12.5 Geometric flows and Ricci flow

Geometric flows are powerful tools in differential geometry that evolve geometric objects over time. They're governed by partial differential equations, typically based on curvature, aiming to deform objects into more desirable shapes while preserving certain properties.

The Ricci flow, introduced by Richard Hamilton in 1982, is a prime example. It evolves Riemannian metrics according to their Ricci curvature. This flow has been instrumental in proving major conjectures in topology and geometry, including the Poincaré conjecture.

Definition of geometric flows

• Geometric flows are a powerful tool in differential geometry that evolve a geometric object over time according to a specific partial differential equation
• The evolution is typically governed by the curvature of the object, with the goal of deforming it into a more desirable or canonical shape while preserving certain properties (topology, symmetries, etc.)

Smooth family of metrics

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• In the context of Riemannian manifolds, a geometric flow is often defined as a smooth family of Riemannian metrics $g(t)$ parameterized by time $t$
• The metrics evolve according to a PDE that depends on the curvature tensor of $g(t)$, such as the Ricci tensor or the scalar curvature
• The initial metric $g(0)$ is given, and the goal is to study the long-time behavior of the flow and its possible singularities

Evolution equations for metrics

• The evolution equation for a geometric flow takes the general form $\frac{\partial}{\partial t}g(t)=F(g(t))$, where $F$ is a smooth function of the metric and its curvature
• The choice of $F$ determines the specific type of geometric flow (Ricci flow, mean curvature flow, Yamabe flow, etc.)
• The evolution equation is a nonlinear PDE that can be challenging to solve explicitly, but its qualitative behavior can often be analyzed using geometric and analytic techniques

Examples of geometric flows

Ricci flow

• The Ricci flow, introduced by Richard Hamilton in 1982, is defined by the evolution equation $\frac{\partial}{\partial t}g(t)=-2\mathrm{Ric}(g(t))$, where $\mathrm{Ric}$ is the Ricci curvature tensor
• It has been used to prove the Poincaré conjecture in dimension 3 and the differentiable sphere theorem in dimension 4
• The Ricci flow tends to smooth out positive curvature and concentrate negative curvature, leading to the formation of singularities in finite time for most initial metrics

Mean curvature flow

• The mean curvature flow deforms a submanifold $M^n$ in a Riemannian manifold $N^{n+k}$ by moving each point in the direction of its mean curvature vector $\vec{H}$
• The evolution equation is $\frac{\partial}{\partial t}F(p,t)=\vec{H}(p,t)$, where $F:M\times[0,T)\to N$ is a smooth family of immersions
• It has applications in geometric topology, image processing, and material science (modeling the motion of interfaces and phase boundaries)

Yamabe flow

• The Yamabe flow is a conformal deformation of a Riemannian metric $g$ that evolves according to the equation $\frac{\partial}{\partial t}g(t)=-R(g(t))g(t)$, where $R$ is the scalar curvature
• It is designed to find a metric of constant scalar curvature within a given conformal class, thus solving the Yamabe problem
• The Yamabe flow always converges to a metric of constant scalar curvature on compact manifolds, but the limiting metric may have singularities

Properties of Ricci flow

Evolution of curvature

• Under the Ricci flow, the Riemann curvature tensor satisfies a nonlinear heat-type equation that involves the full curvature tensor
• The scalar curvature evolves according to $\frac{\partial}{\partial t}R=\Delta R+2|\mathrm{Ric}|^2$, which is similar to a reaction-diffusion equation
• The evolution equations for curvature can be used to derive important estimates and monotonicity formulas for the Ricci flow

Existence and uniqueness

• For compact manifolds, the Ricci flow has a unique solution for a short time starting from any smooth initial metric
• The maximal time of existence depends on the initial metric and the dimension of the manifold
• In higher dimensions ($n\geq 4$), the Ricci flow may develop singularities in finite time even for smooth initial data

Maximum principles

• Various maximum principles hold for the Ricci flow, allowing for the comparison of curvature quantities at different points and times
• For example, the minimum of the scalar curvature is non-decreasing along the flow, and the maximum of the norm of the Riemann curvature tensor is non-increasing
• These principles are crucial for understanding the long-time behavior of the flow and the formation of singularities

Convergence in low dimensions

• In dimension 2, the Ricci flow is equivalent to the heat equation for the Gaussian curvature and always converges to a metric of constant curvature (uniformization theorem)
• In dimension 3, the Ricci flow with surgery, developed by Perelman, can be used to prove the geometrization conjecture for all compact 3-manifolds
• In higher dimensions, the convergence of the Ricci flow is more complicated and may require additional assumptions (e.g., Kähler manifolds, positive curvature, etc.)

Singularities of Ricci flow

Classification of singularities

• Singularities of the Ricci flow can be classified into Type I, Type II, and Type III, depending on the rate of blow-up of the curvature as the singularity time is approached
• Type I singularities are modeled on shrinking self-similar solutions (Ricci solitons) and have a blow-up rate of the form $|\mathrm{Rm}|\leq \frac{C}{T-t}$, where $T$ is the singularity time
• Type II singularities are more complicated and may involve the formation of neck-like regions or the splitting of the manifold into multiple components

Formation of singularities

• In dimension 3, the only way a singularity can form is through the pinching off of a neck-like region, as shown by Hamilton's neck theorem
• In higher dimensions, singularities can also form through the collapse of certain submanifolds or the formation of cusps
• The precise structure of singularities is still not fully understood, especially in dimensions $n\geq 4$

Blow-up analysis

• To study the local behavior of the Ricci flow near a singularity, one can perform a blow-up analysis by rescaling the metric and the time parameter
• The rescaled metrics converge (in a suitable sense) to a self-similar solution of the Ricci flow, called a singularity model
• The classification of singularity models is a crucial step in understanding the global behavior of the flow and in developing a surgery procedure to continue the flow past singularities

Applications of Ricci flow

Uniformization of surfaces

• In dimension 2, the Ricci flow provides a new proof of the uniformization theorem, which states that every compact Riemann surface admits a metric of constant curvature
• The Ricci flow starting from any initial metric converges to a metric of constant curvature, which is unique up to scaling and isometry
• This gives a canonical way to parameterize the moduli space of Riemann surfaces

Geometrization of 3-manifolds

• The geometrization conjecture, formulated by Thurston, states that every compact 3-manifold can be decomposed into pieces that admit one of eight homogeneous geometries
• Perelman used the Ricci flow with surgery to prove the conjecture, by showing that the flow either converges to a metric of constant curvature or collapses the manifold along incompressible tori
• This work revolutionized the field of 3-dimensional topology and earned Perelman the Fields Medal (which he declined)

Canonical metrics on Kähler manifolds

• On a Kähler manifold, the Ricci flow preserves the Kähler condition and is equivalent to a scalar PDE for the Kähler potential
• The Kähler-Ricci flow can be used to find canonical metrics, such as Kähler-Einstein metrics or constant scalar curvature metrics
• It has important applications in complex geometry, algebraic geometry, and mathematical physics (e.g., string theory, mirror symmetry)

Connections to other areas

Relationship with harmonic map flow

• The Ricci flow can be viewed as a generalization of the harmonic map flow between Riemannian manifolds
• If $(M,g)$ and $(N,h)$ are Riemannian manifolds and $f:M\to N$ is a smooth map, the harmonic map flow is defined by $\frac{\partial}{\partial t}f(x,t)=\tau_g f(x,t)$, where $\tau_g$ is the tension field of $f$ with respect to the metric $g$
• When $M=N$ and $f$ is the identity map, the harmonic map flow reduces to the Ricci flow (up to a constant factor)

Analogy with heat equation

• The Ricci flow shares many similarities with the heat equation $\frac{\partial}{\partial t}u=\Delta u$, which describes the diffusion of heat in a medium
• Both equations have smoothing properties, maximum principles, and exhibit finite-time blow-up for certain initial data
• However, the Ricci flow is a nonlinear PDE that involves the curvature of the evolving metric, making its analysis more challenging

Gradient flows vs reaction-diffusion equations

• Geometric flows can be broadly classified into two types: gradient flows and reaction-diffusion equations
• Gradient flows, such as the heat equation or the Yamabe flow, are driven by the gradient of a functional (e.g., the Dirichlet energy or the total scalar curvature) and tend to minimize this functional over time
• Reaction-diffusion equations, such as the Ricci flow or the mean curvature flow, involve terms that can be interpreted as "reaction" terms (e.g., quadratic curvature terms) and may exhibit more complex behavior, such as the formation of singularities or pattern formation