Abstract
In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and \(\lim_{t\rightarrow\infty}L(\gamma_{t})>0\), then for every n > 0, \(\lim_{t\rightarrow\infty}\sup\left(\left|\frac{D^{n}T}{\partial s^{n}}\right|\right)=0\). Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S 1, if γ0 is a ramp, then we have a global flow which converges to a closed geodesic in C ∞ norm. As an application, we prove the theorem of Lyusternik and Fet.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Avoid common mistakes on your manuscript.
References
Abresch U., Langer J. (1986) The normalized curve shortening flow and homothetic solutions. J. Differ. Geom. 23, 175–196
Altschuler S.J. (1991) Singularities of the curve shrinking flow for space curves. J. Differ. Geom. 34, 491–514
Altschuler S.J., Grayson M.A. (1992) Shortening space curves and flow through singularities. J. Differ. Geom. 35, 491–514
Cao F. (2003) Geometric Curve Evolution and Image Processing. Springer, Berlin Heidelberg New York
Carmo M.P.do. (1992) Riemannian geometry. (translated by F. Flaherty, Math. Theory and Appl.) Birkhauser, Boston
Gage M., Hamilton R.S. (1986) The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96
Grayson M.A. (1987) The heat equation shrinks embedded plane curves to round points. J. Differ. Geom. 26, 285–314
Grayson M.A. (1989) Shortening embedded curves. Ann. Math. 129, 71–111
Jost J. (1998) Riemannian Geometry and Geometric Analysis, 2nd edn. Springer, Berlin Heidelberg New York
Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307425 v1
Spanier E. (1966) Algebraic Topology. McGraw Hill, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ma, L., Chen, D. Curve shortening in a Riemannian manifold. Annali di Matematica 186, 663–684 (2007). https://doi.org/10.1007/s10231-006-0025-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-006-0025-y