A singular example for the averaged mean curvature flow. (English) Zbl 0982.53061

It has been conjectured for quite some time that the averaged mean curvature flow can drive embedded plane curves to a loss of embeddedness; such a conjecture enters the literature with M. Gage [Contemp. Math. 51, 51-62 (1986; Zbl 0608.53002)].
In the present paper, using numerical experiments, the author presents an embedded curve in the two-dimensional Euclidean space which, under the averaged mean curvature flow, develops first a self-intersection and then a singularity of blowup type for the curvature, all in finite time.


53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53A04 Curves in Euclidean and related spaces


Zbl 0608.53002
Full Text: DOI EuDML EMIS Link


[1] Angenent S., J. Differential Geom. 33 pp 601– (1991) · Zbl 0731.53002
[2] DOI: 10.1080/10586458.1992.10504253 · Zbl 0769.49033
[3] DOI: 10.1090/S0002-9939-98-04727-3 · Zbl 0909.53043
[4] DOI: 10.1090/conm/051/848933
[5] Gage M. E., J. Differential Geom. 23 (1) pp 69– (1986)
[6] Grayson M. A., J. Differential Geom. 26 (2) pp 285– (1987) · Zbl 0667.53001
[7] Huisken G., J. Reine Angew. Math. 382 pp 35– (1987)
[8] DOI: 10.1017/S0956792599003812 · Zbl 0945.76016
[9] Mayer U. F., Differ. Integral Equations 13 (7) pp 1189– (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.