The entropy formula for the Ricci flow and its geometric applications.

*(English)*Zbl 1130.53001This is the first part of a masterpiece of mathematics which leads to a proof of the Poincaré and geometrization conjectures. Although the most striking application is in dimension 3 a good deal of the results presented here are valid in an arbitrary dimension \(n\). This is the case for sections 1 to 10 of the present paper. It concerns the Ricci flow introduced by Richard Hamilton in the celebrated paper [R. Hamilton, J. Differ. Geom. 17, 255–306 (1982; Zbl 0504.53034)]. Brief descriptions of each section follow. The detailed proofs can be read in [B. Kleiner and J. Lott, “Notes on Perelman’s papers”, arXiv: math.DG/0605667 (2006)], [J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3 (2007; Zbl 1179.57045)] and [H.-D. Cao and X.-P. Zhu, Asian J. Math. 10, No. 2, 165–492 (2006; Zbl 1200.53057)].

Section 1: On a closed manifold \(M\) the Ricci flow is presented as a gradient flow of a functional defined on the couples \((g, f)\), where \(g\) is a Riemannian metric and \(f\) is a function on \(M\). the function is chosen so that \(e^{-f} \,d\operatorname{vol}_g\) is a fixed measure. The functional is \[ \mathcal F(g,f)=\int_M(R+|\nabla f|^2)e^{-f}\,d\operatorname{vol}_g, \] in which \(R\) denotes the scalar curvature of the metric \(g\) and the norm is taken for this metric.

Section 2: This section and the following one present immediate corollaries which we can summarize as follows: On the space obtained by moding out the space of metrics by the action of the diffeomorphisms group and the homotheties there are no closed trajectories of the Ricci flow. In Perelman’s terminology one shows that there are no non trivial breathers. This section is devoted to the proof of this assertion in the case of steady and expanding breathers. This relies on introducing very interesting invariants which are nondecreasing along the flow and stationary on Ricci solitons. One is the infinum on \(f\) of \(\mathcal F(g,f)\) which turns out to be the smallest eigenvalue of \(-4\Delta+R\). The other one is the same eigenvalue normalised by the volume raised at the suitable power in order to get a scale invariant quantity.

Section 3: This section deals with the shrinking breathers which are more difficult to study. A modification of \(\mathcal F\) called \(\mathcal W\) is introduced. It is a function of three parameters: a metric \(g\), a function \(f\) and a number \(\tau\) which is \(t_0 - t\) where \(t\) is the parameter of the Ricci flow. Again \(\mathcal W\) is nondecreasing along the Ricci flow when \(f\) is normalized as in section 1 and is stationary on gradient shrinking solitons.

Section 4: This is the second breakthrough of this paper. It presents a tool that gives its full strength to the Ricci flow, namely a solution on a closed manifold \(M\) is non collapsed at a finite time \(t\). It means there exists a number \(\kappa>0\) such that balls at time \(t\) and of radius \(r\leq\rho\) on which the Riemann curvature is bounded by \(r^{-2}\) have volume bounded below by \(\kappa r^n\). The number , called the scale, is the value below which this property is true. All these quantities are scaled invariant which makes this property true even after rescaling the metrics. This implies a local injectivity radius bound and allows to apply compactness theorems. This is crucial for the singularities analysis.

Section 5 and 6: These two sections are less useful at the moment. They intend to give justifications and explanations for the previous construction either using a statistical mechanics’ approach or the point of view of infinite dimensional geometry.

Section 7: In this very important section the author develops the Morse theory of a functional called \(\mathcal L\)-length, which is a space-time version of the standard length (or energy). \(\mathcal L\)-geodesics, \(\mathcal L\)-Jacobi fields and \(\mathcal L\)-exponential are computed leading to some comparison theorems. The reduced volume is introduced, a monotonic quantity along the Ricci flow which allows to give an alternative proof of the nonlocal collapsing property. This is a key section for the three dimensional applications of the Ricci flow.

Section 8: A refined version of the nonlocal collapsing theorem is proved using the tools developed in the previous section, in particular the reduced volume.

Section 9: A differential Harnack inequality is proved.

Section 10: In this section it is proved that despite the fact that the Ricci flow is not local, it is pseudo-local. Namely, if the curvature is close to zero in a region (and with extra assumption), at least for a short time it remains not too far from zero, i.e., for a short time it is not affected by the possible presence of big curvature elsewhere on the manifold. This fact is made precise.

Section 11: This is a section devoted to the classification of the so-called \(\kappa\)-solutions in dimension 3. They are solutions of the Ricci flow which are ancient (i.e., have an infinite past), non flat, have bounded and nonnegative curvature operator on each time slice and are \(\kappa\)-non-collapsed at all scales for a positive \(\kappa\). They are infinitesimal models for the singularities of a three-dimensional Ricci flow. Indeed, in three dimensions, they are obtained as limits of suitable blow-up around a singularity (points of very large curvature) and at finite time. This classification is necessary in order to understand the Ricci flow with surgery.

Section 12: This section presents another breakthrough, the so-called canonical neighbourhood theorem. This is again specific to the three-dimensional case. It is shown that for a smooth Ricci flow, with normalised initial data, there is a universal number such that if at some point the scalar curvature becomes larger than this value, then a neighbourhood of this point, of controlled size, is, after rescaling, close to a piece of a \(\kappa\)-solution. This describes quite precisely the geometry of the manifold, whose evolution is given by the Ricci flow equation, around points of large curvature. The rest of the section is devoted to the (beginning of the) long term analysis of the Ricci flow.

Section 13: In this section the proof of geometrisation is sketched for manifolds which carry a Ricci flow defined for all time. It is shown that the manifold has a thick-thin decomposition, that the thick part becomes hyperbolic and is bounded by incompressible tori following arguments due to R. Hamilton. The thin part is claimed without proof to be a graph manifold. Some arguments which are adapted from the works of R. Hamilton are more precisely treated in [G. Perelman, “Ricci flow with surgery on three-manifolds”, arXiv: math.DG/0303109 (2003; Zbl 1130.53002)]. Others concerning the case where the flow develops singularities are sketched but not justified. The reader is referred to [Perelman, loc. cit.].

Section 1: On a closed manifold \(M\) the Ricci flow is presented as a gradient flow of a functional defined on the couples \((g, f)\), where \(g\) is a Riemannian metric and \(f\) is a function on \(M\). the function is chosen so that \(e^{-f} \,d\operatorname{vol}_g\) is a fixed measure. The functional is \[ \mathcal F(g,f)=\int_M(R+|\nabla f|^2)e^{-f}\,d\operatorname{vol}_g, \] in which \(R\) denotes the scalar curvature of the metric \(g\) and the norm is taken for this metric.

Section 2: This section and the following one present immediate corollaries which we can summarize as follows: On the space obtained by moding out the space of metrics by the action of the diffeomorphisms group and the homotheties there are no closed trajectories of the Ricci flow. In Perelman’s terminology one shows that there are no non trivial breathers. This section is devoted to the proof of this assertion in the case of steady and expanding breathers. This relies on introducing very interesting invariants which are nondecreasing along the flow and stationary on Ricci solitons. One is the infinum on \(f\) of \(\mathcal F(g,f)\) which turns out to be the smallest eigenvalue of \(-4\Delta+R\). The other one is the same eigenvalue normalised by the volume raised at the suitable power in order to get a scale invariant quantity.

Section 3: This section deals with the shrinking breathers which are more difficult to study. A modification of \(\mathcal F\) called \(\mathcal W\) is introduced. It is a function of three parameters: a metric \(g\), a function \(f\) and a number \(\tau\) which is \(t_0 - t\) where \(t\) is the parameter of the Ricci flow. Again \(\mathcal W\) is nondecreasing along the Ricci flow when \(f\) is normalized as in section 1 and is stationary on gradient shrinking solitons.

Section 4: This is the second breakthrough of this paper. It presents a tool that gives its full strength to the Ricci flow, namely a solution on a closed manifold \(M\) is non collapsed at a finite time \(t\). It means there exists a number \(\kappa>0\) such that balls at time \(t\) and of radius \(r\leq\rho\) on which the Riemann curvature is bounded by \(r^{-2}\) have volume bounded below by \(\kappa r^n\). The number , called the scale, is the value below which this property is true. All these quantities are scaled invariant which makes this property true even after rescaling the metrics. This implies a local injectivity radius bound and allows to apply compactness theorems. This is crucial for the singularities analysis.

Section 5 and 6: These two sections are less useful at the moment. They intend to give justifications and explanations for the previous construction either using a statistical mechanics’ approach or the point of view of infinite dimensional geometry.

Section 7: In this very important section the author develops the Morse theory of a functional called \(\mathcal L\)-length, which is a space-time version of the standard length (or energy). \(\mathcal L\)-geodesics, \(\mathcal L\)-Jacobi fields and \(\mathcal L\)-exponential are computed leading to some comparison theorems. The reduced volume is introduced, a monotonic quantity along the Ricci flow which allows to give an alternative proof of the nonlocal collapsing property. This is a key section for the three dimensional applications of the Ricci flow.

Section 8: A refined version of the nonlocal collapsing theorem is proved using the tools developed in the previous section, in particular the reduced volume.

Section 9: A differential Harnack inequality is proved.

Section 10: In this section it is proved that despite the fact that the Ricci flow is not local, it is pseudo-local. Namely, if the curvature is close to zero in a region (and with extra assumption), at least for a short time it remains not too far from zero, i.e., for a short time it is not affected by the possible presence of big curvature elsewhere on the manifold. This fact is made precise.

Section 11: This is a section devoted to the classification of the so-called \(\kappa\)-solutions in dimension 3. They are solutions of the Ricci flow which are ancient (i.e., have an infinite past), non flat, have bounded and nonnegative curvature operator on each time slice and are \(\kappa\)-non-collapsed at all scales for a positive \(\kappa\). They are infinitesimal models for the singularities of a three-dimensional Ricci flow. Indeed, in three dimensions, they are obtained as limits of suitable blow-up around a singularity (points of very large curvature) and at finite time. This classification is necessary in order to understand the Ricci flow with surgery.

Section 12: This section presents another breakthrough, the so-called canonical neighbourhood theorem. This is again specific to the three-dimensional case. It is shown that for a smooth Ricci flow, with normalised initial data, there is a universal number such that if at some point the scalar curvature becomes larger than this value, then a neighbourhood of this point, of controlled size, is, after rescaling, close to a piece of a \(\kappa\)-solution. This describes quite precisely the geometry of the manifold, whose evolution is given by the Ricci flow equation, around points of large curvature. The rest of the section is devoted to the (beginning of the) long term analysis of the Ricci flow.

Section 13: In this section the proof of geometrisation is sketched for manifolds which carry a Ricci flow defined for all time. It is shown that the manifold has a thick-thin decomposition, that the thick part becomes hyperbolic and is bounded by incompressible tori following arguments due to R. Hamilton. The thin part is claimed without proof to be a graph manifold. Some arguments which are adapted from the works of R. Hamilton are more precisely treated in [G. Perelman, “Ricci flow with surgery on three-manifolds”, arXiv: math.DG/0303109 (2003; Zbl 1130.53002)]. Others concerning the case where the flow develops singularities are sketched but not justified. The reader is referred to [Perelman, loc. cit.].

Reviewer: Gérard Besson (Grenoble)

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

57M40 | Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010) |

57R60 | Homotopy spheres, Poincaré conjecture |