**Background:**


I am reading the book *Variational Problems in Geometry* by Seiki Nishikawa. The main purpose of this book is to prove the existence of harmonic maps $M\to N$ between two compact Riemannian manifolds $M$ and $N$ with the target manifold being nonpositively curved. Since a harmonic map is defined as a minimizer for the energy functional $E$, the idea is to deform an existing map $u:M\to N$ along the gradient lines of $E$, which is equivalent to solving an equation:
$$\frac{\partial u\_t}{\partial t}=\tau(u\_t)\quad\quad\text{Eq.(1)}$$
where $u\_t$ is a smooth variation of some given $u\_0$, and $\tau$ denotes the tension field (along this direction, $E$ decreases fastest). By embedding $N$ into $\mathbb R^q$ using the Nash embedding theorem, we can regard $N$ as a submanifold and $u$ as a vector-valued function to simplify certain statements. After this embedding, the above equation takes a new form:
$$(\Delta-\partial\_t)u\_t=\Pi(u\_t)\quad\quad\text{Eq.(2)}$$
I'll omit the definition of $\Pi(u\_t)$. Just know that they are equivalent in the sense that their solutions coincide, if given identical initial conditions. The difference is:


* Eq.(1) is **intrinsic and is an equation in $\Gamma(u\_t^{-1}TN)$**;
* Eq.(2) is **an equation in $\mathbb R^q$**.




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**Theorem:**


One of the main theorems is the **existence of a solution to any one of the equations above defined for $t\in[0,+\infty)$**, given some $u\_0$ as the initial condition. Suppose now we have already established:


* local (w.r.t time) existence of a solution
* certain estimates


The proof goes as follows:



> 
> *Proof. (The part where there's no problem, so I'll give a sketch only)* Since we have established the local existence, for a given $u\_0$, we can find a smooth $u:M\times[0,T)\to N$ for some $T>0$ such that it solves (1) and (2) and $u(\cdot,0)=u\_0$. Now suppose $T\_0$ is the supremum of all such $T$. We wish to prove $T\_0=+\infty$. If not, we take an increasing sequence $t\_i\to T\_0$. By certain estimates, we know that $\{u\_{t\_i}\}$ and $\{\partial\_tu\_{t\_i}\}$ are uniformly bounded and equicontinuous, respectively in the Holder spaces $C^{2+\alpha}(M,\mathbb R^q)$ and $C^{\alpha}(M,\mathbb R^q)$, for some $0<\alpha<1$.
> 
> 
> *Proof. (The part where I am confused. I'll write exact words)* **By the Ascoli-Arzela theorem, there exists a subsequence $\{t\_{i\_k}\}$ of $\{t\_i\}$ and functions$^1$**
> $$u(\cdot,T\_0)\in C^{2+\alpha}(M,\mathbb R^q)\quad\text{and}\quad\partial\_t u(\cdot,T\_0)\in C^\alpha(M,\mathbb R^q)$$
> such that the subsequences
> $$\{u(\cdot,t\_{i\_k})\}\quad\text{and}\quad\{\partial\_tu(\cdot,t\_{i\_k})\}$$
> respectively, converge uniformly to $u(\cdot,T\_0)$ and $\partial\_tu(\cdot,T\_0)$, as $t\_{i\_k}\to T\_0$. Since for each $t\_{i\_k}$, we have
> $$\partial\_tu(\cdot,t\_{i\_k})=\tau(u(\cdot,t\_{i\_k}))\quad\quad\text{Eq.(3)}$$
> **we also get at$^2$ $T\_0$**
> $$\partial\_tu(\cdot,T\_0)=\tau(u(\cdot,T\_0))\quad\quad\text{Eq.(4)}$$
> **consequently$^3$, we see that (1) has a solution in $M\times[0,T\_0]$**. Using $u(\cdot,T\_0)$ again as the initial condition to solve (1), we extend the solution to $M\times[0,T\_0+\epsilon)$ for some $\epsilon>0$, contradicting $T\_0$ being the supremum. Hence $T\_0=\infty$.
> 
> 
> 




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**Questions:**


1. How do we know that the limits do not depend on the choice of $t\_i$. I think it can be argued using the uniformly boundedness and equicontinuity. Am I right?
2. How can we go from (3) to (4)? Eq. (3) is clearly obtained by (1), which is an equation in $\Gamma(u\_t^{-1}TN)$. However, (3)$\implies$(4) would require the convergence of $\partial\_tu(\cdot,t\_{i\_k})$ to $\partial\_tu(\cdot,T\_0)$. But this convergence is only in $C^{\alpha}(M,\mathbb R^q)$, not in $\Gamma(u\_t^{-1}TN)$. Although I think I can fix this by writing them in the form of (2) to start with.
3. How can we conclude from (4) that (1) has a solution in $M\times[0,T\_0]$. Note that $\partial\_tu(\cdot,T\_0)$ is **so far only a notation** for the limit of $\partial\_tu(\cdot,t\_{i\_k})$, we have not proved that it is actually the (one-sided) derivative of $u$ at $t=T\_0$. How can I fix this?



Possible flaw in the proof of the Eells-Sampson theorem on harmonic maps in Nishikawa's book
$\newcommand{\R}{\mathbb{R}}\newcommand{\pa}{\partial}$Edit: The answer is now LaTeXified.


Below are my notes on this. I reworked the proof:


*Proof*. Let $S:=\big\{T\in[0,\infty):$ the equation has a solution in $C^{2+\alpha,1+\alpha/2}(M\times[0,T],N)\big\}$. Let $T\_0:=\sup S$. By existence of local solution, $T\_0>0$. We claim that $T\_0=\infty$.


Suppose $T\_0<\infty$. By uniqueness of solution and definition of $T\_0$, we have a solution $u\in C^{2,1}(M\times[0,T\_0),N)$. Take $\alpha<\alpha'<1$. By the *a priori* estimate above, $u\_t$ is uniformly bounded in $t$ in $C^{2+\alpha'}(M,\R^L)$.


Define $$u(x,T\_0):=\int\_0^{T\_0}\pa\_tu(x,t)\,dt+u(x,0).$$ For any sequence $t\_k\nearrow T\_0$, $(u\_{t\_k})\_{k=1}^\infty$ has a subsequence that converges in $C^{2+\alpha}(M,N)$ by Arzelà–Ascoli, and its limit is necessarily $u\_{T\_0}$. Thus $u\_{T\_0}\in C^{2+\alpha}(M,N)$, and any such sequence must in fact converge to $u\_{T\_0}$ in $C^{2+\alpha}(M,N)$. In other words, $u\in C^{2+\alpha,0}(M\times[0,T\_0],N)$, or equivalently, $t\mapsto u\_t$ is continuous as a map $[0,T\_0]\to C^{2+\alpha}(M,N)$.


Since $\pa\_tu\_t=\tau(u\_t)$, we see that $t\mapsto\pa\_tu\_t$ has a continuous extension $[0,T\_0]\to C^{\alpha/2}(M,\R^L)$. So in fact $u\in C^{2+\alpha,1+\alpha/2}(M\times[0,T\_0],N)$, i.e., $T\_0\in S$. Now, existence of local solution implies that $u$ can be extended to a solution on $[0,T\_0+\varepsilon]$, which is a contradiction. $\square$


The *a priori* estimate refers to the one as given in the book: For $0<\alpha<1$, $$\sup\_{t\in[0,T)}\Big(\big\|u\_t\big\|\_{C^{2+\alpha}(M,\R^L)}+\big\|\pa\_tu\_t\big\|\_{C^\alpha(M,\R^L)}\Big)\leq C(M,N,f,\alpha).$$


