#!/usr/bin/env python
# coding: utf-8

# # Teichmueller harmonic map flow -- Lecture 2
# 
# > [Peter Topping](http://homepages.warwick.ac.uk/~maseq/), University of Warwick
# 
# #### [Ascona Winter School 2016](http://www.math.uzh.ch/pde16/index-Ascona2016.html), [(alternate link)](http://www.monteverita.org/en/90/default.aspx?idEvent=295&archive=)
# 
# [extended notes on the lectures](http://homepages.warwick.ac.uk/~maseq/ASCONA/)

# 1. [Lecture 1](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/topping-lecture1.ipynb)
# 2. [Lecture 2](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/topping-lecture2.ipynb)
# 3. [Lecture 3](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/topping-lecture3.ipynb)
# 4. [Lecture 4](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/topping-lecture4.ipynb)
# 
# [source](https://github.com/colliand/ascona2016)

# 
# 
# ## The Harmonic Map Flow
# 
# [Eels-Sampson 1964](http://www.math.jhu.edu/~js/Math646/eells-sampson.pdf)
# 
# $$u:(M,g) \longmapsto (N,G)$$
# 
# $$ \frac{\partial u}{\partial t} = \tau (u) = "\Delta u"$$
# 
# $$ \implies \frac{dE}{dt} = - \int |\tau|^2$$.
# 
# 

# ** 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](https://en.wikipedia.org/wiki/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 
# 
# 1. $u(0) = u_0$,
# 2. $E(u(t)) \searrow$ in $t$
# 3. There exists a sequence of times $t_i \rightarrow \infty$ and a smooth harmonic map $u_\infty : M \rightarrow N$ such that $u(t_i) \rightharpoonup u_\infty$ in $W^{1,2} (M,N)$ and $u(t_i) \rightarrow u_\infty$ in $W^{2,2}_{loc} (M\backslash S_2, N)$ finite set.

# 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?
# 
# ## Finding higher genus BMIs

# Now, we need to look for critical points of $E(u,g)$. 
# 
# * try gradient flow for $E(u,g)$. What does this mean? $L^2$-gradient flow? Move $u$ in the direction of the tension field and the $g$ should move in the direction of $\Re( \Phi(u,g) )$. There's maybe a theory here but it's tricky and hard to analyze. You could normalize by adding a negative Ricci term...seems bad.
# * We proceed differerntly. Exploit the conformal invariance of the energy $E(u,g) = E(u, fg)$ with $f>0$. We can therefore choose a nice conformal representative of $g$. This allows us to restrict $g$ to metrics with constant curvature. We can even do a bit better than that...super important result
# 
# **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.

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