Ricci-flow as an “intrinsic-Ricci-flat” Space-time.

A Ricci flow solution {(M^m, g(t)), t ∈ I ⊂ R} is a smooth family of metrics satisfying the evolution equation

∂/∂t g = −2Rc —– (1)

where M^m is a complete manifold of dimension m. We assume that sup_M |Rm|_g(t) < ∞ for each time t ∈ I. This condition holds automatically if M is a closed manifold. It is very often to put an extra term on the right hand side of (1) to obtain the following rescaled Ricci flow

∂/∂t g = −2 {Rc + λ(t)g} —– (2)

where λ(t) is a function depending only on time. Typically, λ(t) is chosen as the average of the scalar curvature, i.e. , 1/m ∱Rdv or some fixed constant independent of time. In the case that M is closed and λ(t) = 1/m ∱Rdv, the flow is called the normalized Ricci flow. Starting from a positive Ricci curvature metric on a 3-manifold, Richard Hamilton showed that the normalized Ricci flow exists forever and converges to a space form metric. Hamilton developed the maximum principle for tensors to study the Ricci flow initiated from some metric with positive curvature conditions. For metrics without positive curvature condition, the study of Ricci flow was profoundly affected by the celebrated work of Grisha Perelman. He introduced new tools, i.e., the entropy functionals μ, ν, the reduced distance and the reduced volume, to investigate the behavior of the Ricci flow. Perelman’s new input enabled him to revive Hamilton’s program of Ricci flow with surgery, leading to solutions of the Poincaré conjecture and Thurston’s geometrization conjecture.

In the general theory of the Ricci flow developed by Perelman in, the entropy functionals μ and ν are of essential importance. Perelman discovered the monotonicity of such functionals and applied them to prove the no-local-collapsing theorem, which removes the stumbling block for Hamilton’s program of Ricci flow with surgery. By delicately using such monotonicity, he further proved the pseudo-locality theorem, which claims that the Ricci flow can not quickly turn an almost Euclidean region into a very curved one, no matter what happens far away. Besides the functionals, Perelman also introduced the reduced distance and reduced volume. In terms of them, the Ricci flow space-time admits a remarkable comparison geometry picture, which is the foundation of his “local”-version of the no-local-collapsing theorem. Each of the tools has its own advantages and shortcomings. The functionals μ and ν have the advantage that their definitions only require the information for each time slice (M, g(t)) of the flow. However, they are global invariants of the underlying manifold (M, g(t)). It is not convenient to apply them to study the local behavior around a given point x. Correspondingly, the reduced volume and the reduced distance reflect the natural comparison geometry picture of the space-time. Around a base point (x, t), the reduced volume and the reduced distance are closely related to the “local” geometry of (x, t). Unfortunately, it is the space-time “local”, rather than the Riemannian geometry “local” that is concerned by the reduced volume and reduced geodesic. In order to apply them, some extra conditions of the space-time neighborhood of (x, t) are usually required. However, such strong requirement of space-time is hard to fulfill. Therefore, it is desirable to have some new tools to balance the advantages of the reduced volume, the reduced distance and the entropy functionals.

Let (M^m, g) be a complete Ricci-flat manifold, x₀ is a point on M such that d(x₀, x) < A. Suppose the ball B(x₀, r₀) is A⁻¹−non-collapsed, i.e., r^−m₀|B(x₀, r₀)| ≥ A⁻¹, can we obtain uniform non-collapsing for the ball B(x, r), whenever 0 < r < r₀ and d(x, x₀) < Ar₀? This question can be answered easily by applying triangle inequalities and Bishop-Gromov volume comparison theorems. In particular, there exists a κ = κ(m, A) ≥ 3^−mA^−m−1 such that B(x, r) is κ-non-collapsed, i.e., r^−m|B(x, r)| ≥ κ. Consequently, there is an estimate of propagation speed of non-collapsing constant on the manifold M. This is illustrated by Figure

We now regard (M, g) as a trivial space-time {(M, g(t)), −∞ < t < ∞} such that g(t) ≡ g. Clearly, g(t) is a static Ricci flow solution by the Ricci-flatness of g. Then the above estimate can be explained as the propagation of volume non-collapsing constant on the space-time.

However, in a more intrinsic way, it can also be interpreted as the propagation of non-collapsing constant of Perelman’s reduced volume. On the Ricci flat space-time, Perelman’s reduced volume has a special formula

V((x, t)r²) = (4π)^-m/2 r^-m ∫_M e^{-d2(y, x)/4r2} dv_y —– (3)

which is almost the volume ratio of B_g(t)(x, r). On a general Ricci flow solution, the reduced volume is also well-defined and has monotonicity with respect to the parameter r², if one replace d²(y, x)/4r² in the above formula by the reduced distance l((x, t), (y, t − r²)). Therefore, via the comparison geometry of Bishop-Gromov type, one can regard a Ricci-flow as an “intrinsic-Ricci-flat” space-time. However, the disadvantage of the reduced volume explanation is also clear: it requires the curvature estimate in a whole space-time neighborhood around the point (x, t), rather than the scalar curvature estimate of a single time slice t.

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