I would like to know if the following holds:
Let $(M,g_1)$ be a smooth Riemannian manifold, $f:M \longrightarrow \mathbb{R}$ a smooth function with gradient $\nabla^1f$ and $x:\mathbb{R} \longrightarrow M$ a curve satisfying the negative gradient flow equation, i.e. $\dot x(s) = -\nabla^1f(x(s)).$ Furthermore, let's assume that $x(s) \xrightarrow{s \to \infty} p$, where $p$ is a critical point of $f$. Let now $g_2$ be another Riemannian metric on $M$ which might be completely unrelated to $g_1$. $f$ has a different gradient under $g_2$ denoted by $\nabla^2f$. My conjecture is now the following: For $s$ large enough, there is a constant $C >0$ such that $$||\nabla^2f(x(s))||_2^2 \leq C ||\nabla^1f(x(s))||_1^2,$$ where $|| \cdot ||_i$ denotes the norm with respect to $g_i$. I would be glad if someone could prove this (sketch should be enough) or give a counter example.