[Paper Review] Ricci flow coupled with harmonic map flow

Reto Müller|arXiv (Cornell University)|Jan 1, 2012

Geometric Analysis and Curvature Flows45 references103 citations

TL;DR

This paper introduces a coupled Ricci-harmonic map flow that evolves a Riemannian metric and a map from a manifold to a target space simultaneously. By introducing a positive coupling constant α, the system prevents energy concentration in the harmonic map component and ensures curvature-controlled regularity. The key contribution is the derivation of monotonicity formulas for energy, entropy, and reduced volume, which rule out finite-time singularities such as non-trivial breathers and geometric collapsing.

ABSTRACT

We investigate a new geometric flow which consists of a coupled system of the Ricci flow on a closed manifold M with the harmonic map flow of a map phi from M to some closed target manifold N with a (possibly time-dependent) positive coupling constant alpha. This system can be interpreted as the gradient flow of an energy functional F_alpha which is a modification of Perelman's energy F for the Ricci flow, including the Dirichlet energy for the map phi. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of phi a-priori - without any assumptions on the curvature of the target manifold N - by choosing alpha large enough. Moreover, if alpha is bounded away from zero it suffices to bound the curvature of (M,g(t)) to also obtain control of phi and all its derivatives - a result which is clearly not true for alpha = 0. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of an entropy functional W_alpha similar to Perelman's Ricci flow entropy W and of so-called reduced volume functionals. We then apply these monotonicity results to rule out non-trivial breathers and geometric collapsing at finite times.

Motivation & Objective

• To study a coupled system of Ricci flow and harmonic map flow on compact manifolds.
• To analyze the regularity and singularity formation in the coupled system, especially when the harmonic map component might otherwise concentrate.
• To establish monotonicity of energy, entropy, and reduced volume functionals under the coupled flow.
• To apply monotonicity to rule out non-trivial breathers and geometric collapsing at finite time.
• To show that bounded curvature implies long-time existence and control of all derivatives of the map φ via a positive coupling constant α.

Proposed method

• Define the (RH)α flow: ∂tg = −2Rc + 2α∇φ⊗∇φ and ∂tφ = τgφ, with α ≥ ᾱ > 0.
• Use DeTurck's trick to transform the weakly parabolic system into a strictly parabolic one via diffeomorphism gauge fixing.
• Derive evolution equations for curvature tensors, Ricci curvature, and gradient of φ using commutator identities and curvature formulas.
• Introduce a modified energy functional Fα(g, φ, f) and show it is non-decreasing under the flow, with constancy iff a steady gradient soliton.
• Construct a backwards reduced volume functional ˜Vk(t) and prove its monotonicity using Lb-geodesics and Jacobian estimates.
• Apply maximum principle and barrier methods to control the growth of |∇φ|2 and Riemann curvature tensor.

Experimental results

Research questions

• RQ1 Can the coupled Ricci-harmonic map flow prevent energy concentration in the harmonic map component?
• RQ2 Does bounded curvature of (M, g(t)) imply control of all derivatives of φ when α ≥ ᾱ > 0?
• RQ3 Can monotonicity of energy, entropy, and reduced volume be established for the coupled system?
• RQ4 Does the monotonicity of these functionals rule out non-trivial breathers and geometric collapsing at finite time?
• RQ5 Under what conditions does the (RH)α flow admit long-time existence?

Key findings

• For any α > 0 large enough, energy concentration of the harmonic map φ is ruled out a-priori.
• If the curvature |Rm| is bounded along the flow, then |∇φ|2 and all higher derivatives of φ remain uniformly bounded.
• The modified energy functional Fα(g, φ, f) is non-decreasing along the flow, and constant if and only if (g(t), φ(t)) is a steady gradient soliton.
• The reduced volume functional ˜Vk(t) is non-increasing in backwards time, and its limit behavior is used to rule out finite-time singularities.
• The existence of a uniform Riemann curvature bound implies long-time existence of the (RH)α flow.
• The flow rules out non-trivial breathers and geometric collapsing at finite time, as shown by contradiction using the monotonicity of ˜Vk(t).