0

Let $H: \mathbb{R}^d\to [0,\infty)$ be a (smooth) Morse function and satisfies some growth condition like $$\vert p\vert ^2-C_1\le \vert H(p)\vert \le \vert p\vert^\alpha+C_2.$$ (Feel free to assume similar bounds for the gradient). I want to consider $$ \dot{y}=-\nabla H(y)\\ y(0) = y_0 $$ I know, that as $t\to \infty$ $y$ cannot escape to $\infty$ and thus will converge to some point $x$.

Question: I want to estimate the length of the curve $y$.

My approach: For any $T>0$ I can get $$ 1/2\int_0^T \vert\dot{y}\vert^2\le H(y_0)- H(y(t)), $$ by taking $\partial_t H(y(t))$ and applying Young'S inequality. From this I can get, by Cauchy-Schwarz, $$ \int_0^T\vert \dot{y}\vert\leq \sqrt{T}\,(H(y_0)- H(y(T))) $$ My problem is that this bound doesn't really help for evaluating the integral $$ \ell(y)=\int_0^\infty \vert \dot{y}\vert $$

Rodrigo de Azevedo
  • 23,223
C3r6eru5
  • 43
  • 1
    I would not expect this kind of growth condition to control the length of gradient flow curves - you can always get arbitrarily short by starting near a local minimum, and you can get arbitrarily long by designing some kind of spiral groove in a bounded region. I also don't think you're guaranteed convergence to point - see https://math.stackexchange.com/questions/501007/when-does-gradient-flow-not-converge – Anthony Carapetis Mar 24 '25 at 04:56
  • @Anthony Carapetis, I implicitely assumend that $F$ is a Morse function, for which the convergence should be fine, I'll edit the question. But nonetheless, I agree that the bound shouldn't be enough, as it "doesn't see" whether $F$ is Morse or not. – C3r6eru5 Mar 24 '25 at 10:17

1 Answers1

0

There is Proposition 8.13 in these notes, which is concerned with compact manifolds $M$. There we get that any GF curve is bounded in length by $$ \frac{1}{m}(\sup_{x\in M} H(x)-\inf_{x\in M}H(x))+ 2n \sum_i R_i, $$ Where $R_i$ are the radii of some balls $B_i$ around the $n$ critical points $p_i$ and $m=\min_{x\not\in \bigcup_i B_i} \vert \nabla H(x)\vert>0$.

In this case here, we take our compact manifold to be a closed sublevel-set of $H$, e.g., $\{x: H(x)\le H(y_0\}$ which must be compact because $H$ growth at least quadratically.

Taking $\vert y_0\vert$ large enough, we get a bound of the form $C\vert y_0\vert^\alpha.$

C3r6eru5
  • 43