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 LeviCivita connection of $g(0)$ and assume that for every $m\geq 0$
$\int_0^T \sup_MD_0^m \frac{\partial}{\partial t} g(t)_{g(0)} dt< \infty$, then why $g(t)$ converges in $C^{\infty}$ to a smooth tensor?
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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 $$\sup_MD_0^m[g(T)g(t_0)]=\sup_M\int\limits_{t_0}^TD_0^m[\tfrac{\partial}{\partial t}g]\\,dt\to 0\ \ \text{as}\ \ t_0\to T.$$ 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 $t\to T$. 

