**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 $\mathbb R^q$**.

**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 $T_0=\infty$.

**Questions:**

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?

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.

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?