Gaussian densities and stability for some Ricci solitons

Huai-Dong Cao^{∗} Lehigh University Richard Hamilton Columbia University

Tom Ilmanen^{†} ETH-Zürich and Columbia University

April 7, 2004

Perelman [Pe02] has discovered a remarkable variational structure for the Ricci flow: it can be viewed as the gradient flow of the entropy functional λ. There are also two monotonicity formulas of shrinking or localizing type: the shrinking entropy ν, and the reduced volume. Either of these can be seen as the analogue of Huisken’s monotonicity formula for mean curvature flow [Hu90]. In various settings, they can be used to show that centered rescalings converge subsequentially to shrinking solitons, which function as idealized models for singularity formation.

In this note, we exhibit the second variation of the λ and ν functionals, and investigate the linear stability of examples. We also define the “central density” of a shrinking Ricci soliton (shrinker) and compute its value for certain examples in dimension 4. Using these tools, one can sometimes predict or limit the formation of singularities in the Ricci flow. In particular, we show that certain Einstein manifolds are unstable for the Ricci flow in the sense that generic perturbations acquire higher entropy and thus can never return near the original metric. A detailed version of the calculations summarized in this announcement will follow in [CHI].

In §1, we investigate the stability of Perelman’s λ-functional. Its critical points are steady solitons (Ricci flat in the compact case). We compute the second variation D^{2}λ; the corresponding Jacobi field operator L is a degenerate negative elliptic integro-differential operator. In fact, L equals half the Lichnerowicz Laplacian Δ_{L} on divergence-free symmetric tensors, and zero on Lie derivatives. This fact and further investigations of the second variation have been reported by Perelman [Pe03]. We call a steady soliton linearly stable if L ≤ 0, otherwise linearly unstable. If g is linearly unstable, then g can be perturbed so that λ(g) > 0, which will destabilize it utterly: it will decay into a cacophony of shrinkers and disappear in finite time. One observes that λ(g) ≤ 0 for any metric on the torus T^{n}, n ≤ 7; in fact this is equivalent to the positive mass theorem. By Guenther, Isenberg and Knopf [GIK02] every K3 surface is linearly stable; more generally, by Dai, Wang and Wei [DWW04] any manifold with a parallel spinor is linearly stable. Other cases are open.

∗With partial support from NSF Grant DMS-0206847.

†With partial support from Schweizerische Nationalfonds Grant 21-66743.01.

In §2, we investigate the stability of the ν-functional, whose critical points are shrinkers. The Jacobi field operator N of ν is like L but with lower order terms. We call a shrinker linearly stable if N ≤ 0. Again, N is closely related to the Lichnerowicz Laplacian. We observe that ℂℙ^{N} (with the standard metric) is linearly stable, but all other compact complex surfaces with c_{1} > 0 are linearly unstable. Using results of Gasqui and Goldschmidt [GG96, GG91] the complex hyperquadric Q^{3} (a Hermitian symmetric space) is linearly unstable. This implies that Q^{3} is irremediably unstable in the sense that a generic (non-Kähler!) perturbation of Q^{3} will never approach the original geometry of Q^{3} at any scale or time. On the other hand, the hyperquadric Q^{4} is linearly stable. Other cases are open.

A notion of central density (or gaussian density) of a shrinker can be defined from either of Perelman’s monotonicity formulas; we call these notions Θ and ν. On a shrinker, the two definitions are equivalent via Θ = e^{ν}. For a general solution, e^{ν} is a lower bound for the central density of any shrinker that arises later as a singularity model, which restricts the shrinkers that may occur later. This is presented in §3.

The central density of certain standard 4-dimensional examples are exhibited in a table in §4.

We are grateful to Hugh Bray, Robert Bryant, Hubert Goldschmidt, Dan Knopf, and John Morgan for illuminating conversations.

1 Second Variation of the Entropy λ

The second variation of the Einstein functional is positive in the conformal direction but negative in all other directions. The glory of Perelman’s entropy is that there is a preliminary minimization over scalar functions that absorbs nearly all the positive directions: the Jacobi field operator is a linear integrodifferential operator with nonpositive symbol.^{1}

Fix a compact manifold (M, g). Define [Pe02]

F(g, f) := ∫ e^{−f}(|Df|^{2} + R).

Define the entropy

λ(g) := inf{F(g, f) : f ∈ C_{c}^{∞}(M), ∫ e^{−f} = 1}

1It is convenient that this happens without changing the metric via a conformal change.

So the scalar variations do not affect the background geometry. Instead they satisfy a linear PDE.

The infimum is achieved by a function f solving

−2Δf + |Df|^{2} − R = λ(g).

Now consider variations g(s) = g + sh. Following Perelman, the first variation D_{g}λ(h) of λ is given by

\frac{d}{ds}|_{s=0} λ(g(s)) = ∫ e^{−f}(−Rc − D^{2}f) : h,

where f is the minimizer. A stationary point satisfies

Rc + D^{2}f = 0,

which implies that g is a (gradient) steady soliton, that is, the Ricci flow with initial condition g satisfies

g(t) = ϕ_{t}^{∗}(g)

where ϕ_{t} is a family of diffeomorphism generated by the gradient vector field Df. In fact, any compact steady is Ricci flat with f = 0, λ = 0.

Note by diffeomorphism invariance of λ that D_{g}λ vanishes on any Lie derivative h = L_{x}g. From this, by inserting h = −2(Rc+D^{2}f) one recovers Perelman’s wonderful result that λ(g(t)) is nondecreasing on any Ricci a Ricci flow, and is constant if and only if g(t) is a steady soliton.

We prove the following. Write Rm(h, h) := R_{ijkl}h_{ik}h_{jl}, div ω := D_{i}ω_{i}, (div h)_{i} := D_{j}h_{ji}, (div^{∗} ω)_{ij} = −(D_{i}ω_{j} + D_{j}ω_{i})/2 = −(1/2)L_{ω^{#}}g_{ij}.

1.1 Theorem The second variation D_{g}^{2}λ(h, h) of λ on a compact Ricci flat manifold is given by

\frac{d^{2}}{ds^{2}} |_{s=0} λ(g(s)) = ∫ −\frac{1}{2}|Dh|^{2} + | div h|^{2} − \frac{1}{2}|Dv_{h}|^{2} + Rm(h, h)
= ∫ Lh : h,

where

Lh := \frac{1}{2}Δh + div^{∗} div h + \frac{1}{2}D^{2}v_{h} + Rm(h, ·),

and v_{h} satisfies

Δv_{h} = div div h.

The symbol of L in the direction ξ ∈ T_{x}^{∗}M is

σ_{ξ}(h) = −π_{ξ^{⊥}}(h),

where π_{ξ^{⊥}}(h) restricts h to the hyperplane ξ^{⊥}. So the operator L is degenerate negative elliptic, and has a discrete spectrum with at most a finite-dimensional space of positive eigenfunctions.

Decompose C^{∞}(Sym^{2}(T^{∗}M)) as

ker div ⊕ im div^{∗} .

One verifies that L vanishes on im div^{∗}, that is, on Lie derivatives. On ker div one has

L = \frac{1}{2}Δ_{L}

where

Δ_{L}h := Δh + 2Rm(h, ·) − Rc · h − h · Rc

is the Lichnerowicz Laplacian on symmetric 2-tensors.

We call a critical point g of λ linearly stable if L ≤ 0, and a maximizer if λ(g_{1}) ≤ λ(g) for all g_{1}. A compact Ricci flat metric is a maximizer if and only if it admits no metric of positive scalar curvature. (This follows from Schoen’s solution of the Yamabe problem [S84].) Evidently a maximizer is stable. If g is not stable, then a slight perturbation will develop λ > 0 and R > 0 and (in principle) disappear in finite time as positive manifolds do. A good question is whether any Ricci-flat manifold is unstable. We call this the positive mass problem for Ricci flat manifolds.

1.2 Example T^{n} admits no metric of positive scalar curvature by the positive mass theorem, so λ(g) ≤ 0 for all g on T^{n}.

1.3 Example A Calabi-Yau K3 surface and more generally, any manifold with a parallel spinor has Δ_{L} ≤ 0 [GIK02, DWW04]. So these manifolds are linearly stable in the sense presented here.

1.4 Example Let g be compact and Ricci flat. Following [B84, GIK02] we examine conformal variations. It is convenient to replace ug by

h = Su := (Δu)g − D^{2}u

which differs from the conformal direction only by a Lie derivative and is divergence free. We have

Δ_{L}Su = (SΔu)g,

so Δ_{L} has the same eigenvalues as Δ. In particular, N ≤ 0 in the conformal direction. This contrasts with the Einstein functional.

2 Second Variation of the Shrinker Entropy ν

Fix a complete manifold (M, g). Define

W(g, f, τ) := \frac{1}{(4πτ)^{n/2}} ∫ e^{−f} [τ(|Df|^{2} + R) + f − n] dV.

Define the shrinker entropy by

ν(g) := inf\{W(g, f, τ) : f ∈ C_{c}^{∞}(M), τ > 0, \frac{1}{(4πτ)^{n/2}} ∫ e^{−f} = 1\}

Assume that M is compact or is asymptotic at infinity to a metric cone over a smooth, compact Riemannian manifold. One checks that ν(g) is realized by a pair (f, τ) that solve the equations

τ(−2Δf + |Df|^{2} − R) − f + n + ν = 0, \frac{1}{(4πτ)^{n/2}} ∫ fe^{−f} = \frac{n}{2} + ν,

and f grows quadratically.

Consider variations g(s) = g + sh where h is smooth of compact support. Following Perelman, one calculates the first variation D_{g}ν(h) to be

\frac{d}{ds}|_{s=0} ν(g(s)) = \frac{1}{(4πτ)^{n/2}} ∫ e^{−f}(τ(−Rc − D^{2}f) + g/2) : h.

A stationary point of ν satisfies

D^{2}f + Rc − \frac{g}{2τ} = 0  (1)

which says that g is a (gradient) shrinker, that is, its Ricci flow g(t) has the form

g(t) := (T − t)ψ_{t}^{∗}(g)), t < T,

where ψ_{t} are the diffeomorphisms generated by −Df, and τ = T − t.

As before, D_{g}ν vanishes on Lie derivatives. By scale invariance it vanishes on multiplies of the metric. Inserting h = −2(Rc + D^{2}f − g/2τ), one recovers Perelman’s brilliant formula that finds that ν(g(t)) is monotone on a Ricci flow, and constant if and only if g(t) is a gradient shrinker.

A positive Einstein manifold is a shrinker with f ≡ n/2, normalized by Rc = g/2τ. We compute:

2.1 Theorem Let (M, g) be a positive Einstein manifold. The second variation D_{g}^{2}ν(h, h) is given by

\frac{d^{2}}{ds^{2}} |_{s=0} ν(g(s)) = \frac{τ}{vol(g)} ∫ −\frac{1}{2}|Dh|^{2} + | div h|^{2} − \frac{1}{2}|Dv_{h}|^{2} + Rm(h, h) + \frac{v_{h}^{2}}{4τ}

−\frac{1}{2n} (\frac{1}{vol(g)} ∫ tr_{g} h)^{2} ,

= \frac{τ}{vol(g)} ∫ Nh : h,

where

Nh := \frac{1}{2}Δh + div^{∗} div h + \frac{1}{2}D^{2}v_{h} + Rm(h, ·) − \frac{g}{2nτ vol(g)} ∫ tr_{g} h.

and v_{h} is the unique solution of

Δv_{h} + \frac{v_{h}}{2τ} = div div h, ∫ v_{h} = 0.

There is a strictly more complicated formula in the case of non-Einstein shrinkers.

As in the previous case, N is degenerate negative elliptic and vanishes on im div^{∗}. Write

ker div = (ker div)_{0} ⊕ ℝg

where (ker div)_{0} is defined by ∫ tr_{g} h = 0. Then on (ker div)_{0} we have

N = \frac{1}{2} (Δ_{L} − \frac{1}{τ} )

where Δ_{L} is the Lichnerowicz Laplacian. So the linear stability of a shrinker comes down to the (divergence free) eigenvalues of the Lichnerowicz Laplacian. Let us write μ_{L} for the maximum eigenvalue of Δ_{L} on symmetric 2-tensors and μ_{N} for the maximum eigenvalue of N on (ker div)_{0},

2.2 Example The round sphere is geometrically stable (i.e. nearby metrics are attracted to it up to scale and gauge) by the results of Hamilton [Ha82, Ha86, Ha88] and Huisken [Hu88]. In particular it is linearly stable: μ_{N} = −2/(n − 1)τ < 0.

2.3 Example For ℂℙ^{N}, the maximum eigenvalue of Δ_{L} on (ker div)_{0} is μ_{L} = 1/τ by work of Goldschmidt [G04], so ℂℙ^{N} is neutrally linearly stable, i.e. the maximum eigenvalue of N on (ker div)_{0} is μ_{N} = 0.

Any product of two nonflat shrinkers N_{1}^{n_{1}} × N_{2}^{n_{2}} is linearly unstable, with μ_{N} = 1/2τ. The destabilizing direction h = g_{1}/n_{1} − g_{2}/n_{2} corresponds to a growing discrepancy in the size of the factors.

More generally, any compact Kähler shrinker with dim H^{1,1}(M) ≥ 2 is linearly unstable. Again, this can be seen directly: a small perturbation into a non-canonical Kähler class will move in a straight line nearly toward the vertex of the Kähler cone, hence away from the canonical class (in a scale invariant sense). If M is Kähler-Einstein, we compute μ_{N} as follows. Let σ be a harmonic 2-form and h be the corresponding metric perturbation; then Δ_{L}h = 0, and if σ is chosen perpendicular to the Kähler form, then as above we obtain μ_{N} = 1/2τ.

A complete list of compact complex surfaces with c_{1} > 0 is ℂℙ^{2}, ℂℙ^{1} × ℂℙ^{1}, and ℂℙ^{2}#k(−ℂℙ^{2}), k = 1, … 8. Each of these has a unique Kähler shrinker metric (Kähler-Einstein unless k = 1, 2). By the above, all are linearly unstable except ℂℙ^{2}.

Let Q^{N} denote the complex hyperquadric in ℂℙ^{N+1} defined by

∑_{i=0}^{N+1} z_{i}^{2} = 0,

a Hermitian symmetric space of compact type, hence a positive Kähler-Einstein manifold. Then Q^{2} is isometric to ℂℙ^{1} ×ℂℙ^{1}, the simplest example of the above instability phenomenon.

2.4 Example Consider Q^{3}. It has dim H^{1,1}(Q^{3}) = 1, so the above discussion does not apply. But the maximum eigenvalue of Δ_{L} on (ker div)_{0} is μ_{L} = −2/3τ by work of Gasqui and Goldschmidt [GG96] (or see [GG04]). The proximate cause is a representation that appears in the sections of the symmetric tensors but not in scalars or vectors. Therefore, Q^{3} is linearly unstable with

μ_{N} = \frac{1}{6τ} .

Since ν increases along the Ricci flow, this implies that a generic small perturbation of the Einstein metric ḡ will grow and g(t) will never return near ḡ at any scale or time. We say that ḡ is geometrically unstable. Now imagine that we start with a random metric on Q^{3} and propose to use the Ricci flow to find an Einstein metric or other canonical geometry for Q^{3}. Assuming there are no other critical points, we find that the flow combusts in singularities of more elementary type and the topology of the underlying manifold simplifies drastically, unless it happens to get hung up at the Einstein metric ν. So the Ricci flow has fundamentally more complicated behavior than in dimension three, as one expects. Further exploration of this example will appear in [BGIM].

2.5 Example Let Q^{4} be the 4-dimensional hyperquadric. The maximum eigenvalue of Δ_{L} on symmetric tensors is μ_{L} = −1/τ by work of Gasqui and Goldschmidt [GG91] (or see [GG04]). So Q^{4} is neutrally linearly stable: μ_{N} = 0.

Let g be a positive Einstein metric, and let us examine conformal variations. As before, without loss replace ug by the divergence-free variation

h = Su := (Δu)g − D^{2}u + \frac{ug}{2τ} .

As before, Δ_{L}Su = (SΔu)g. Thus Δ_{L} has the same eigenvalues as Δ_{fns}|(ker S)^{⊥}. But ker S is empty except on round S^{n}, which is linearly stable. Note that μ_{fns} ≤ −n/2(n − 1)τ with equality only on round S^{n}. So we have:

2.6 Proposition A positive Einstein metric is linearly unstable for conformal variations if and only if the maximum eigenvalue of Δ on functions satisfies

−\frac{1}{τ} < μ_{fns} < −\frac{n}{2(n − 1)τ} .

We do not know whether this inequality can ever be satisfied on a positive Einstein manifold.

3 The Central Density of a Shrinker

Our aim in this section is to define the central density of a gradient shrinker. First we define a suitable class of gradient shrinkers, then we review the two Perelman monotonicity formulas of shrinking type and apply them by taking the center point to be the parabolic vertex of the shrinker. Our principal result is that the two notions of density coincide.

A gradient shrinker solves

∂g/∂t = −2Rc,  g(t) := −tψ_{t}^{∗}(g(−1)),  t < 0,

where ψ_{t} are the diffeomorphisms generated by the gradient of a function F (x, t). Differentiating the above expression yields

D^{2}F + Rc − \frac{g}{2τ} = 0,  (2)

where τ = −t. Normalizing F by adding a time-dependent constant, we obtain

\frac{∂F}{∂τ} + |DF |^{2} = 0.  (3)

Differentiating (2), taking the trace two ways, and applying Bianchi II and commutation rules yields D(|DF |^{2} + R − F/τ) = 0. Adding a further global constant to F leads to the classical auxiliary equation

|DF |^{2} + R − \frac{F}{τ} = 0.  (4)

Equations (2)-(4) are the fundamental equations for a gradient shrinker. Combining (2) and (3) yields the backward heat equation

\frac{∂F}{∂τ} = ΔF − |DF |^{2} + R − \frac{n}{2τ} .  (5)

In order to prove our results we need some analytic hypotheses on the metric of M. We assume that M is complete, connected, and κ-noncollapsed at all scales. We also assume that the curvature decays quadratically as x → ∞. (This is satisfied, for example, by the blowdown shrinker L(N, −1) [FIK04].) However, many of our results hold under the weaker hypothesis of bounded curvature. Under the quadratic decay hypothesis, g(t) converges in the Gromov-Hausdorff sense as t ↗ 0 to a metric cone C which is smooth except at the vertex, which we call 0. The convergence is smooth except on a compact set, which falls into the vertex, which we call 0. For a proof and further details, see [I].

We now wish to define a gaussian density centered at the parabolic vertex (y, s) = (0, 0) of the spacetime M := (M × (−∞, 0)) ∪ (C × {0}). We may do this in two ways: via the shrinking entropy or the reduced volume, both due to Perelman [Pe02].

The reduced volume generalizes Bishop volume monotonicity to the spacetime setting. For a smooth point (y, s) in any Ricci flow, define the reduced distance ℓ = ℓ_{y,s} by

ℓ(x, t) := \frac{1}{2\sqrt{τ}} inf_{γ} ∫_{0}^{τ} \sqrt{σ} (|\frac{dγ}{du}|^{2} + R) dσ,  t < s, x ∈ M,  (6)

where the infimum is taken over all paths (γ(u), u), t ≤ u ≤ s that connect (x, t) to (y, s). The reduced volume centered at (y, s) is defined by

θ_{y,s}(t) := \frac{1}{(4πτ)^{n/2}} ∫_{M} e^{−ℓ(x,t)} dV_{t}(x).

Perelman wonderfully shows that θ_{y,s}(t) is increasing in t and is constant precisely on a gradient shrinker. Now define ℓ_{0,0} by passing smooth points (y_{i}, s_{i}) to (0, 0). We have:

3.1 Proposition ℓ_{0,0} is well-defined and is locally Lipschitz on M. For t < 0, θ_{0,0}(t) is well defined, constant, and contained in (0,1].

This constant value we call the central density of (M,g(t)) and denote

Θ(M) = Θ(M, g(·)) := θ_{0,0}(t), t < 0.

Next we turn to the shrinking entropy ν. Let (M, g(t)) be a smooth Ricci flow existing up to t = s and set τ := s − t. Let f solve the heat equation

\frac{∂f}{∂t} = Δf − |Df|^{2} + R − \frac{n}{2τ} ,

that came up for the soliton potential of a shrinker (5). Define u by

u := \frac{e^{−f}}{(4πτ)^{n/2}} .

Remarkably, u solves the adjoint heat equation

\frac{∂u}{∂t} = Δu − Ru.  (7)

This leads to the conservation law

\frac{1}{(4πτ)^{n/2}} ∫ e^{−f} = ∫ u = 1 for t < s.

Perelman has shown [Pe02]

\frac{∂}{∂t}W(s − t, f(t), g(t)) = \frac{1}{(4πτ)^{n/2}} ∫ 2e^{−f} |D^{2}f + Rc − \frac{g}{2τ} |^{2} dV_{t} ≥ 0.

and the right hand side vanishes precisely when g(t) is a gradient shrinker and f is its soliton potential. (This shows, as mentioned above, that ν increases in general and is constant on a shrinker.)

If u emerges from a dirac source at a smooth point (y, s), we write u = u_{y,s}, f = f_{y,s} and define the shrinker entropy centered at (y, s) by

ϕ_{y,s}(t) := W(s − t, f_{y,s}(t), g(t))

Passing smooth points (y_{i}, s_{i}) to (0, 0), we prove:

3.2 Proposition u_{0,0} is well-defined, smooth, and positive on M × (−∞, 0) and solves equation (7). It satisfies

∫_{M} u_{0,0} ≡ \frac{1}{(4πτ)^{n/2}} ∫ e^{−f_{0,0}} = 1,  t < 0.  (8)

Also, ϕ_{0,0}(t) is well-defined and lies in (−∞, 0] for all t < 0. In fact, it is constant, with

ϕ_{0,0}(t) = ν(M), t < 0.

Since ϕ_{0,0}(t) is constant, f_{0,0} is a soliton potential and so

f_{0,0} = F + C  (9)

for some constant C depending only on M.

We now wish to relate Θ(M) and ν(M) via F. In the process we determine the value of C, and sharpen on a shrinker the general Perelman relation ℓ_{0,0} ≤ f_{0,0} [Pe02].

We begin with Θ(M). Using the symmetry of the shrinker and a simple comparison argument, one checks:

3.3 Proposition The integral curves of F are minimizing L-geodesics emanating from (0, 0).

Then using the homothetic time-symmetry of the shrinker, one obtains after a straightforward computation:

ℓ_{0,0} = τ(|DF |^{2} + R) = F = f_{0,0} − C,

and thus by the definition and (8), one gets:

Θ(M) = e^{C}.

Next, we evaluate ν. Compute

ν(M) = ϕ_{0,0}(t)

= \frac{1}{(4πτ)^{n/2}} ∫ e^{−f_{0,0}} [τ(|Df_{0,0}|^{2} + R) + f_{0,0} − n] dV_{t}

= \frac{1}{(4πτ)^{n/2}} ∫ e^{−f_{0,0}} [τ(Δf_{0,0} + R) + f_{0,0} − n] dV_{t}

= \frac{1}{(4πτ)^{n/2}} ∫ e^{−f_{0,0}} [f_{0,0} − n/2] dV_{t}

by integrating by parts and the trace of (2). On the other hand, by (4) and (9), the first integral expression also equals

ν(M) = \frac{1}{(4πτ)^{n/2}} ∫ e^{−f_{0,0}} [f_{0,0} − C + f_{0,0} − n] dV_{t}

We conclude that ν(M) = 2ν(M) − C, so ν(M) = C. We summarize these results in a theorem.

3.4 Theorem On a shrinker satisfying the above assumptions, we have

ℓ_{0,0} = F = f_{0,0} − ν(M)

and

Θ(M) = e^{ν(M)}.

Details will appear in [CHI].

4 Table of Values

In this section we calculate Θ for some standard shrinkers. The computations are simplified by several observations. Normalize positive Einstein manifolds by Rc = g/2τ, τ = 1/2(n − 1), so that S^{n} has radius 1.

(1) For any shrinker M, Θ(M) ≤ 1 with equality if and only if M = ℝ^{n}.

(2) Let M be a Ricci flat cone with g = dr^{2} + r^{2}g_{Σ} where Σ is positive Einstein. Then M is a shrinker (with interior singularity), and

Θ(M) = \frac{vol(Σ)}{vol(S^{n})}.

(3) If M is a positive Einstein manifold, then (for any τ)

Θ(M) = (\frac{1}{4πτe})^{n/2} vol_{τ} (M) ≤ Θ(S^{n}),

with equality if and only if M = S^{n}.

(4) Θ(S^{n}) = (\frac{n − 1}{2πe})^{n/2} vol(S^{n}). By way of comparison, note that for

mean curvature flow, Θ_{MCF} (S^{n}) = (\frac{n}{2πe})^{n/2} vol(S^{n}).^{2}

(5) Θ(ℂℙ^{N}) = (\frac{N + 1}{πe})^{N} \frac{vol(S^{2N+1})}{2π}.

(6) The positive Kähler-Einstein manifold M = ℂℙ^{2}#k(−ℂℙ^{2}), k = 0, 3, …, 8, has Θ(M) = (9 − k)/2e^{2}.

2Following an observation of White, we note (tantalizingly) that the respective limits as n → ∞ are \sqrt{2/e} and \sqrt{2}.

(7) Θ(M × N) = Θ(M)Θ(N).

We say that one shrinker decays to another if there is a small perturbation of the first whose Ricci flow develops a singularity modelled on the second. Because the ν-invariant is monotone during the flow, decay can only occur from a shrinker of lower density to one of higher density. This creates a “decay lowerarchy”. (It should be a partial order.)

We have computed the following density values in dimension 4. Note that the conclusion of Theorem 3.4 holds for all our examples, though not all are smooth enough to satisfy the hypotheses.

Shrinker  Type  Θ  Θ
ℝ^{4}  flat  1  1.000
S^{4}  positive Einstein  6/e^{2}  .812
S^{3} × ℝ  product  2 (π/e^{3})^{1/2}  .791
S^{2} × ℝ^{2}  product  2/e  .736
L(2, −1)  blowdown shrinker [FIK04] e^{\sqrt{2}−2}(1 + \sqrt{2})/2 .672
ℂℙ^{2}  positive Einstein  9/2e^{2}  .609
S^{2} × S^{2}  product  4/e^{2}  .541
ℂℙ^{2}#(−ℂℙ^{2}) Koiso metric [K90, C94]  3.826/e^{2}  .518
ℂℙ^{2}#(−ℂℙ^{2}) Page metric [Pa78]  3.821/e^{2}  .517
C(ℝℙ^{3})  Ricci flat cone  1/2  .500
C(ℝℙ^{2}) × ℝ  product  1/2  .500
ℝℙ^{4}  positive Einstein  3/e^{2}  .406
ℂℙ^{2}#3(−ℂℙ^{2}) positive Einstein  3/e^{2}  .406
ℝℙ^{3} × ℝ  product  (π/e^{3})^{1/2}  .396
ℝℙ^{2} × ℝ^{2}  product  1/e  .368
ℂℙ^{2}#4(−ℂℙ^{2}) positive Einstein  5/2e^{2}  .338
C(S^{3}/ℤ_{3})  Ricci flat cone  1/3  .333
ℂℙ^{2}#5(−ℂℙ^{2}) positive Einstein  2/e^{2}  .271
ℂℙ^{2}#6(−ℂℙ^{2}) positive Einstein  3/2e^{2}  .203
ℂℙ^{2}#7(−ℂℙ^{2}) positive Einstein  1/e^{2}  .135
ℂℙ^{2}#8(−ℂℙ^{2}) positive Einstein  1/2e^{2}  .068

All manifolds in the table are created from Einstein manifolds except for L(2, −1) and the Koiso metric. The computations for these metrics will be detailed in [CHI]. The volume of the Page metric is computed in [Pa78].

The blowdown shrinker L(n, −1) is a Kähler shrinker defined on the total space of the tautological holomorphic line bundle o_{N−1}(−1) over ℂℙ^{N−1}, that is, on ℂ^{N} blown up at z = 0. The metric of L(N, −1) is U(N) invariant, complete, and conelike at infinity, satisfying quadratic decay for the curvature. As t ↗ 0, the exceptional divisor ℂℙ^{N−1} shrinks to a point and elsewhere the metric converges smoothly to a cone metric on ℂ^{N} \{0} whose metric completion has a vertex at 0. For positive time, the flow can continue by a smooth, U(N)- invariant Kähler expander on ℂ^{N}. See [FIK].

The Koiso metric [K90, C94] and the Page metric [Pa78] are both U(2)- invariant metrics on ℂℙ^{2}#(−ℂℙ^{2}). The former, but not the latter, is Kähler. The remarks following Example 2.3 show that the Koiso metric has one direction of instability (in a Kähler direction). On the other hand, the Page metric may well decay to the Koiso metric. By the discussion in [FIK04], this leads us to conjecture that either metric decays to ℂℙ^{2} via a ℂℙ^{1} pinches off.

References

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