# Properties of the affine curve-shortening flow

The affine curve-shortening flow (ACSF) seems to satisfy the following properties, even if the initial curve(s) is (are) self-intersecting (at least as long as they are "nice enough"):

(1) The length of the curve strictly decreases with time.

(2) The number of self-intersections of a curve never increases. Similarly, the number of intersections between two curves never increases.

(3) If the curve is simple (no self-intersections) then the number of inflection points never increases.

(4) The total absolute curvature decreases strictly until it settles at $$2\pi$$.

Are there any references for these properties?

Regarding (1), this property is stated in the Wikipedia for the CSF, but not for the ACSF.

Regarding (2), we know by the "avoidance principle" that if the number of intersections or self-intersections is 0 then it stays 0. Furthermore, Angenent et al.  in Section 10 address the case of intersections between two curves. But I'm not sure if they assume the curves to be simple throughout their paper.

Regarding (3), I think this is stated in  at the beginning of Section 9, when they say that the number of sign changes of kappa does not increase with time. Do I understand correctly? However, property (3) is clearly false if the curve self-intersects, because the disappearance of a self-intersection might create inflection points. The fact that  do not qualify their statement makes me think that they are only dealing with simple curves.

Regarding (4), Theorem 15.1 in  just says that the total absolute curvature tends to $$2\pi$$. But Lemma 9.6 earlier seems to say that the total absolute curvature is decreasing. Am I correct? And is the property true even for self-intersecting curves?

 Angenent, Sigurd; Sapiro, Guillermo; Tannenbaum, Allen, On the affine heat equation for non-convex curves, J. Am. Math. Soc. 11, No. 3, 601-634 (1998). ZBL0902.35048.

## 1 Answer

From what Sergey Avvakumov and I can see, for the CSF, it has been shown that one can continue the flow past singularities, which occur when a self-intersection collapses to a cusp (see refs. 2, 3). Actually the analysis of ref. 3 applies not only to the CSF but to a wide range of flows. Unfortunately, according to ref. 1 (cited in the question), the ACSF is not included in this range. Hence, it seems that continuing the ACSF past singularities is an open problem.

Not being experts in differential geometry, we are not sure that we understood the situation correctly. So we would be glad if some expert in the field would confirm this answer or provide a more precise answer.

References:

1. Altschuler, Steven J.; Grayson, Matthew A., Shortening space curves and flow through singularities, J. Differ. Geom. 35, No. 2, 283-298 (1992). ZBL0782.53001.

2. Angenent, Sigurd, Parabolic equations for curves on surfaces. II: Intersections, blow-up and generalized solutions, Ann. Math. (2) 133, No. 1, 171-215 (1991). ZBL0749.58054.