
The Ricci flow for three-and four-dimensional nilmanifolds is fairly well understood. In dimension three, there is a single simply connected nonabelian nilpotent Lie group, the Heisenberg group H 3 . It is shown in [IJ92] that under the Ricci flow, any initial left-invariant metric g 0 on H 3 collapses to a flat metric on R 2 . In dimension four, there is a single simply connected nilpotent Lie group that is not a product of lower-dimensional nilpotent Lie groups, the filiform group L 4 . It was shown in [IJL06] that under the Ricci flow, any initial metric g 0 that is diagonal with respect to a special basis collapses. As time goes to infinity, any initial metric on H 3 and the special metrics on L 4 can be viewed in the appropriate framework as asymptotically projectively approaching the unique soliton metric on H 3 or L 4 , respectively (See [Lot07], [GIK06]). Now we summarize the main results in this paper. Following this section, in Section 2 we establish the necessary background and preliminaries involving nilmanifold geometry and the Ricci flow. The Ricci flow on the space of left-invariant metrics on a simply connected Lie group G can be converted to a flow on the space of inner products on the corresponding Lie algebra g. The flows for all individual metric Lie algebras can be combined to define a Ricci flow on the space of metric Lie algebras, and this flow projects to a projectivized Ricci flow on the space of all volume-normalized metric Lie algebras. We define what it means for for a metric Lie algebra to projectively approach another metric Lie algebra, and what it means for a metric Lie algebra to collapse under the Ricci flow. We define a Lie bracket flow for a Lie algebra that describes how the Lie bracket relative to a moving orthonormal basis changes under the Ricci flow, and we define a projectivization of this flow.
In Theorem 3.1 of Section 3, we set up systems of O.D.E.'s for the Ricci flow and the Lie bracket flow for a single nilpotent Lie algebra (n, Q). Theorem 3.6 gives O.D.E.'s for the projectivized Lie bracket flow. Proposition 3.9 describes invariant quantities for the Ricci flow for nilpotent metric Lie algebras.
In Section 4, we find some explicit solutions for soliton trajectories for the Ricci flow for nilpotent metric Lie algebras; these are presented in Theorem 4.2. The theorem only gives solutions for nilpotent metric Lie algebras admitting a special kind of basis. We describe some broad conditions under which these bases exist.
Finally, in Section 5, we give some examples. We consider the Lie bracket flow for Heisenberg Lie algebras and a Lie algebra that does not admit a soliton metric.
The author is grateful to Jim Isenberg for his suggestion of studying the Ricci flow for nilmanifolds. She thanks him, Jorge Lauret, Peng Lu and Dave Glickenstein for helpful discussions. This work was supported by NSF ADVANCE grant #SBE-0620073.

Preliminaries
2.1 Structure of metric nilpotent Lie algebras indexes the set of nonzero structure constants α l jk for g relative to B without repetitions due to skew-symmetry.
Suppose that g is nonabelian, so Λ B is nonempty. Let {e i } n i=1 be the standard orthonormal basis for R n . For 1 ≤ j, k, l ≤ n, define the 1 × n row vector y l jk to be e T j + e T k − e T l . We call a vector y l jk , where (j, k, l) ∈ Λ B , a root vector for (g, Q) relative to the basis B. Let y 1 , y 2 , . . . , y m (where m = |Λ B |) be an enumeration of the root vectors y l jk for (g, Q) relative to B, in the dictionary order on the integer triples (j, k, l). Define the root matrix Y for (g, Q) relative to B to be the m × n matrix whose rows are the root vectors y 1 , y 2 , · · · , y m . When g is nonabelian, the Suppose that the basis B is orthonormal, so that the inner product Q can be written as . . , n. The structure constants for g relative to B are given by . . , q n > 0, yields a family of inner products on g. For each such Q, there is an orthonormal basis B Q = B obtained by rescaling each basis vector in B by its length; elements of B are the vectors x i = 1 x i x i , for i = 1, . . . , n. The structure constants for g relative to the orthonormal basis B are Note that the set Λ B defined by B is the same as the set Λ B defined by B, and hence the set of root vectors is the same for B and B. Consequently B and B have the same root matrix Y and the same Gram matrix U. We define the structure vector for (g, Q) relative to the orthogonal basis B to be the m × 1 vector a having as entries the squares of the nonzero structure constants relative to the orthonormal basis B. More precisely, the entries of a are the numbers q l q j q k (α l jk ) 2 , for (j, k, l) in Λ B , listed in the dictionary order on (j, k, l).

Curvatures of nilpotent metric Lie algebras
We say that a basis B = {x i } n i=1 of a metric Lie algebra (g, Q) is a Ricci-diagonal if the Ricci form for (g, Q) is diagonal when represented with respect to B. The 1 × n vector Ric B defined by will be called the Ricci vector for (n, Q) relative to the basis B. Together, an orthogonal Ricci-diagonal basis B for (n, Q), its Ricci vector, and the set of lengths of the basis vectors determine the inner product and Ricci form for (n, Q). The next theorem gives easy-to-compute formulas for the Ricci form and Ricci vector for a metric nilpotent Lie algebra (n, Q). Recall that the Lie bracket of a metric Lie algebra (g, Q) is encoded in the linear map J : g → End(g) defined by J x (y) = ad * y x, for x and y in g. The inner product Q(·, ·) = ·, · on g induces an inner product on the tensor algebra of g, which we also denote by ·, · .
Theorem 2.1 (Theorems 6 and 8, [Pay]). Let (n, Q) be a nonabelian metric nilpotent Lie algebra. Then the Ricci form for (n, Q) is given by where x and y are in n. Let B be an orthogonal Ricci-diagonal basis with associated root matrix Y and structure vector a. The Ricci vector for (n, Q) relative to B may be written as Remark 2.2. From Equation (4) it follows that an orthogonal basis Soliton inner products on nilpotent Lie algebras may be characterized algebraically by the property that the Ricci endomorphism differs from a scalar multiple of the identity automorphism by a derivation D = Ric Q −β Id of the Lie algebra ( [Lau01]). The constant β is called the soliton constant for (n, Q). The soliton constant is always negative when n is nonabelian, and the eigenvalues for D are positive and rational (See [Heb98], [Lau01]).
The next theorem gives an easily checked linear condition that is equivalent to an inner product Q on a nilpotent Lie algebra n being a soliton inner product. We conclude the section with some examples that illustrate the definitions and theorems presented thus far. First we will consider the indecomposable nilpotent Lie algebras in dimensions three and four.
In the next example we consider the five-dimensional Heisenberg Lie algebra.

The Ricci flow on the space of metric Lie algebras
Isometries are preserved under the Ricci flow, so that as an initial metric g 0 on a manifold M evolves under the Ricci flow, the metric g t at time t has the isometry group Iso(g 0 ) of the initial metric as a subgroup of its isometry group Iso(g t ).
Therefore, the set of homogeneous metrics on M is invariant under the Ricci flow. Let (G, g) be an n-dimensional simply connected Lie group endowed with leftinvariant metric g and with corresponding metric Lie algebra (g, Q). Since the group G acts simply transitively on itself by isometries, there exists an G-invariant global framing of G. The Ricci flow at any point can therefore be expressed in terms of the Ricci flow at the tangent space to the identity. We view the Ricci flow as a flow φ t on the space P + (g) of inner products on the vector space g. We will call this flow the Ricci flow for the Lie algebra g, and we will write the solution for the initial condition Q 0 as Q t .
The Ricci flow can be thought of as a flow on the space of metric Lie algebras (modulo isometry). We now describe the structure of that space. After fixing a basis for a Lie algebra g of dimension n, the Lie algebra can be identified with a point µ in Λ 2 V * ⊗ V, where V is an n-dimensional vector space. We use g µ to denote the algebra with underlying space V and its multiplication defined by µ.
A metric Lie algebra (g, Q) of dimension n is a point in while a metric nilpotent Lie algebra of dimension n is a point in Note that the Jacobi identity and the nilpotency conditions for a Lie algebra g µ are both polynomial constraints on the structure constants so these sets are algebraic subsets.
Define an equivalence relation ∼ on X n and N n such that (g µ , Q 1 ) ∼ (g ν , Q 2 ) if and only if (g µ , Q 1 ) and (g ν , Q 2 ) are isometric. Then X n = X n /∼ parametrizes the spaces of metric Lie algebras, and N n = N n /∼ parametrizes the spaces of metric nilpotent Lie algebras.
E. Wilson showed that the simply connected nilmanifolds corresponding to metric Lie algebras (n µ , Q 1 ) and (n ν , Q 2 ) are isometric if and only if there is an isometric isomorphism mapping one to the other ([Wil82]). Hence, the space N n can be identified with the quotient space for the natural action of O(n) on N n . This action is the restriction of the natural action of GL n (R) on Λ 2 V * ⊗ V × P + (V ), and is defined as follows: for g in GL n (R), where gµ ∈ Λ 2 g * ⊗ g is given by (gµ)(x, y) = gµ(g −1 x, g −1 y), and the inner product gQ satisfies gQ(x, y) = Q(gx, gy).

The projectivized Ricci flow on the space of volume-normalized metric Lie algebras
Let P + (g)/∼ denote the space of volume-normalized inner products on a Lie algebra g, obtained through the equivalence relation Q ∼ λQ for Q in P + (g) and λ in R + . Denote the equivalence class of inner product Q by Q. The Ricci flow φ t for the Lie algebra g projects to a flow φ t : P + (g)/∼ → P + (g)/∼ on P + (g)/∼ because φ t (λQ) = λφ t (Q) for any Q in P + (g) and any λ > 0. We call this flow the projectivized Ricci flow for g, and we will write φ t (Q 0 ) as Q t , for Q 0 in P + (g)/∼ . The space P + (g)/∼ has a natural closure, the compact set P ≥0 (g)/∼, where P ≥0 (g) is the set of nontrivial positive semidefinite symmetric bilinear forms on g, and the equivalence relation ∼ is defined as before.
We say that a metric Lie algebra (g, Q) collapses under the Ricci flow if the limit Q ∞ = lim t→∞ Q t exists in P ≥0 (g)/∼ but is in the boundary of P + (g)/∼; that is, any representative Q ∞ for the limiting normalized symmetric bilinear form Q ∞ is not positive definite. We will then say that Q and Q collapse to Q ∞ . Let X n and N n be as defined previously. Define a new equivalence relation ∼ on These spaces parametrize the families of nonabelian volume-normalized metric Lie algebras and nonabelian volume-normalized nilpotent metric Lie algebras, respectively. We will represent the equivalence class of (g, Q) in X n or N n by (g, Q). Note We extend the projectivized Ricci flow for a single Lie algebra to a flow, also denoted by φ t , on the spaces X n and N n . It is defined by where Q t is the solution φ t (Q) to the Ricci flow for the nilpotent Lie algebra g with initial condition Q. The flow is well-defined since the Ricci flow commutes with homotheties. For a metric Lie algebra (g, Q), φ t (g, Q) can be identified with (g, φ t (Q)), where φ t is the projectivized Ricci flow for the single Lie algebra g. Note the the algebraic structure is constant under the flow: for all finite t ≥ 0, the underlying nilpotent Lie algebras for (g, Q) and for ψ t ((g, Q)) are isomorphic.
The next simple example illustrates how a metric Lie algebra can collapse under the Ricci flow while having a nondegenerate limit point for the projectivized Ricci flow.
Example 2.7. It was shown in [IJ92] that the Ricci flow for the three-dimensional Heisenberg Lie algebra h 3 is given by The positive constant c is the structure constant α 3 12 for B. All inner products Q on h 3 are homothetic, so the space N 3 consists of a single point: the equivalence class of (h 3 , Q 0 ). At all finite times, the metric nilpotent Lie algebras (h 3 , Q t ) that are solutions to the Ricci flow with initial condition Q 0 project to the same point (h 3 , Q 0 ) in N 3 . Therefore, the point (h 3 , Q 0 ) is a fixed point for the projectivized Ricci flow on N 3 .
Yet the inner product Q 0 collapses under the flow, because forcing lim t→∞ Q t to be in the boundary of P + (h 3 )/∼ .

The Lie bracket flow and the projectivized Lie bracket flow
Let (g, Q) be a metric Lie algebra, and let B = {x i } n i=1 be an orthonormal basis for g, with respect to which Q is written as It is often possible to find stably Ricci-diagonal bases. All three-dimensional unimodular metric Lie algebras and many four-dimensional metric Lie algebras have such bases ( [IJ92], [IJL06]).
If the basis B is a stably Ricci-diagonal basis for (g, Q), both the inner product and the Ricci form remain diagonal with respect to B under the Ricci flow, and the positive functions q 1 , q 2 , . . . , q n encode the solution Q t = n i=1 q i dx i ⊗ dx i to the Ricci flow for g with initial condition Q 0 = Q. For 1 ≤ j, k, l ≤ n, define the function a l jk of t by where α l jk is the structure constant for (g, Q 0 ) relative to the Q 0 -orthonormal basis B as defined in Equation (1). These functions represent structure constants for (g, Q t ) relative to the Q t -orthonormal basis B Qt defined in Section 2.1.
Let a 1 , . . . , a m be an enumeration of such functions a l jk for (j, k, l) in Λ B , in dictionary order. Define the structure vector function a : R → R m by for each t; this is just the m × 1 structure vector relative to the basis B Qt for (g, Q t ).
Observe that the vector a t is positive at all times t ≥ 0. We call the flow a t the Lie bracket flow.
The structure vector function a t defined in Equation (7) takes values in R m . We are interested in the asymptotic behavior of a t modulo rescaling; that is, the values of [a t ] in projective space P m−1 (R). Since entries of a t are positive for all t > 0, we will describe the flow of [a t ] using the homogeneous variables s 1 , . . . , s m−1 given by These coordinates parametrize the affine algebraic subset A m−1 = {(s 1 : s 2 : · · · : s m−1 : 1)} of P m−1 (R).

Limits for the Ricci flow and the Lie bracket flow as time goes to infinity
In general, if [a t ] converges to [a ∞ ] under the projectivized Lie bracket flow (relative to some stably Ricci-diagonal basis), the Ricci flow does not necessarily limit on a Lie group endowed with a left-invariant metric whose corresponding metric Lie algebra has structure constants equal to [a ∞ ]. In general, caution must be exercised in extracting limits for the projectivized Ricci flow from limits for the projectivized Lie bracket flow. However, in the case of the projectivized Ricci flow on the space of volume-normalized metric nilpotent Lie algebras, problems do not arise: one can say that the structure constants for the metric Lie algebra associated to the limiting homogeneous space are given by [a ∞ ]. It was demonstrated in [Wil82] that the isometry group of a simply connected nilmanifold N associated to metric nilpotent Lie algebra (n, Q) is equal to the semidirect product K ⋉ N of translations from N and a compact isotropy group K, and the Lie algebra of K is equal to Aut ((n, Q)) . Suppose that (n, Q t ) converges to (n ∞ , Q ∞ ). If f ∈ Aut ((n, Q t )) for all t, then f ∈ Aut ((n ∞ , Q ∞ )) , where we identify f with a linear map of the vector space V used to define N n . In contrast, whenever n ∼ = n ∞ , the n-dimensional group of translational isometries for finite time does not coincide with the group of translational isometries for the limiting nilmanifold.
There is a one-to-one correspondence between Lie brackets in Λ 2 V * ⊗ V, modulo the action of GL n (R), and metric nilpotent Lie algebras (n, Q) in N n . The space N n is closed and invariant for the flow φ t , and if a t is nilpotent for all t, and . Then there is a corresponding limit of metric nilpotent Lie algebras lim t→∞ (n, Q t ) = (n ∞ , Q ∞ ) in N n . At all finite times, the geometry of the nilmanifold class corresponding to (n, Q t ) is completely determined by left-multiplication of the inner product Q t over the Lie group N = exp(n), while the geometry of the nilmanifold class corresponding to (n ∞ , Q ∞ ) is completely determined by left-multiplication of the inner product Q ∞ over the Lie group N ∞ = exp(n ∞ ). Hence, the local geometry (modulo homothety), of the nilmanifolds for (n, Q t ) converges to the local geometry (modulo homothety) for the nilmanifold class defined by (n ∞ , Q ∞ ).
Observe that the vector a t determines the metric nilpotent Lie algebra (n, Q t ) up to isometry because it tells us the Lie algebra structure's structure constants relative to an orthonormal basis. Hence the Lie bracket flow determines the Ricci flow and the two flows are equivalent.
Suppose that one could solve for a t exactly. Equation (5) describes the Ricci vector in terms of a t , which in turn gives the Ricci form ric Qt as a function of time. Using the equation for the Ricci flow, one can solve for the functions q 1 , . . . , q m giving the inner product Q t as a function of time by integrating −2ric Qt with respect to t. Also note that since the connection and curvatures for (n, Q t ) depend only on the structure constants relative to an orthonormal basis, these geometric quantities are determined by the Lie bracket flow. In order to compute or estimate geometric quantities, it is sufficient to consider only the Lie bracket flow, which in many cases is easier to work with than the Ricci flow.

Systems of ordinary differential equations
In this section, we set up systems of ODE's for the Ricci flow and for the projectivized bracket flow. Theorem 3.1. Let (g, Q) be a metric Lie algebra with stably Ricci-diagonal basis Let Y be the root matrix and let U be the Gram matrix for (n, Q) relative to B. Let Q t be the solution to the Ricci flow for g with initial condition Q 0 = Q, and let the functions q 1 , . . . , q n be defined by Q t = n i=1 q i dx i ⊗ dx i . Let a t = (a i (t)) be the structure vector function for (g, Q t ) as defined in Equation (7). and In the case that g is nonabelian and nilpotent, and Remark 3.2. Observe that the O.D.E.'s for a 1 , a 2 , . . . , a m are quadratic, and they can always be put in log-linear form: for example, Equation (10) Changing variables to ln q i , for i = 1, . . . , m, yields (ln q 1 , . . . , ln q n ) ′ = −2 Ric B .
For (l 4 , Q) as in Example 2.5, when we let a 1 = q 3 q 1 q 2 and a 2 = q 4 q 1 q 3 , the system of equations from Equation (12) is , which decouples, so it is not hard to solve for a 1 and a 2 , and then q 1 , q 2 , q 3 and q 4 . For the metric Heisenberg algebra (h 5 , Q) as in Example 2.6, after letting a 1 = q 5 q 1 q 2 and a 2 = q 5 q 3 q 4 , we get the system a ′ 1 = −3a 2 1 − a 1 a 2 (15) a ′ 2 = −a 1 a 2 − 3a 2 2 , which, in contrast to the first two examples, has no simple explicit general solution. Instead, it is necessary to make a qualitative analysis of a projectivization of such a system; this is the goal of the next section.

O.D.E.'s for the projectivized Lie bracket flow
Now we define some matrices and hyperplanes that are needed to state the main theorem of the section. Define the (m − 1) × m matrix P by The next proposition describes some properties of these hyperplanes and their normal vectors, when U is the Gram matrix for a metric nilpotent Lie algebra (n, Q) relative to an orthogonal basis B. Proof of Theorem 3.6. Using the quotient rule, we get By Theorem 3.1, for i = 1, . . . , m − 1, the function a ′ i is equal to −a i times the product of the ith row of U with a, so The direction of the trajectories for the two systems is the same because a m > 0.
Although the system in Equation (18) may be quite difficult to solve exactly, as the functions η i , i = 1, . . . , m are linear in the variables s i , the system in Equation (19) is more tractable.
In the next example, we illustrate the definitions of the vectors n i and hyperplanes H 0 i , and we give an application of the previous theorem. Example 3.8. Let (n, Q) be the five-dimensional metric Lie algebra that with respect to an orthogonal basis B = {x i } 5 i=1 has the following bracket relations: The kernel of P U is spanned by v = (2, 2, 1) T ; note that U v = 7(1, 1, 1) T . Therefore, by Theorem 2.3, if the inner product Q which has B as an orthogonal basis has structure vector a = ( q 4 q 1 q 3 , q 5 q 1 q 4 , q 5 q 2 q 3 ) equal to (2, 2, 1) T , then it is soliton with soliton constant β = −7/2. In that case the Ricci vector is − 1 2 (4, 1, 3, 0, −3), and the derivation D = Ric −β Id is represented by [D] B , the diagonal matrix diag(3, 6, 4, 7, 10). This example will be revisited later in Example 4.3, and soliton inner products will be described.
(Compare to the arrows drawn on the axes, and Part 2 of Proposition 3.5.) It can be seen from the slope field that for any initial point s 0 > 0, eventually the trajectory enters the cone η 1 ( s)η 2 ( s) > 0, and once inside that cone, asymptotically approaches (2, 2). We note that the fixed points on the axes correspond to soliton metric nilpotent Lie algebras whose underlying algebra is not isomorphic to n.

Invariants for the Ricci flow
In previous studies of the Ricci flow for homogeneous spaces, the systems of ODE's for the Ricci flow are solved by finding invariant quantities under the flow. The next proposition gives an easy way to find some of these invariant quantities: vectors in the kernel of the root matrix Y for (n, Q) relative to orthogonal Ricci-diagonal basis B yield conserved monomial quantities for the Ricci flow for n. Proposition 3.9. Let (n, Q) be a nonabelian metric nilpotent Lie algebra with orthogonal Ricci-diagonal basis B. Suppose that there is a stably Ricci-diagonal basis with respect to which the solution Q t to the Ricci flow is expressed as Q t = n i=1 q i dx i ⊗ dx i . Let Y be the root matrix for (n, Q) relative to B. Let a t be as defined in Equation (6). d = (d 1 , . . . , d n ) satisfies the condition that dY T a t = 0 for all t if and only if the quantity q d 1 1 q d 2 2 · · · q dn n is preserved under the Ricci flow. In particular, if dY T = 0 then q d 1 1 q d 2 2 · · · q dn n is preserved.

The constant vector
Proof. The quantity q d 1 1 q d 2 2 · · · q dn n is preserved if and only if its natural logarithm n i=1 d i ln q i is preserved. This is true if and only if for all t > 0, Rewriting (ln q 1 , . . . , ln q n ) ′ using Equation (11), we see that q d 1 1 q d 2 2 · · · q dn n is preserved if and only if (d 1 , . . . , d n )Y T a = 0.

The phase portrait for the projectivized Lie bracket flow
The structure vector a t for a metric nilpotent Lie algebra (n, Q) evolves under the Ricci flow according to the law in Equation (12). The projection of the a t flow on R m to the flow [a t ] on P m−1 (R), as represented by the coordinates s t in A m−1 , obeys Equation (18); and by Remark 3.7, has the same trajectories as the solutions to Equation (19). We describe the properties of a flow satisfying Equation (19). The fixed points in this satisfy s i = 0 for i ∈ M, and for η i ( s t ) = 0 for i ∈ M. After renaming s i , i ∈ M, as t 1 , . . . , t k for some k, these can be written as P U ′ t = 0 for the k × k Gram matrix that is a minor of U obtained by crossing out rows and columns with indices in M. By Proposition 3.5, solutions satisfy U t = −2β ′ [1] for some β < 0. By Theorem 2 of [Pay], the set of such t in A M is compact.  The next example illustrates the phase portrait for the projectivized Lie bracket flow when there is no positive solution v to U v = [1]. To our knowledge, the smallest Gram matrix U ′ arising from a metric nilpotent Lie algebra is 8 × 8 (see Example 5.2). For the sake of simplicity, in our example we have chosen a 3 × 3 matrix U that does not necessarily come from a nilpotent Lie algebra, but for which the a t flow defined by U and Equation (12) has the same qualitative features as for the a t flow defined by U ′ . Let us consider the evolution of [a t ] under time, as measured by (s 1 , s 2 ) coordinates in A 2 . It is easy to see that η 1 ((0, 0)) < 0 and η 2 ((0, 0)) < 0 as stated in Lemma 3.5. As in Example 3.8, properties of the slope field imply that any point in ( s > 0) must approach the fixed point b + , corresponding to the point (1 : 0 : 1) in P 2 (R).

Soliton trajectories for the Ricci flow 4.1 Some solutions of nilsoliton trajectories
Before stating the theorem, we note that all known soliton metrics on nilpotent Lie algebra admit stably Ricci-diagonal bases. It follows from Theorem 2.1 that any time an orthogonal basis B = {X i } has the property that the sets {J X i } and {ad X i } are orthogonal, the basis is stably Ricci-diagonal. In many other cases a stably Ricci-diagonal basis is guaranteed to exist: Proposition 4.1. Let (n, Q) be a soliton metric nilpotent Lie algebra with associated semisimple derivation D = Ric Q −β Id . Suppose that the eigenspaces for D are all one-dimensional, and let B be a set of orthogonal eigenvectors. The set B is a stably Ricci-diagonal basis.
Proof. From [Heb98] it is known that the eigenvalues of D are positive and rational for all i = 1, . . . , n.
where D(x i ) = λ i x i , with λ 1 < · · · < λ n . Because D is a derivation, for all i < j, the bracket [x i , x j ] is in the one-dimensional eigenspace for λ i + λ j , so is a multiple of x k for some k > j. Hence the sets {J x i } and {ad x i } are orthogonal, and by Remark 2.2, the Ricci form is diagonal.
Theorem 4.2. Let (n, Q) be a metric nilpotent Lie algebra with stably Ricci-diagonal basis B = {x i } n i=1 with |Λ B | = m > 0. Suppose that Q is a soliton inner product with soliton constant β. Let Ric B = (r 1 , . . . , r n ) denote the Ricci vector for (n, Q) relative to B.
Let Q t = n i=1 q i dx i ⊗ dx i be the solution to the Ricci flow with initial condition Q. Let a t = (a 1 (t), . . . , a m (t)) be the structure vector for (n, Q t ) as in Equation (6). Denote a 1 , . . . , a m also by a l jk , where (j, k, l) is in Λ B . 1. For i = 1, . . . , m, the function a i is the solution The ray Ra 0 is invariant under the flow, with a t = a 1 (t) a 1 (0) a 0 .
2. The Ricci form ric Qt for (n, Q t ) is diagonal relative to B with diagonal entries given by the Ricci vector for Q t : for j = 1, . . . , n.
4. Let E min denote the eigenspace for the minimal eigenvalue of ric Q . The inner product Q collapses under the Ricci flow to the equivalence class Q ∞ , where Q ∞ is a semidefinite symmetric bilinear form that is positive definite on E min and is degenerate on any subspace properly containing E min .

Now we prove the theorem.
Proof of Theorem 4.2. Since (n, Q) is soliton, by Theorem 2.3, U a 0 = −2β[1] m×1 for some β < 0. By Theorem 3.1, . . , m. At the point a = λa 0 , for any λ > 0 and any i = 1, . . . , m, Therefore a ′ (t) = 2βλa t at all points λa 0 . Thus, the ray R + a 0 is invariant under the flow. Since for i = 1, . . . , m, the function a i is given by a i (t) = a i (0) a 1 (0) a 1 (t) for all t ≥ 0, a t = a 1 (t) a 1 (0) a 0 for all t ≥ 0. At the point a t = λa 0 , the value of λ is a 1 (t) a 1 (0) , so Equation (20) becomes Solutions are a i (t) = a i (0) (−2βt + 1) −1 , for i = 1, . . . , m. This proves Part 1. Part 2 is an immediate consequence of Equation (5) in Theorem 2.1. Now we consider Part 3 of the theorem. In order to compute q j (t), for j = 1, . . . , m, by Theorem 3.1, we need to solve the system of differential equations (ln q j ) ′ = −2(Ric B ) j , where j = 1, . . . , n. Substituting the expressions for for a i (t) and Ric B from Parts 1 and 2, we have Integrating, we find that ln q j is equal to After exponentiating both sides and using that by Theorem 2.1, the desired expression for q j (t) is obtained. Because inner product Q is soliton, the Lie algebra n is orthogonally N-graded by the eigenspaces of the derivation associated to Q. Without loss of generality, let r 1 be the minimal eigenvalue of the Ricci form for Q, and let E min be the corresponding eigenspace. Then lim t→∞ t −r 1 /β Q t is positive definite on E min but is degenerate on any subspace properly containing E min . This completes the proof of the theorem.

Heisenberg metric Lie algebras
The Heisenberg metric Lie algebras are among the most symmetric nilpotent Lie algebras.
Example 5.1. Let Q be an inner product on the (2r + 1)-dimensional Heisenberg algebra h 2r+1 . From Equation (4), it is easy to see that the center z is the single positive eigenspace for the Ricci endomorphism, and the Ricci form is negative definite on z ⊥ . A vector z spanning the center is an eigenvector for the Ricci endomorphism. By orthogonally block diagonalizing the nondegenerate skew-symmetric endomorphism J z : h 2r+1 → h 2r+1 into one block of form [0] and r blocks of form ( 0 c −c 0 ), it is possible to find an orthogonal Ricci eigenvector basis B = {x i } 2r+1 i=1 of h 2r+1 such that Λ B = {(2i − 1, 2i, 2r + 1) | i = 1, . . . , r}. The Gram matrix U = (u ij ) for h 2r+1 relative to B is the positive definite r × r matrix defined by The solution to U v = [1] r×1 is a scalar multiple of b = [1] r×1 . Therefore, any inner product Q ⋆ with structure vector that is a scalar multiple of [1] is soliton. The (r − 1) × r matrix P U = (b ij ) is of form and by Remark 3.7, after a change of variables to s, the trajectories for the projectivized Ricci flow for h 2r+1 are encoded in the system of differential equations (ln s i ) ′ = 2(1 − s i ), i = 1, . . . , r − 1, which has solution s i (t) = e 2t C i + e 2t , i = 1, . . . , r − 1.
Clearly, s t converges to [1] as t → ∞ for all initial conditions. At the limit point, all values of the structure constants are equal, so that the limiting volume-normalized metric Lie algebra is the Heisenberg Lie algebra endowed with a volume-normalized soliton inner product.

When the Lie algebra does not support a soliton inner product
Next is an example of a seven-dimensional nilpotent metric Lie algebra (n, Q) such that the limit (n ∞ , Q) of (n, Q) under the projectivized Ricci flow ψ t : N 7 → N 7 has a limiting Lie algebra n ∞ that is not isomorphic to the initial Lie algebra n. where α i+1 1,i = 0 for i = 2, . . . , 6 and α i+2 2,i = 0 for i = 2, . . . , 5. No Lie algebra of this form admits a soliton inner product (Theorem 34, [Pay]).
Take an initial inner product Q that is diagonal with respect to B. By Theorem 2.1, B is a Ricci-diagonal basis, and by Remark 2.2 B remains Ricci-diagonal under rescalings of Q, so it is stably Ricci-diagonal. Let a t denote the structure vector for the solution Q t to the Ricci flow at time t. Recall that the entries of a i are the squares of the nonzero structure constants α l jk . Since no Lie algebra with a t > 0 can be soliton, the limit a ⋆ = lim t→∞ [a t ] = (a 1 : a 2 : · · · : a 8 ) in projective space must have some entry a i equal to zero.
The only elements x in n such that the endomorphism ad x has rank three or more lie in span{x 1 , x 2 }. For this kind of element x = c 1 x 1 + c 2 x 2 , where c 1 , c 2 ∈ R, the adjoint map is represented with respect to B by the matrix In particular, there exist elements x 1 and x 2 such that the ranks of ad x 1 and ad x 2 are five and three respectively. We claim that if any structure constant becomes zero as [a t ] approaches its limit, the limiting Lie algebra n ∞ no longer has this property and therefore is not isomorphic to the original Lie algebra n. The only way that n ∞ can have an element such that the rank of ad c 1 x 1 +c 2 x 2 is five is if α i+1 1,i = 0 for i = 2, . . . , 6. In order to have an additional element c 1 x 1 + c 2 x 2 with the rank of ad c 1 x 1 +c 2 x 2 equal to three, it is necessary α i+2 2,i = 0 for i = 3, 4, 5. But then a ⋆ > 0, a contradiction. Therefore, n ∞ can not be isomorphic to n. [Pera] Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:mathDG/0211159. [Perb] , Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:mathDG/0307245. [Perc] , Ricci flow with surgery on three-manifolds, arXiv:mathDG/0303109.