# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The shape of a figure-eight under the curve shortening flow. (English) Zbl 0669.53003
Main Theorem: The isoperimetric ratio ${L}^{2}/A$ converges to infinity under the curve shortening flow if and only if the loops of the original figure-eight curve bound regions of equal area.
Reviewer: D.Ferus
##### MSC:
 53A04 Curves in Euclidean space
##### References:
 [1] Abresch, U., Länger, J.: The normalized curve shortening flow and homothetic solutions. J. Differ. Geom.23, 175-196 (1986) [2] Angenent, S.: The zeroset of a solution of a parabolic equation. Preprint [3] Epstein, C., Weinstein, M.: A stable manifold theorem for the curve shortening equation. Commun. Pure Appl. Math.40, 119-139 (1987) · Zbl 0602.34026 · doi:10.1002/cpa.3160400106 [4] Gage, M.: An isoperimetric inequality with application to curve shortening. Duke Math. J.50, 1225-1229 (1983) · Zbl 0534.52008 · doi:10.1215/S0012-7094-83-05052-4 [5] Gage, M.: Curve shortening makes convex curves circular. Invent. Math.76, 357-364 (1984) · Zbl 0542.53004 · doi:10.1007/BF01388602 [6] Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom.23, 69-96 (1986) [7] Grayson, M.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom.26, 285-314 (1987) [8] Grayson, M.: Shortening embedded curves. Ann. Math. (to appear)