Open AccessSoliton solutions to the curve shortening flow on the sphere
Author(s) -
Hiuri dos Reis,
Keti Tenenblat
Publication year - 2019
Publication title -
proceedings of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.968
H-Index - 84
eISSN - 1088-6826
pISSN - 0002-9939
DOI - 10.1090/proc/14607
Subject(s) - geodesic , tangent , mathematics , curvature , geometry , equator , convergence (economics) , tangent vector , algorithm , physics , economic growth , economics , astronomy , latitude
It is shown that a curve on the unit sphere is a soliton solution to the curve shortening flow if and only if its geodesic curvature is proportional to the inner product between its tangent vector and a fixed vector of R 3 \mathbb {R}^3 . Using this characterization, we describe the geometry of such a curve on the sphere, we study its qualitative behavior, and we prove the convergence of the curve to the equator orthogonal to the fixed vector.