Hi, I have the following question: take a Riemannian manifold M, with a family of smooth metrics $g(t)$ in $[0,T)$, call $D\_0$ the Levi-Civita connection of $g(0)$ and assume that for every $m\geq 0$  

$\int\_0^T \sup\_M|D\_0^m \frac{\partial}{\partial t} g(t)|\_{g(0)} dt< \infty$, then why $g(t)$ converges in $C^{\infty}$ to a smooth tensor?



convergence of metrics
Sounds like a home work problem?


Note that 
$$g(T)=\lim\_{t\to T-}g(t)=g(0)+\int\limits\_0^T\tfrac{\partial}{\partial t}g$$
Then you get 
$$|D\_0^m g(T)|\le \mathrm{Const}(m)$$
and
$$\sup\_M|D\_0^m[g(T)-g(t\_0)]|=\sup\_M\int\limits\_{t\_0}^TD\_0^m[\tfrac{\partial}{\partial t}g]\\,dt\to 0\ \ \text{as}\ \ t\_0\to T-.$$
One can cover $M$ by charts with bounded $g(0)$-Christoffel symbols in each.
Then the above inequalities imply that $g(T)$ is $C^\infty$-smooth and 
$g(t)\to g(T)$ in $C^\infty$-topology as $t\to T-$. 


