(This is an edited version of Pin's presentation on Michael Struwe's "On the evolution of harmonic mappings of Riemannian surfaces", 1985.)

Outline

• Define the harmonic map flow
• Gagliardo-Nirenberg inequality
• Energy estimates: global/local
• ε-regularity
• Putting it all together via Moser iteration to get criterion for "global existence"
• Deduce the main theorem: convergence to harmonic map and classification of singularities (i.e. bubbling)

[Rmk: the G-N inequality is a pure analysis fact, the energy estimates are generic facts concerning evolutionary partial differential equations. The ε-regularity result, however, is more specifically a phenomenon associated to geometric equations.]

harmonic map flow
We consider where Σ is a two-dimensional compact Riemannian manifold, N is a compact manifold that can be isometrically embedded (via Nash's theorem) in Rⁿ for sufficiently high dimension n. We abuse the notation u to simultaneously mean the map from Σ to N as well as the map from Σ to the embedding of N in Rⁿ. For a map u, we can define its energy density which is a coordinate independent scalar on Σ define the total energy functional on the class of H¹ maps from Σ to N.

A harmonic map is a critical point of E[u]. By taking the first variation, we see that the Euler-Lagrange equation implies that a harmonic map must satisfy the equation where A is the second fundamental form associated to the Nash embedding of N into Rⁿ. Since both N and Σ are compact, the right-hand-side is morally speaking just a quadratic form on the gradient of u (the variable coefficient is not so important when we study the problem as a p.d.e., since we'll just take the L^∞ which is guaranteed to exist finitely by virtue of the smoothness of the metric γ and g, as well as the smoothness of the Nash embedding). A important feature of the second fundamental form that carries into the equation is that the second fundamental form is, by definition, orthogonal to the tangent bundle of N as a subset of the tangent bundle of Rⁿ.

The harmonic heat flow (HH in the sequel) is defined to be the gradient flow for the energy functional E[u]. Formally the equation for u (which we now enlarge to be a map from I×Σ to N, where I is an interval in R₊; the coordinate for I shall be t) is Perhaps the most widely known result related to HH is that of Eells-Sampson, which states that given an arbitrary continuous map f from M to N, where M and N are compact Riemannian manifolds, there exists a harmonic map u homotopic to f as long as the sectional curvature for N is non-positive. The technique employed is basically by setting f to be the initial data for the heat flow and showing convergence at time infinity.

The result of Struwe which is discussed here has a slightly stronger corallary: let Σ be a surface, and f a continuous map from Σ to N, then there exists a harmonic map u homotopic to f as long as π₂(N) = 0 (the second fundamental group vanishes). Notice that by Cartan-Hadamard, the non-positivity of sectional curvature implies the universal cover of N is diffeomorphic to Euclidean space and so has vanishing second fundamental group and so N must also have vanishing second fundamental group.

Statement of main theorem

Let u(x,t) be H¹(Σ,N) a solution of HH in the sense of distributions for I=[0,T), then the following are true

• u extends to a solution on R₊×Σ (i.e. u is global in time)
• The energy is monotonic: E[u](t) ≤ E[u](s) whenever t ≥ s
• There exists a finite set of points (t_k,x_k) such that away from those points, u is smooth
• As time t approaches ∞, u(x,t) converges in some sense to a smooth harmonic map u_∞: Σ to N
• For each blow-up singularity (t_k,x_k) defined above, we can constructed naturally an associated nontrivial harmonic map u_k from S² to N, a phenomenon called bubbling.

Gagliardo-Nirenberg
We begin with a lemma. Suppose u is H¹(R²). Let R > 0 be arbitrary. Let ψ be a cut-off function associated to R: that |ψ| ≤ 1, |∇ψ| ≤ 4/R, and ψ supported in a ball of radius R around the origin. Then we have the inequality for some universal constant C.

The proof of the lemma is a simple application of the classical G-N inequality on a compact domain for functions with compact support.

Energy identities
These are a priori estimates, so we will assume in this part that u is a smooth solution of HH. First we have the global energy estimate Its proof is simple: we multiply the HH equation by ∂u/∂t. Notice that ∂u/∂t lives pointwise in the tangent space of N, so its inner product with the second fundamental form on the RHS must be 0. After integrating over Σ, we can integrate the Laplace-Beltrami operator by parts, and another integral over the time interval [0,t] gives the desired estimate. The whole proof is nearly identical to the energy decay estimate for the linear heat equation.

We also have a local energy growth control: assume R < (injectivity radius of Σ)/2 is small. We fix a point x and assume that where E[u,Ω] means the integral of the energy density over the set Ω, then we have the following control on a slightly smaller ball where C is a universal constant. The proof is slightly more involved. Let ψ be a cut-off function that is equal to 1 on the ball or radius R and equals 0 outside the ball of radius 2R, with its gradient bounded by 2/R. Recall the differential form of the energy identity and we multiply it by ψ². Since ψ is constant in time, it commutes with time derivatives on the left hand side. We integrate the whole thing over Σ and take the absolute value. The LHS is manifestly positive. The RHS we integrate by parts where in the first inequality we used Cauchy-Schwarz, and the second we used Young's inequality (which is basically just Cauchy-Schwarz combined with the arithmetic-geometric mean inequality). Notice that the last term on the RHS is identical to the first term on the LHS. So by choosing λ small enough, we can absorb it on the left. Noticing the positivity of (∂_tu)^2, we have Notice that the integral on the RHS is equivalent (up to a constant factor) to E[u, B_2R], and hence is bounded by E₀ from the hypothesis. Now we just integrate in time from 0 to T and get the desired local energy control.

(To Be Continued ...)