Peter Topping, University of Warwick
.
** Theorem (Eels-Sampson, Hartman):** If $(N,g)$ has nonpositive sectional curvature, then given $u:M \longmapsto N ~ \exists$ smooth flow $u$ with $u(0) = u_0$ and $u(t) \rightarrow u_\infty$ in $C^k$ as $ t \rightarrow \infty$ where $u_\infty$ is a harmonic map $M \longmapsto N$, momostopic to $u_0$.
Hartman's contribution was to show that the convergence is uniform. Hartman also showed that the destination map is "essentially" unique.
Let's switch back to two dimensions but release the condition that the curvature is nonpositive. We can them embed $(N,G)$ isometrically into $\mathbb{R}^n$ using the Nash Embedding Theorem.
The simplest thing you might do is to calcluate the Laplacian and then project it into the direction of the tension field. Harmonic map flow: $$ \frac{\partial u}{\partial t} = (\Delta u)^T. $$
The reason for doing this embedding is that it allows us to beging talking about rough embeddings right away. The embedding allows us to discuss Sobolev regularity, for example.
Theorem (Struwe 85): Given $ u \in W^{1,p}(M,N)$ there exists weak solution $u \in W^{1,2}_{loc} (M \times [0, \infty), N)$ which is smooth except at finitely many points $S_1 \subset M \times (0, \infty)$ such that
We can't omit the $S_1, S_2$ singular sets.
Theorem (Eels-Wood): There does not exist a harmonic map of degree 1 from $T^2 \longmapsto S^2$.
Maps like this exist but they can not be deformed into harmonic maps.
At points in $S_1$ and $S_2$, we encounter the phenomena of bubbling. A quantum of energy would be flowing into a small region of space. There is only one strategy to try and understand what's going on in this scenario: rescale!
Theorem (Struwe): Suppose $u$ as in preceding theorem. Then WLOG $\{ t_i \}$ also satisfies: $\forall ~x_0 \in S_2$, when we pick local isothermal coordinates near $x_0$ there exists rescaling points $a_i \rightarrow 0 \subset ~\mathbb{R}^2$ and scales $\lambda_i \searrow 0$ and nonconstant harmonic map $\omega: S^2 \longmapsto (N,G)$ such that $u(a_i + \lambda_i y, t_i) \longmapsto \omega (y)$ in $W^{2,2}_{loc} (\mathbb{R}^2, N).$
You can also perform this rescaled bubble analysis at finite time.
That's a way of extracting a bubble. The energy will be lost into the bubble. The next part of the story involves a conisderation of different rescaling centers and scales which may capture other mechanisms for the loss of energy. We know by the Eels-Wood result that we can't have $S_1$ and $S_2$ both be empty. Some examples by Cheng-Ding and others show that the various possible scenarios are actually realized.
Consequence: If there does not exist a bubble in $(N,g)$ then the flow is globally smooth and converges to $u_\infty$ uniformly.
Lemma: There does not exist a non-constant harmonic map $S^2 \rightarrow (N,G)$ if $(N,G)$ has nonpositive curvature.
Why? Suppose you do have such a map. Take your target and lift it to the universal cover. You can then lift the map so this gives a nonconstant harmonic map from $S^2$ into the universal cover of a nonpositively curved manifold. On such a manifold, the distnace from any point defines a convex function. You can compose the distance function with the harmonic map and then deduce, using the maximum principle, that the harmonic map must be constant.
Theorem: Suppose the closed target manifold satisfies $\Pi_2 (N) \neq \{0 \}$. Then $\forall ~G$ there exists a BMI (branched minimal immersion) with $S^2 \longmapsto (N, G)$.
Proof: Take a nontrivial $u_0: S^2 \longmapsto N$. Flow using the harmonic map flow. Then, there is a dichotomy. Either $S_1 = \emptyset, ~S_2 = \emptyset$ and $u_\infty$ is that BMI, OR a bubble develops and $\omega$ is that BMI.
You can often find harmonic maps from $S^2$. What would you do if you want to build a high genus harmonic map? How can we find high genus branched minimal immersions?
Now, we need to look for critical points of $E(u,g)$.
Lemma: Any surface $(N,g)$ can be conformally deformed to a surface of constant Gauss curvature with value $+1, 0, -1$. $M=S^2$ corresponds to case +1. $M=\mathbb{T}^2$ corresponds to case 0. $genus(M) \geq 2 $ is in the case -1.
Proof of Lemma: Run Ricci flow. $\frac{\partial g}{\partial t} = (k_0 - k) g$ where $k_0 = \{ +1, 0 , -a \}$.
Idea of the Teichmller Harmonic map flow is to restrict $g$ to $\mathcal{M}_c$. The next task is to begin to understand this set of constant curvature metrics.