Moment estimates, exponential integrability, concentration inequalities and exit times estimates on evolving manifolds

On a smooth (not necessarily compact) manifold $M$ equipped with a ${\mathsf{C}}^{\mathsf{1}}$-family of complete Riemannian metrics $g(t)$ and a ${\mathsf{C}}^{\mathsf{1},\mathsf{\infty }}$-family of vector fields $Z(t)$ both indexed by the real interval $[0,T)$ where $T\in (0,\mathrm{\infty }]$, we prove moment estimates, exponential integrability, concentration inequalities and exit times estimates for diffusions on complete evolving Riemannian manifolds.