1
$\begingroup$

Is there any existence theory applicable to general geometric flows of space curves in the following form? $$ \partial_t \gamma = v_t t + v_n n + v_b b $$ Here $\gamma$ is the evolving curve, $t$, $n$ and $b$ are the Frenet vectors and $v_t$, $v_n$ and $v_b$ are functions of the curvature $\kappa$ and torsion $\tau$.

I am familiar with results related to the curve shortening flow, where $v_n = \kappa$ and $v_b = 0$. But are there other examples with at least short time existence results?

Improve this question
$\endgroup$
3
  • $\begingroup$ Your question is perhaps a little bit too vague to give you completely satisfactory answer. Let me make a few comments, though: some functions (for example $v_n = -\kappa$) will make your problem ill-posed. Apart from the curve shortening flow, I believe e.g. the elastic flow is also actively studied. As far as I understand this is basically the 'curve analogue' of the Willmore flow. $\endgroup$
    – Leo Moos
    Commented Feb 8, 2021 at 19:45
  • $\begingroup$ See section 2 of Gage and Hamilton's "The heat equation shrinking convex plane curves" as an application of Hamilton's general short-time existence result in his "Three-manifolds with positive Ricci curvature." Other examples would follow in the same way $\endgroup$
    – Quarto Bendir
    Commented Feb 8, 2021 at 20:14
  • $\begingroup$ If $v_b=\kappa$ and $v_t=v_n=0$, then this is the vortex filament equation, which is a completely integrable dispersive equation. (See iveyt.people.cofc.edu/papers/contemp3.pdf for other geometric flows of curves of this type). $\endgroup$
    – RBega2
    Commented Feb 8, 2021 at 20:45

0

Reset to default

You must log in to answer this question.