Given a compact Riemannian manifold and a Morse function on it. Is there a bound on the length of the Morse trajectories? (You can assume the Morse-Smale condition if helpful.)

EDIT (In response to Dick's answer): The Morse function and the metric are fixed. I am just looking for something like $\int_{-\infty}^{\infty}\| \nabla f(\phi_t(p))\|dt\leq C$ and $C=C(f,g)$ where $f$ is the Morse function in question, $g$ stands for the metric and $\phi$ denotes the flow of $-\nabla f$. Note here that the constant is independent of the starting point $p$, as it is easy to see that such a constant additionally depending on $p$ exists (you use the hyperbolicity of $\nabla f$ to deduce exponential convergence towards a critical point). Furthermore it is also easy to see that the above integral is bounded if you include a $2$ in the exponent of the norm (a.k.a. $L^2$), as $\| \nabla f(\phi_t(p))\|^2=-\frac{d}{dt}f(\phi_t(p)).$

Given a compact Riemannian manifold (with a fixed metric) and a Morse function on it (also fixed). Is there a bound (depending on the metric and the Morse function) on the length of the Morse trajectories? (You can assume the Morse-Smale condition if helpful.)

EDIT (In response to Dick's answer): The Morse function and the metric are fixed. I am just looking for something like $\int_{-\infty}^{\infty}\| \nabla f(\phi_t(p))\|dt\leq C$ and $C=C(f,g)$ where $f$ is the Morse function in question, $g$ stands for the metric and $\phi$ denotes the flow of $-\nabla f$. Note here that the constant is independent of the starting point $p$, as it is easy to see that such a constant additionally depending on $p$ exists (you use the hyperbolicity of $\nabla f$ to deduce exponential convergence towards a critical point). Furthermore it is also easy to see that the above integral is bounded if you include a $2$ in the exponent of the norm (a.k.a. $L^2$), as $\| \nabla f(\phi_t(p))\|^2=-\frac{d}{dt}f(\phi_t(p)).$

EDIT2 (In response to Bill's answer): I changed the statement to make it abundantly clear that the bound may depend on the metric and the Morse function. Maybe I was somewhat unclear in my formulation - sorry for that.

Given a compact Riemannian manifold and a Morse function on it. Is there a bound on the length of the Morse trajectories? (You can assume the Morse-Smale condition if helpful.)

Given a compact Riemannian manifold and a Morse function on it. Is there a bound on the length of the Morse trajectories? (You can assume the Morse-Smale condition if helpful.)

EDIT (In response to Dick's answer): The Morse function and the metric are fixed. I am just looking for something like $\int_{-\infty}^{\infty}\| \nabla f(\phi_t(p))\|dt\leq C$ and $C=C(f,g)$ where $f$ is the Morse function in question, $g$ stands for the metric and $\phi$ denotes the flow of $-\nabla f$. Note here that the constant is independent of the starting point $p$, as it is easy to see that such a constant additionally depending on $p$ exists (you use the hyperbolicity of $\nabla f$ to deduce exponential convergence towards a critical point). Furthermore it is also easy to see that the above integral is bounded if you include a $2$ in the exponent of the norm (a.k.a. $L^2$), as $\| \nabla f(\phi_t(p))\|^2=-\frac{d}{dt}f(\phi_t(p)).$