# Inequality for gradients under different metrics

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.

Your conjecture is true. Recall that for any differentiable function $$f:M\to\mathbb{R}$$ and $$p\in M$$ we have $$\|\nabla f(p)\|=\|df(p)\|,$$ where $$df$$ denotes the derivative of $$f$$ and the norm $$\|\cdot\|$$ is induced by a Riemannian metric $$g$$. Hence, the inequality you are after can be stated as $$\|df(x(s))\|_2^2\leq C\|df(x(s))\|_1^2.$$ Now, as all norms on a finite dimensional space are equivalent to one another, for $$p\in M$$ there is some $$C_p>0$$, such that $$\|\cdot\|_{1,p}\leq C_p\|\cdot\|_{2,p}.$$ Furtheremore, for a compact subset $$S\subset M$$ the constant $$C$$ can be chosen uniformly. Your trajectory converges to a point, and so its image indeed lies in some compact subset of the limit point, at least for $$s$$ large enough.