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2 clarified: integral curve -> trajectory of gradient

When a function is very close to flat, the trajectories tend to be very erratic and wandering. The answer is NO, even if linear bounds are specified on the derivatives of the function.

To be more specific: in the unit sphere, draw any smooth path $\gamma$ starts from the south pole, bottom, ends at the north pole, and weaves and wanders as much as you like, for as long as you like. Now define a function $f$ on a neighbhood of $\gamma$ that is 0 at the bottom, $1$ at the top, and for which $\gamma$ is an integral curve a trajectory of the gradient, and the north pole and south pole are critical points. You can do this by using closest-point projection in a regular neighborhood. Smooth functions on a regular neighborhood can be extended to $C^\infty$ funtions on $S^2$, and a smooth extension can be perturbed away from $\gamma$ to be a Morse function, so the particular curve is a gradient line of a Morse function with arbitrary length.

Funtions that have a path like $\gamma$ in their gradient flow are obviously very inefficient. If you want functions with more efficiency, you could look at linear combinations of eigenfunctions of the Laplacian with small eigenvalue. In those cases, I think you can get reasonable inequalities concerning the average length of gradient flow lines; this is related to the known and widely used Cheeger type relationships between diameter of manifolds (or graphs), size of separators, and eigenvalues of the Laplacaian. I'm not sure what you can conclude about the maximum length of gradient flow lines, but I suspect something could be done, and may well be known.

1

When a function is very close to flat, the trajectories tend to be very erratic and wandering. The answer is NO, even if linear bounds are specified on the derivatives of the function.

To be more specific: in the unit sphere, draw any smooth path $\gamma$ starts from the south pole, bottom, ends at the north pole, and weaves and wanders as much as you like, for as long as you like. Now define a function $f$ on a neighbhood of $\gamma$ that is 0 at the bottom, $1$ at the top, and for which $\gamma$ is an integral curve and the north pole and south pole are critical points. You can do this by using closest-point projection in a regular neighborhood. Smooth functions on a regular neighborhood can be extended to $C^\infty$ funtions on $S^2$, and a smooth extension can be perturbed away from $\gamma$ to be a Morse function, so the particular curve is a gradient line of a Morse function with arbitrary length.

Funtions that have a path like $\gamma$ in their gradient flow are obviously very inefficient. If you want functions with more efficiency, you could look at linear combinations of eigenfunctions of the Laplacian with small eigenvalue. In those cases, I think you can get reasonable inequalities concerning the average length of gradient flow lines; this is related to the known and widely used Cheeger type relationships between diameter of manifolds (or graphs), size of separators, and eigenvalues of the Laplacaian. I'm not sure what you can conclude about the maximum length of gradient flow lines, but I suspect something could be done, and may well be known.
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