convergence of metrics

Question

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?

Answer 1

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 $t\to T-$.